From 52e1732d5a0dd5f630e2440eb4ec87a7005ede00 Mon Sep 17 00:00:00 2001 From: Five Grant <5@fivegrant.com> Date: Wed, 18 Oct 2023 11:37:01 -0500 Subject: [PATCH] Include outputs in repo --- .gitignore | 2 - outputs/notice.txt | 1 - outputs/ta1/report_20231017_194227.json | 7069 +++++++++++++++++++++++ outputs/ta3/report_20230908_100637.json | 53 + 4 files changed, 7122 insertions(+), 3 deletions(-) delete mode 100644 outputs/notice.txt create mode 100644 outputs/ta1/report_20231017_194227.json create mode 100644 outputs/ta3/report_20230908_100637.json diff --git a/.gitignore b/.gitignore index ad7faf5..68bc17f 100644 --- a/.gitignore +++ b/.gitignore @@ -158,5 +158,3 @@ cython_debug/ # and can be added to the global gitignore or merged into this file. For a more nuclear # option (not recommended) you can uncomment the following to ignore the entire idea folder. #.idea/ - -outputs/ta* diff --git a/outputs/notice.txt b/outputs/notice.txt deleted file mode 100644 index 0b9a1a5..0000000 --- a/outputs/notice.txt +++ /dev/null @@ -1 +0,0 @@ -Place your local reports in `outputs/ta1` and `outputs/ta3`. diff --git a/outputs/ta1/report_20231017_194227.json b/outputs/ta1/report_20231017_194227.json new file mode 100644 index 0000000..49cd450 --- /dev/null +++ b/outputs/ta1/report_20231017_194227.json @@ -0,0 +1,7069 @@ +{ + "scenarios": { + "SIDARTHE": { + "success": false, + "description": "In Italy, 128,948 confirmed cases and 15,887 deaths of people who tested positive for SARS-CoV-2 were \nregistered as of 5 April 2020. Ending the global SARS-CoV-2 pandemic requires implementation of multiple \npopulation-wide strategies, including social distancing, testing and contact tracing. We propose a new model \nthat predicts the course of the epidemic to help plan an effective control strategy. The model considers eight \nstages of infection: susceptible (S), infected (I), diagnosed (D), ailing (A), recognized (R), threatened (T), \nhealed (H) and extinct (E), collectively termed SIDARTHE. Our SIDARTHE model discriminates between infected \nindividuals depending on whether they have been diagnosed and on the severity of their symptoms. The distinction \nbetween diagnosed and non-diagnosed individuals is important because the former are typically isolated and hence \nless likely to spread the infection. This delineation also helps to explain misperceptions of the case fatality \nrate and of the epidemic spread. We compare simulation results with real data on the COVID-19 epidemic in Italy,\nand we model possible scenarios of implementation of countermeasures. Our results demonstrate that restrictive \nsocial-distancing measures will need to be combined with widespread testing and contact tracing to end the ongoing \nCOVID-19 pandemic.\n\n", + "steps": { + "pdf_extraction": { + "id": "extraction-b105d6b6-64ba-4a70-9617-d08c4c03408f", + "status": "finished", + "result": { + "created_at": "2023-10-17T19:35:42.309956", + "enqueued_at": "2023-10-17T19:35:42.310103", + "started_at": "2023-10-17T19:35:42.317700", + "job_result": { + "extraction_status_code": 200, + "extraction": [ + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -2.5192012787, + "content": "", + "postprocess_score": 0.9981482029, + "detect_cls": "Section Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 92, + 152, + 334, + 168 + ], + "detect_score": -5.1885566711, + "content": "N E T W O R K S C I E N C E", + "postprocess_score": 0.9791147709, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 94, + 197, + 1099, + 290 + ], + "detect_score": -2.5088102818, + "content": "Lack of practical identifiability may hamper reliable predictions in COVID-19 epidemic models", + "postprocess_score": 0.9595780373, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 93, + 313, + 825, + 339 + ], + "detect_score": -3.683866024, + "content": "predictions in COVID-19 epidemic models Luca Gallo1,2, Mattia Frasca3,4*, Vito Latora1,2,5,6, Giovanni Russo7", + "postprocess_score": 0.5951064229, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 91, + 372, + 1198, + 581 + ], + "detect_score": -1.8331866264, + "content": "Compartmental models are widely adopted to describe and predict the spreading of infectious diseases. The unknown parameters of these models need to be estimated from the data. Furthermore, when some of the model variables are not empirically accessible, as in the case of asymptomatic carriers of coronavirus disease 2019 (COVID-19), they have to be obtained as an outcome of the model. Here, we introduce a framework to quantify how the uncertainty in the data affects the determination of the parameters and the evolution of the unmeasured variables of a given model. We illustrate how the method is able to characterize different regimes of identifiability, even in models with few compartments. Last, we discuss how the lack of identifiability in a realistic model for COVID-19 may prevent reliable predictions of the epidemic dynamics.", + "postprocess_score": 0.998639524, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 91, + 656, + 741, + 1511 + ], + "detect_score": -2.3022465706, + "content": "INTRODUCTION The pandemic caused by severe acute respiratory syndrome coronavirus-2 is challenging humanity in an unprecedented way (1), with the disease, which in a few months has spread around the world, affecting large parts of the population (2,\u00a03) and often requir- ing hospitalization or even intensive care (4,\u00a05). Mitigating the impact of coronavirus disease 2019 (COVID-19) urges synergistic efforts to understand, predict, and control the many, often elusive, facets of the complex phenomenon of the spreading of a previously unknown virus, from RNA sequencing to the study of the virus pathogenicity and transmissibility (6,\u00a0 7) to the definition of suitable epidemic spreading models (8) and the investigation of nonpharmaceutical intervention policies and containment measures (9\u201312). In particu- lar, a large number of epidemic models have recently been proposed to describe the evolution of COVID-19 and evaluate the effectiveness of different counteracting measures, including social distancing, testing, and contact tracing (13\u201319). However, even the adoption of well-consolidated modeling techniques, such as the use of mecha- nistic models at the population level based on compartments, poses fundamental problems. First of all, the very same choice of the dynamical variables to use in a compartmental model is crucial; as such, variables should adequately capture the spreading mecha- nisms and need to be tailored to the specific disease. This step is not straightforward, especially when the spreading mechanisms of the disease are still unknown or only partially identified. In addition, some of the variables considered might be difficult to measure and track as, for instance, in the case of COVID-19, it occurred in the number of individuals showing mild or no symptoms. Second, compartmental models, usually, involve a number of parameters, including the initial values of the unmeasured variables, which are not known and need to be estimated from data.", + "postprocess_score": 0.9998086095, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 91, + 1591, + 739, + 1794 + ], + "detect_score": -3.3431372643, + "content": "1Department of Physics and Astronomy, University of Catania, Catania 95125, Italy. 2INFN Sezione di Catania, Via S. Sofia, 64, Catania 95125, Italy. 3Department of Elec- trical, Electronics and Computer Science Engineering, University of Catania, Catania 95125, Italy. 4Istituto di Analisi dei Sistemi ed Informatica \"A. Ruberti,\" Consiglio Nazionale delle Ricerche (IASI-CNR), 00185 Roma 00185, Italy. 5School of Mathe- matical Sciences, Queen Mary University of London, London E1 4NS, UK 6Complexity Science Hub Vienna, A-1080 Vienna, Austria. 7Department of Mathematics and Computer Science, University of Catania, Catania 95125, Italy. *Corresponding author. Email: mattia.frasca@dieei.unict.it", + "postprocess_score": 0.9676234126, + "detect_cls": "Other", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -3.6390016079, + "content": "", + "postprocess_score": 0.9992510676, + "detect_cls": "Body Text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 769, + 655, + 1451, + 1796 + ], + "detect_score": -0.809820056, + "content": "Having at disposal a large amount of data, unfortunately, does not simplify the problem of parameter estimation and prediction of unmeasured states. Once a model is formulated, it may occur that some of its unknown parameters are intrinsically impossible to determine from the measured variables or that they are numerically very sensitive to the measurements themselves. In the first case, it is the very same structure of the model to hamper parameter estima- tion as the system admits infinitely many sets of parameters that fit the data equally well; for this reason, this problem is referred to as structural identifiability (20,\u00a021). In the second case, although under ideal conditions (i.e., noise-free data and error-free models) the problem of parameter estimation can be uniquely solved for some trajectories, it may be numerically ill conditioned, such that from a practical point of view, the parameters cannot be determined with precision even if the model is structurally identifiable (22). This situation typically occurs when large changes in the parameters entail a small variation of the measured variables, such that two similar trajectories may correspond to very different parameters (23). The term practical identifiability is adopted in this case. Identifiability in general represents an important property of a dynamical system, as in a nonidentifiable system, different sets of parameters can produce the same or very similar fits of the data. Consequently, predictions from a nonidentifiable system become unreliable. In the context of epidemics forecasting, this means that even if the model considered is able to reproduce the measured variables, a large uncertainty may affect the estimated values of the parameters and the predicted evolution of the unmeasured variables (24). The problem of practical identifiability of model parameters has been investigated using different methodologies based on Fisher's information theory (25,\u00a026), profile likelihood (27), Monte Carlo simulations (28), and other computational approaches (29). How- ever, the lack of practical identifiability can also affect the reliability of the prediction of the unmeasured variables dynamics (27), an issue of utmost importance in the context of COVID-19, which nevertheless still requires a systematic investigation. In particular, an approach to simultaneously characterize the problem of sensitivity to parameters and that of the reliability of predictions of unmeasured variables is still missing. In more detail, in this paper, we investigate the problem of the practical identifiability of dynamical systems whose state includes", + "postprocess_score": 0.9999736547, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 1247, + 157, + 1416, + 451 + ], + "detect_score": -4.8968873024, + "content": "Copyright \u00a9 2022 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution License 4.0 (CC BY).", + "postprocess_score": 0.6939247847, + "detect_cls": "Equation label", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 1362, + 1826, + 1416, + 1840 + ], + "detect_score": -5.3873357773, + "content": "1 of 13", + "postprocess_score": 0.9996168613, + "detect_cls": "Page Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -5.0522069931, + "content": "", + "postprocess_score": 0.9983310103, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 91, + 152, + 741, + 482 + ], + "detect_score": -3.0875182152, + "content": "not only measurable but also hidden variables, as is the case of com- partment models for COVID-19 epidemic. We present a general framework to quantify not only the sensitivity of the measured variables of a given model on its parameters but also the sensitiv- ity of the unmeasured variables on the parameters and on the measured variables. This will allow us to introduce the notion of practical identifiability of the hidden variables of a model. As a relevant and timely application, we show the variety of different regimes and levels of identifiability that can appear in epidemic models, even in the simplest case of a four compartment system. Last, we study the actual effects of the lack of practical identifiability in more sophisticated models introduced for COVID-19.", + "postprocess_score": 0.9908027649, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 92, + 545, + 738, + 650 + ], + "detect_score": -4.3164615631, + "content": "RESULTS Dynamical systems with hidden variables Consider the n-dimensional dynamical system described by the following equations", + "postprocess_score": 0.9807692766, + "detect_cls": "Section Header", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 341, + 675, + 736, + 733 + ], + "detect_score": -4.1081390381, + "content": "m\u202f\u0307 = f(m, h, q ) , (1) h\u202f\u0307 = g(m, h, q) m \u2208 \u211dnm that can be empirically accessed (measurable variables)", + "postprocess_score": 0.9940838218, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 90, + 767, + 741, + 1293 + ], + "detect_score": 1.2131830454, + "content": "where we have partitioned the state variables into two sets: the variables m \u2208 \u211dnm that can be empirically accessed (measurable variables) The dynamics of the system is governed by the two Lipschitz- and those h \u2208 \u211dnh, with nm + nh = n, that cannot be measured (hidden). continuous functions f and g, which also depend on a vector of system in eq. 1 are uniquely determined by the structural parameters structural parameters q \u2208 \uf057q \u2282 \u211dnq. The trajectories m(t) and h(t) of q and by the initial conditions m(0) = m0, h(0) = h0. Here, we assume that some of the quantities q are known, while the others are not known and need to be determined by fitting the trajectories of measurable variables m(t). identify the trajectories, which comprises the unknown terms of q We denote by p \u2208 \uf057p \u2282 \u211dnp, the set of unknown parameters that and the unknown initial conditions h0. The initial values of the hidden variables are not known and act indeed as parameters for the tra- jectories generated by system in eq. 1. The initial conditions of the measurable variables m0 may be considered fitting parameters as well. System in eq. 1 is said to be structurally identifiable when the measured variables satisfy (21)", + "postprocess_score": 0.9998364449, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 256, + 1324, + 736, + 1347 + ], + "detect_score": -4.3372335434, + "content": "(2) m(t, \u02c6 p ) = m(t, p ) , \u2200 t \u2265 0 \u21d2 \u02c6 p = p", + "postprocess_score": 0.993761003, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 91, + 1381, + 741, + 1628 + ], + "detect_score": -2.0305085182, + "content": "for almost any p \u2208 \uf057p. Notice that, as a consequence of the existence and uniqueness theorem for the initial value problem, if system in eq. 1 is structurally identifiable, the hidden variables can also be uniquely determined. Structural identifiability guarantees that two different sets of parameters do not lead to the same time course for the measured variables. When this condition is not met, one cannot uniquely associate a data fit to a specific set of parameters or, equivalently, recover the parameters from the measured variables (23).", + "postprocess_score": 0.999979496, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 91, + 1660, + 739, + 1796 + ], + "detect_score": -4.7516450882, + "content": "Assessing the practical identifiability of a model Structural identifiability, however, is a necessary but not sufficient condition for parameters estimation, so that when it comes to use a dynamical system as a model of a real phenomenon, it is fundamen- tal to quantify the practical identifiability of the dynamical system.", + "postprocess_score": 0.9580492973, + "detect_cls": "Reference text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -5.282356739, + "content": "", + "postprocess_score": 0.9997695088, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 769, + 150, + 1417, + 421 + ], + "detect_score": -2.1444900036, + "content": "To do this, we consider a solution, m\u202f\u0304 (t ) = m(t, p\u202f\u0304 ) and h\u202f\u0304 (t ) = h(t, p\u202f\u0304 ) , obtained from parameters p = p\u202f\u0304 , and we explore how much the func- tions m(t) and h(t) change as we vary the parameters p\u202f\u0304 by a small amount \uf064p. To first order approximation in the perturbation of the parame- ters, we have \uf064m = \ud835\udedbm\u202f_ \ud835\udedbp\u202f \uf064p + O(\u2225 \uf064p \u2225 2 ) and \uf064h = \ud835\udedbh\u202f_ \ud835\udedbp \uf064p + O(\u2225 \uf064p \u2225 2 ) . Hence, by dropping the higher order terms, we have \u221e \u221e \u2223 \uf064h \u2223 2 dt = \uf064p T H\uf064p , \u2225 \uf064m \u2225 2 = \u222b 0 \u2223 \uf064m \u2223 2 dt = \uf064 p T M\uf064p and \u2225 \uf064h \u2225 2 = \u222b 0 where the entries of the sensitivity matrices M = M( p\u202f\u0304 ) \u2208 \u211d n p \u00d7 n p and defined as H = H( p\u202f\u0304 ) \u2208 \u211d n p \u00d7 n p for the measured and unmeasured variables are", + "postprocess_score": 0.9973409772, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 769, + 445, + 1451, + 1231 + ], + "detect_score": 0.010581227, + "content": "H = H( p\u202f\u0304 ) \u2208 \u211d n p \u00d7 n p for the measured and unmeasured variables are \u221e \u221e \u2202 m T \u2500 \u2202 h T \u2500 (3) (M) ij = \u222b \u2202 p i \u2202 m\u202f\u2500 \u2202 p j dt ;\u00a0 (H) ij = \u222b \u2202 p i \u2202 h\u202f\u2500 \u2202 p j dt 0 0 Note that these matrices are positive semidefinite by con- struction. The smallest change in the measured variables m(t) will take place if \uf064p is aligned along the eigenvector v1 of M correspond- ing to the smallest eigenvalue \uf06c1(M). Hence, we can consider \uf073 = _ \uf06c 1 (M) to quantify the sensitivity of the measured variables to the \u221a parameters. Practical identifiability requires high values of \uf073 as these indicate cases where small changes in the parameters may produce considerable variations of the measurable variables, and therefore, the estimation of the model parameters from fitting is more reliable. Suppose now we consider a perturbation, \uf064p1, of the parameters aligned along the direction of v1. We can evaluate the change in h(t) due to this perturbation by \uf064 p 1 T H\uf064 p 1 \u2500 (4) \uf068 2 = \uf064 p 1 T \uf064 p 1 The value of \uf068 quantifies the sensitivity of the hidden variables to the parameters of the model, when these parameters are estimated from the fitting of the observed variables since \u2225\uf064h \u2225 = \uf068 \u2225 \uf064p1\u2225. Notice that in this case, and differently from \uf073, lower values of \uf068 are desirable because they imply a better prediction on the hidden variables. Last, with the help of the sensitivity matrices defined above, we can also evaluate the sensitivity of the hidden variables to the mea- sured variables as", + "postprocess_score": 0.9987471104, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 995, + 1264, + 1414, + 1327 + ], + "detect_score": -2.4051089287, + "content": "\uf064 p T H\uf064p\u202f\u2500 \uf06d 2 = max (5) \u2225\uf064p\u2225=1 \uf064 p T M\uf064p", + "postprocess_score": 0.9999140501, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 769, + 1353, + 1419, + 1796 + ], + "detect_score": 3.1673007011, + "content": "This parameter is of particular relevance here, since it provides a bound on how the uncertainty on the measured variables affects the evolution of the hidden variables. In addition, the parameter \uf06d2 can be efficiently computed as it corresponds to the maximum general- ized eigenvalue of matrices (H, M), as shown in Materials and Methods. The sensitivity matrices are useful in studying the effect of changing the number of hidden variables and unknown parameters on the practical identifiability of a model. Assume that we have access to one more variable, thus effectively increasing the size of the set of measured variables to nm\u2032 = nm + 1 and, correspondingly, reducing that of the unmeasured variables to nh\u2032 = nh \u2212 1. This cor- responds to considering new variables m\u00b4 and h\u00b4. From the defini- tion in Eq. 3, the new sensitivity matrix can be written as M\u2032 = M + M1, where M1 is the sensitivity matrix for the newly measured variable. Given Weyl's inequality [page 239 of (30)], we have that", + "postprocess_score": 0.9999639988, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 1361, + 1826, + 1416, + 1840 + ], + "detect_score": -4.4724731445, + "content": "2 of 13", + "postprocess_score": 0.9992861152, + "detect_cls": "Equation label", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -2.6963226795, + "content": "", + "postprocess_score": 0.9989008904, + "detect_cls": "Equation label", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 91, + 149, + 741, + 924 + ], + "detect_score": 0.713558197, + "content": "\uf06c1(M\u2032) \u2265 \uf06c1(M) + \uf06c1(M1) and since M1 is also positive semidefinite, \uf06c1(M\u2032) \u2265 \uf06c1(M). This means that measuring one further variable (or more than one) of the system increases the practical identifiability of a model, as expected. As H\u2032 = H \u2212 M1, it is also possible to demon- strate that \uf06d(M\u2032) \u2264 \uf06d(M) (see Materials and Methods). Let us now consider a different scenario: Suppose we have a priori knowledge of one of the model parameters so that we do not need to estimate its value by fitting the model to the data. In this case, we can define unmeasured variables, respectively. Given the Cauchy's interlacing new sensitivity matrices \u02dc M , \u02dc H \u2208 \u211d ( n p \u22121)\u00d7( n p \u22121) for the measured and theorem [page 242 of (30)], we have that \uf06c 1 ( \u02dc M ) \u2265 \uf06c 1 (M) , which im- plies that practical identifiability is improved by acquiring a priori in- formation on some of the model parameters. For instance, in the context of COVID-19 models, one may decide to fix some of the parameters, such as the rate of recovery, to values derived from medical and biological knowledge (24,\u00a0 31\u201333) and to determine from fitting the more elusive parameters, such as the percentage of asymptomatic individuals or the rates of transmission. The sensitivity measures we have introduced point out that prior knowledge of some of the parameters, or a larger set of measurable variables, reduces the sensitivity of the measured variables to the parameters and that of the hidden variables to a variation in the measured ones. However, gathering further knowledge can be diffi- cult or even not possible so that these results, which should not be interpreted as an oversimplified solution to the problem of identifi- ability, have to be considered in the light of practical issues that might arise in the measurement of the model variables and pa- rameters.", + "postprocess_score": 0.9999562502, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 91, + 962, + 741, + 1796 + ], + "detect_score": 2.4364526272, + "content": "The sensitivity measures reveal different regimes of identifiability As a first application, we study the practical identifiability of a four compartment mean-field epidemic model (34) in the class of SIAR models (35), developed to assess the impact of asymptomatic carriers of COVID-19 (8,\u00a036,\u00a037) and other diseases (38\u201340). In such a model (Fig.\u00a01), a susceptible individual (S) can be infected by an infectious individual who can either be symptomatic (I) or asymptomatic (A). The newly infected individual can either be symptomatic (S \u2192 I) or asymptomatic (S \u2192 A). Furthermore, we also consider the possibility that asymptomatic individuals develop symptoms (A \u2192 I), thus accounting for the cases in which an individual can infect before and after the onset of the symptoms (41). Last, we suppose that individuals cannot be reinfected as they acquire a permanent immunity (R). One of the crucial aspects of COVID-19 is the presence of asymptomatic individuals who are difficult to trace as the individuals themselves could be unaware about their state. Consequently, we assume that the fraction of asymptomatic individuals, a(t), is not measurable, while the fractions of symptomatic, \uf069(t), and recovered, r(t), are measured variables, that is, m \u2261 [\uf069, r] and h \u2261 [s, a]. As mentioned above, practical identifiability is a property of the trajec- tories of the system, which are uniquely determined by the values of the unknown parameters p. Here, we illustrate how the sensitivity of both measured and unmeasured variables changes with the probability \uf067 that a newly infected individual shows no symptoms when all the other parameters of the model are fixed (to the values reported in Materials and Methods). Concerning the choice of vector p, here, we consider the following case. First, as the number of symptomatic infectious and recovered individuals are supposed", + "postprocess_score": 0.9999735355, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -3.4730391502, + "content": "", + "postprocess_score": 0.9999371767, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 774, + 153, + 1412, + 514 + ], + "detect_score": 0.2607835531, + "content": "", + "postprocess_score": 0.9998978376, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 770, + 538, + 1416, + 605 + ], + "detect_score": -0.9148697257, + "content": "Fig. 1. Graphical representation of a SIAR model in which infectious individuals can either be symptomatic (I) or asymptomatic (A) (see also Eq. 21 in Materials and Methods).", + "postprocess_score": 0.9996759892, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 769, + 655, + 1451, + 1796 + ], + "detect_score": -0.3447470367, + "content": "to be measurable, we have assumed the initial conditions \uf069(0), r(0), and the recovery rate of the symptomatic individuals, i.e., \uf061IR,to be known quantities. Second, we assume to be able to measure, for instance, through backward contact tracing, the rate at which asymptomatic individuals develop symptoms, i.e., \uf061AI. Hence, the parameters to be determined are the remaining ones, i.e., p = [a(0), \uf062I, \uf062A, \uf067, \uf061AR]. Figure\u00a02A shows a nontrivial nonmonotonic dependence of our sensitivity measures, \uf073 and \uf068, on \uf067. The value of \uf073 has a peak at \uf067 = 0.51, in correspondence of which \uf068 takes its minimum value. This represents an optimal condition for practical identifiability, as the sensitivity to parameters of the measured variables is high, while that of the unmeasured ones is low, and this implies that the unknown quantities of the system (both the model parameters and the hidden variables) can be estimated with small uncertainty. On the contrary, for \uf067 = 0.86, we observe a relatively small value of \uf073 and a large value of \uf068, meaning that the measured variables are poorly identifiable, and the unmeasured variables are sensitive to a variation of parameters. This is the worst situation in which the estimated parameters may substantially differ from the real values, and the hidden variables may experience large variations even for small changes in the parameters. Furthermore, the quantity \uf06d, which measures the sensitivity of the hidden variables to the measured ones, reported in Fig.\u00a02B, exhibits a large peak at the value of \uf067 for which \uf073 is minimal. This is due to the fact that the vector that determines \uf06d is almost aligned with v1. When this holds, we have that \uf06d = \u03b7/\uf073, which explains the presence of the spike in the \uf06d curve. Similarly, the sensitivity \uf06d takes its minimum almost in correspondence of the maximum of \uf073. The behavior of the model for \uf067 = 0.86 is further illustrated in Fig.\u00a02C, where the trajectories obtained in correspondence to the unperturbed values of the parameters, i.e., m(t, p) and h(t, p) (solid lines), are compared with the dynamics observed when p undergoes a perturbation with \u2225\uf064p\u2225 = 0.3\u2225p\u2225 along v1 (dashed lines). The small sensitivity \uf073 of the measured variables \uf069(t, p) and r(t, p) to parameters is reflected into perturbed trajectories that remain close to the unperturbed ones, whereas the large sensitivity \uf068 of the unmeasured variables s(t, p) and a(t, p) yields perturbed trajectories that significantly deviate from the unperturbed ones. We now illustrate the different levels of identifiability that appear in the SIAR model for diverse settings of the parameters. Its analysis, in fact, fully depicts the more complete perspective on the", + "postprocess_score": 0.9999833107, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 1361, + 1826, + 1416, + 1840 + ], + "detect_score": -6.8431921005, + "content": "3 of 13", + "postprocess_score": 0.9998441935, + "detect_cls": "Other", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -1.1529467106, + "content": "", + "postprocess_score": 0.9997326732, + "detect_cls": "Equation label", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 209, + 1451, + 1844 + ], + "detect_score": -0.3209046125, + "content": "Fig. 2. Practical identifiability of the SIAR model in Fig. 1 as a function of the fraction \uf067 of asymptomatic new infectious individuals. (A) Sensitivity \uf073 and \uf068 of measured and hidden variables, respectively, to the parameters of the model. (B) Sensitivity \uf06d of the hidden variables to the measured ones. (C) State variables for unper- turbed values of parameters (with \uf067 = 0.86, solid line) and for a perturbation with \u2225\uf064p \u2225 = 0.3 \u2225 p\u2225 along the first eigenvector of M (dashed lines). problem of practical identifiability offered by simultaneously inspecting the sensitivity measures, \uf073 and \uf068. As the two sensitivity measures are not necessarily correlated, there can be cases for which a high identifiability of the measured variables to the parameters, i.e., large values of \uf073, corresponds to either a low or a high identifi- ability of the hidden variables to the parameters. Analogously, for other system configurations, in correspondence of small values of \uf073, namely, to nonidentifiable parameters, one may find large values of \uf068, meaning that the hidden variables are nonidentifiable as well or, on the contrary, small values of \uf068, indicating that the hidden variables are poorly sensitive to parameter perturbations. Together, four distinct scenarios of identifiability can occur, and all of them effectively appear in the SIAR model (Fig. 3): (A) low identifiability of the model parameters p and high identifiability of the hidden variables h, (B) high identifiability of both p and h, (C) low identifiability of both p and h, and (D) high identifiability of p and low identifiability of h. To illustrate them, we have considered four distinct configura- tions of the model (with parameters as given in Table\u00a03 and illus- trated in Materials and Methods) and, for each case study, compared the unperturbed trajectories to the perturbed ones, with the vector of parameters undergoing a variation \u2225\uf064p \u2225 = 0.3 \u2225 p\u2225 along v1. Fig. 3. Four scenarios of identifiability for the SIAR model of Fig. 1. All panels As regard cases (A) and (C), we have considered the vector of param- show the system dynamics (solid line) and the evolution of the system when the eters to determine to be p = [\uf069(t), a(0), r(t), \uf062I, \uf062A, \uf067, \uf061IR, \uf061AR, \uf061AI], vector of parameters undergoes a variation \uf064p such that \u2225\uf064p \u2225 = 0.3 \u2225 p\u2225 along the while for the cases (B) and (D), we have p = [a(0), \uf062I, \uf062A, \uf067, \uf061AR], first eigenvector of M (dashed lines). (A) and (C) display configurations for which which is the same choice of p adopted in Fig.\u00a02. Figure\u00a03 shows the the observed variables (\uf069, r) are not sensitive to the variation, i.e., the model parame- results obtained for each parameter configuration. In each panel, ters are not identifiable, while (B) and (D) show the opposite case. Furthermore, (A) the solid lines represent the unperturbed trajectories, while the and (B) present scenarios for which the unobserved variables (s, a) are insensitive dashed lines correspond to the perturbed dynamics. In cases (A) and to the variation, meaning that they are predictable; vice-versa, (C) and (D) show the case in which the variables s and a are sensitive. (B), we see that, under the variation \uf064p, the perturbed trajectories of 4 of 13", + "postprocess_score": 1.0, + "detect_cls": "Figure", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -3.1195502281, + "content": "", + "postprocess_score": 0.9914628863, + "detect_cls": "Section Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 91, + 152, + 741, + 455 + ], + "detect_score": 1.5236207247, + "content": "the hidden variables remain close to the unperturbed dynamics. Hence, the hidden variables are highly identifiable. Conversely, in cases (C) and (D), the perturbed trajectories substantially differ from the unperturbed dynamics, meaning that the hidden variables are poorly identifiable as they are sensitive to a variation of the model parameters. As concerns the measured variables, in cases (A) and (C), the perturbed trajectories slightly differ from the unperturbed dynamics. Therefore, as the measured variables are insensitive to the perturbation \uf064p, the model parameters have a low degree of identifiability. On the other hand, in cases (B) and (D), the perturba- tion of the parameters significantly affects the trajectories of the", + "postprocess_score": 0.9998476505, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 91, + 541, + 739, + 789 + ], + "detect_score": -6.910340786, + "content": "Table 1. Values of \uf073, \uf068, and \uf06d for the four configurations of the SIAR model shown in Fig. 3. Case A Case B Case C Case D \uf073 0.0096 0.15 0.013 0.091 \uf068 0.012 0.16 0.36 1.4 \uf06d 34 5.2 29 15", + "postprocess_score": 0.9030431509, + "detect_cls": "Page Footer", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 207, + 813, + 741, + 1660 + ], + "detect_score": -4.2762527466, + "content": "", + "postprocess_score": 0.997459352, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 1433, + 712, + 1451, + 1206 + ], + "detect_score": -4.3379187584, + "content": "from data. In these conditions, a small uncertainty in the measurable variables due to the presence of noise in the data can propagate", + "postprocess_score": 0.5828691721, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 767, + 813, + 1302, + 1660 + ], + "detect_score": -3.1280417442, + "content": "", + "postprocess_score": 0.9994435906, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 769, + 487, + 1417, + 790 + ], + "detect_score": 1.3488147259, + "content": "Poorly identifiable models may provide unreliable predictions when the parameters are estimated from data So far, we have illustrated how variations on the parameters affect the trajectories of measurable and hidden variables under different degrees of identifiability. However, a high sensitivity of the hidden variables to measured ones has relevant practical consequences, especially when the parameters are unknown and need to be fitted from data. In these conditions, a small uncertainty in the measurable variables due to the presence of noise in the data can propagate markedly and make the prediction of the hidden variables unreliable. Hence, in this section, we study how the lack of practical identifiability", + "postprocess_score": 0.9994488358, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 769, + 152, + 1418, + 455 + ], + "detect_score": 1.7274762392, + "content": "measured variables, meaning that the set of parameters reproducing the observed data is more identifiable. Last, Table\u00a01 illustrates the values of the sensitivity measures \uf073, \uf068, and \uf06d for each case. In particular, case (C) represents the worst scenario as the value of \uf073 is relatively small, meaning that the model parameters p are poorly identifiable, and the value of \uf068 is large, indicating a high sensitivity of the hidden variables to the parameters. Conversely, the best scenario is represented by case (B), for which both the model parameters and the hidden variables are highly identifiable as the value of \uf073 is large compared to the other cases, while the value of \uf068 remains relatively small.", + "postprocess_score": 0.9999136925, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 92, + 1700, + 1416, + 1794 + ], + "detect_score": 3.594250679, + "content": "Fig. 4. Dynamics of the SIAR model when two different methods to estimate the model parameters from data are used. (A to D) display the results obtained by a least square error minimization procedure, while (E to H) show the outcome of Bayesian inference. The time evolution of both measured and hidden variables (solid lines) is reported together with the data. Data are shown with different markers if they pertain to measured variables (full circles, used for fitting) or unmeasured ones (empty circles, not used for fitting).", + "postprocess_score": 0.9999787807, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 92, + 1826, + 1416, + 1844 + ], + "detect_score": -5.0760269165, + "content": "5 of 13", + "postprocess_score": 0.9991152883, + "detect_cls": "Figure", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -4.6529726982, + "content": "", + "postprocess_score": 0.9998381138, + "detect_cls": "Figure Caption", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 298, + 153, + 1210, + 624 + ], + "detect_score": -0.0541875176, + "content": "", + "postprocess_score": 0.9999186993, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 92, + 648, + 1256, + 665 + ], + "detect_score": -2.1938490868, + "content": "Fig. 5. Graphical representation of a nine-compartment model for the propagation of COVID-19 (see also Eq. 24 in Materials and Methods).", + "postprocess_score": 0.9999555349, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 711, + 741, + 1796 + ], + "detect_score": 2.1195061207, + "content": "can affect the predictions of the SIAR model when this is fitted to empirical data. The reliability of the model predictions has been investigated by means of two different fitting techniques: a least square error minimization procedure (42), which provides a point estimate of the parameters, and a Bayesian inference approach (43), which, conversely, gives an estimate of the probability distribution in the parameter space. To carry out the numerical analysis, we consider the model under the same settings (i.e., fixing the values of the six parameters and the initial values of three variables) as those adopted in case (C) in the previous section, which correspond to the case of low identifiability of both the parameters and the hidden variables. We then generate from such a model a synthetic dataset of trajectories, which we fit using the two approaches mentioned above (see Materials and Methods for further details). All the model parameters are considered unknown and thus need to be deter- mined through the fit. First, we consider the least square error minimization approach. To show how, because of the lack of identifiability of the model, significant variations in the dynamics of the hidden variables can be obtained when fitting the measured variables, we have performed the following analysis. As the estimation procedure (based on a nonlinear optimization algorithm; see Materials and Methods) starts from an initial guess of the fitting parameters, indicated as p0, instead of fitting a single set of values, we have repeated the proce- dure under the very same conditions of the algorithm, for 500 runs, randomly selecting p0 from a Gaussian distribution centered on a fixed point of the parameter space and with variance equal to 0.25. We then discarded those runs yielding a fitting error d > 0.015, which corresponds to a relative error of 2.5%, thus keeping a total of 65 sets of parameters fitting the measured variables with a similar value of the error. The fact that different sets of parameters are obtained in this way may indicate that the error function has several local minima. Figure\u00a04\u00a0(A\u00a0to\u00a0D) displays the average trajectories (over the 65 sets of parameters) of the four state variables (solid lines) and the respective regions where 95% of the trajectories lie (shadowed area). While the dynamics of the measured variables (Fig.\u00a04,\u00a0C\u00a0and\u00a0D) produced by the SIAR model are in very good agreement with the data, the same is not true for the temporal evolution of the hidden variables (Fig.\u00a04,\u00a0A\u00a0and\u00a0B). In particular, the", + "postprocess_score": 0.9999719858, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -3.0004971027, + "content": "", + "postprocess_score": 0.9999166727, + "detect_cls": "Body Text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 769, + 711, + 1451, + 1796 + ], + "detect_score": -3.2431461811, + "content": "number of newly asymptomatic infected individuals is substantially overestimated. We now consider the Bayesian inference approach. To imple- ment it, we used the Delayed Rejection Adaptive Metropolis (DRAM) (44), a Markov Chain Monte Carlo (MCMC) algorithm, with set- tings as described in Materials and Methods. In particular, since we have here assumed to have no a priori information on the value of the model parameters, we have considered uniform prior probability distributions, representing the less informative conditions for the model (further details are given in Materials and Methods). Figure\u00a04\u00a0(E\u00a0to\u00a0H) shows the temporal evolution of the SIAR variables. Solid lines represent the average trajectory obtained by sampling 500 sets of parameters from the posterior distributions, while the shadowed areas indicate the regions where 95% of the trajectories lie. Similarly, to the case of the least square minimization, while the dynamics of the measured variables (Fig.\u00a04,\u00a0G\u00a0and\u00a0H) is in a good agreement with the synthetic data, the prediction of the hidden variables (Fig.\u00a04,\u00a0E\u00a0and\u00a0F) is not. At variance with the previous example, the number of newly asymptomatic infected individuals is here largely underestimated. Hence, these results indicate that the lack of identifiability can lead to unreliable results even when a Bayesian approach is adopted. In the analysis presented in this section, we have assumed to have no a priori information on the values of the model parameters. Therefore, when performing the least square error minimization, we have extracted all the parameters from fitting, while, following the same reasoning, we have chosen a uniform prior probability distribution in the Bayesian approach. When instead we have strong a priori knowledge of a disease, this can be used to inform the models, by fixing the values of certain parameters while estimating the others, in the case of the least square error method, or by considering more informative prior distributions, in the case of Bayesian inference. When a priori information on the values of the parameters can be obtained, for instance, through medical and biological studies, the model predictions are expected to become less affected by uncertainty. The analysis of the sensitivity matrices (see also Materials and Methods) confirms this expectation as we have demonstrated that additional knowledge of the parameters, or", + "postprocess_score": 0.9999821186, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 1361, + 1826, + 1416, + 1840 + ], + "detect_score": -7.0740022659, + "content": "6 of 13", + "postprocess_score": 0.9993032217, + "detect_cls": "Body Text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -2.7868821621, + "content": "", + "postprocess_score": 0.9986153841, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 207, + 153, + 733, + 358 + ], + "detect_score": -3.6461851597, + "content": "", + "postprocess_score": 0.9988308549, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 217, + 374, + 733, + 579 + ], + "detect_score": -4.6077470779, + "content": "", + "postprocess_score": 0.9993357062, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 785, + 374, + 1302, + 579 + ], + "detect_score": -3.6906137466, + "content": "", + "postprocess_score": 0.9977636337, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 785, + 153, + 1302, + 358 + ], + "detect_score": -2.8508985043, + "content": "", + "postprocess_score": 0.998703599, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 91, + 595, + 1451, + 1796 + ], + "detect_score": -2.134550333, + "content": "Fig. 6. Modeling the COVID-19 outbreak in Italy. The evolution of both measured (A to D) and hidden variables (E to H) of the model in Fig. 5 (solid lines) is reported together with the official data from the Civil Protection Department (circles). (D) individuals. Following the study of Giordano et al. (16), to account the measurement of a hidden variable, can reduce the sensitivity to for the different nonpharmaceutical interventions and testing strat- the measured variables, thus improving the reliability of the predic- egies issued during the COVID-19 outbreak in Italy (47,\u00a048), the tion. However, there are cases in which a priori knowledge is not model parameters have been considered piece-wise constant and available, and the analysis of identifiability becomes crucial. As an estimated using nonlinear optimization by fitting of the official example, one could estimate the percentage of asymptomatic indi- data provided by the Civil Protection Department (49). As the data- viduals according to serological surveys or to longitudinal studies. set provides the evolution in time of the daily number of home iso- However, these data can be unavailable at an early stage of an lated, hospitalized, detected recovered, and deceased individuals, epidemic outbreak (45,\u00a046), preventing their use to inform the we have considered four measured and five hidden variables in epidemiological models. These considerations hallmark once again the need for a synergistic approach to study newly discovered infec- the model, namely, m \u2261 [H, T, Rd, D] and h \u2261 [S, E, IA, IS, Ru]. It is here worth discussing an important issue that concerns the tious diseases and stress the importance of assessing the reliability model parameters. We have considered that different policy strategies of mathematical modeling when the amount of available informa- affect, according to their nature, only specific parameters. In particu- tion is limited. lar, we have assumed that a change in the containment strategy Lack of identifiability in COVID-19 modeling prevents leads to a variation in the transmission rates \uf062, while an adjustment reliable predictions in the testing strategy affects the values of the parameters \uf061ISH, \uf061HT, As a second application, we show the relevance of the problem of and \uf061HRd (further details are reported in Materials and Methods). As both the containment and the testing strategies in Italy have practical identifiability in the context of COVID-19 pandemic frequently changed during the pandemic, most of the parameters to modeling. We consider a realistic model (Fig.\u00a05) of the disease estimate consist of transmission and detection rates. While other propagation, that is a variant of the SIDARTHE model (16) and is parameters, such as the death or the recovery rates, can be derived characterized by nine compartments accounting respectively for from the current literature (50), very limited information is available susceptible (S), exposed (E), undetected asymptomatic (IA), un- on the transmission and detection rates that are difficult to measure detected symptomatic (IS), home isolated (H), treated in hospital (T), undetected recovered (Ru), detected recovered (Rd), and deceased directly and, therefore, need to be estimated by fitting available", + "postprocess_score": 1.0, + "detect_cls": "Body Text", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -4.5371718407, + "content": "", + "postprocess_score": 0.9639735818, + "detect_cls": "Page Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 1361, + 1826, + 1416, + 1840 + ], + "detect_score": -3.5708231926, + "content": "7 of 13", + "postprocess_score": 0.9947144389, + "detect_cls": "Page Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -2.3884868622, + "content": "", + "postprocess_score": 0.9993494153, + "detect_cls": "Equation label", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 152, + 741, + 818 + ], + "detect_score": -1.4408782721, + "content": "data. Hence, as in (16), we have assumed all the model parameters to be unknown. To show how significant variations in the evolution of the hidden variables can arise when fitting the measured variables, we have performed a numerical analysis similar to the one of the previous section. Again, rather than fitting a single set of values, we have repeated the minimization procedure under the same conditions of the algorithm, for 500 runs, randomly selecting the initial guess p0 from a Gaussian distribution centered on a fixed point of the pa- rameter space, with variance equal to 0.25. We discarded the runs yielding a fitting error e > 900, corresponding to a relative error of 1.4%, thus keeping a total of 40 sets of parameters. Figure\u00a06 shows the dynamical trajectories that we have obtained for each of the 40 sets of parameters (solid lines). Both measured (Fig.\u00a06,\u00a0A\u00a0to\u00a0D) and hidden (Fig.\u00a06,\u00a0E\u00a0to\u00a0H) variables are reported. While the time evolution of the measured variables produced by the model is in very good agreement with the empirical data, reported as circles in Fig.\u00a06, significant differences in the trend of the hidden variables appear. A large variability is observed, confirming that the lack of identifiability yields a high sensitivity of the hidden variables to the measured one. These findings have relevant implications. The large uncertainty on the size of the asymptomatic population makes questionable the use of the model as a tool to decide the policies to adopt.", + "postprocess_score": 0.999920249, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 880, + 741, + 1796 + ], + "detect_score": -2.3098857403, + "content": "DISCUSSION The practical identifiability of a dynamical model is a critical, but often neglected, issue in determining the reliability of its predic- tions. In this paper, we have introduced a novel framework to quantify: (i) the sensitivity of the dynamical variables of a given model to its parameters, even in the presence of variables that are difficult to access empirically and (ii) how changes in the measured variables affect the evolution of the unmeasured ones. The measures we have proposed are easy to compute and enable to assess, for instance, if and when the model predictions on the unmeasured variables are reliable or not, even in the cases in which the parameters of the model can be fitted with high accuracy from the available data. As we have shown with a series of case studies, practical identifi- ability can critically affect the predictions of even very refined epidemic models introduced for the description of COVID-19, where dynamical variables, such as the population of asymptomatic individuals, are impossible or difficult to measure. This by no means should question the importance of these models\u2014in that they enable a scenario analysis, otherwise impossible to carry out, and a deeper understanding of the spreading mechanisms of a novel disease\u2014but should hallmark the relevance of a critical analysis of the results that takes into account sensitivity measures. It also high- lights the importance of cross-disciplinary efforts that can provide a priori information on some of the parameters, ultimately improv- ing the reliability of a model (8,\u00a024). A problem related to the one studied in our paper is that of observability, which investigates how to reconstruct the internal state of a system from measurements on the input and output, under the hypothesis that the model and its parameters are known (51,\u00a052). Techniques based on the observability problem are clearly extremely important and may be applied, for instance, to derive the time evolution of asymptomatic individuals from measurements on", + "postprocess_score": 0.9999864101, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -3.7020123005, + "content": "", + "postprocess_score": 0.9997121692, + "detect_cls": "Equation label", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 770, + 152, + 1417, + 203 + ], + "detect_score": -1.7074959278, + "content": "infected and recovered individuals, when it is possible to develop a fully observable model with known parameters.", + "postprocess_score": 0.9710310102, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 770, + 265, + 1359, + 343 + ], + "detect_score": -2.7602171898, + "content": "MATERIALS AND METHODS The sensitivity matrices and their properties The sensitivity matrices considered in this paper are given by", + "postprocess_score": 0.9148218632, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 918, + 374, + 1414, + 430 + ], + "detect_score": 2.6263093948, + "content": "\u221e \u221e \u2202 m T \u2500 (6) M ij = \u222b 0 \u2202 p i \u2202m\u202f\u2500 \u2202 p j dt ;\u00a0 H ij = \u222b \u2202 h T \u2500 \u2202 p i \u2202h\u202f\u2500 \u2202 p j dt 0", + "postprocess_score": 0.9999818802, + "detect_cls": "Equation", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 769, + 460, + 1417, + 538 + ], + "detect_score": 0.229545176, + "content": "where the vector functions m = m(t, p) and h = h(t, p) are obtained integrating system in eq. 1. The derivative of measurable and hidden variables with respect to the parameters p", + "postprocess_score": 0.9996961355, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 966, + 570, + 1220, + 593 + ], + "detect_score": -0.6075288057, + "content": "m i \u2261 \u2202 m / \u2202 p i ,\u00a0 h i \u2261 \u2202 h / \u2202 p i", + "postprocess_score": 0.9982327223, + "detect_cls": "Equation", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 770, + 626, + 1170, + 649 + ], + "detect_score": -2.2754375935, + "content": "can be obtained by integrating the system", + "postprocess_score": 0.9394777417, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 769, + 686, + 1451, + 1349 + ], + "detect_score": -1.2977672815, + "content": "= \u2202 f\u202f\u2500 \u2202 m\u202f \u00b7 m i + \u2202 f\u202f\u2500 \u2202 h\u202f \u00b7 h i + \u2202 f\u202f\u2500 \u2202 p i d m i \u2500 dt (7) = \u2202 g\u202f\u2500 \u2202 m\u202f \u00b7 m i + \u2202 g\u202f\u2500 \u2202 h\u202f \u00b7 h i + \u2202 g\u202f\u2500 \u2202 p i d h i \u2500 dt where i = 1, \u2026np. The numerical evaluation of the sensitivity matrices is carried out first by integrating system in eq. 7 (for this step we use a fourth-order Runge-Kutta solver with adaptive step size control), resampling the trajectories with a sampling period of 1 day, and then performing a discrete summation over the sampled trajectories. Moreover, inte- gration is carried out over a finite time interval [0, \uf074], with large enough \uf074. In the context of our work, as we have considered SIR (susceptible infected removed)-like epidemic models, we set the value of \uf074 such that the system has reached a stationary state, i.e., the epidemic outbreak has ended, as every infected individual has eventually recovered (or dead, depending on the model). We now present an important property of the sensitivity matri- ces. We will only take into account the set of measured variables m, as similar considerations can be made for the hidden variables. Let us assume to be able to measure only a single variable, so that the vector m collapses into a scalar function, which we call m1(t). In this case, the element Mij of the sensitivity matrix would be simply given by", + "postprocess_score": 0.9999060631, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 996, + 1386, + 1414, + 1439 + ], + "detect_score": 1.3249536753, + "content": "\u221e (8) (M) ij = \u222b \u2202 m 1 \u2500 \u2202 p i \u2202 m 1 \u2500 \u2202 p j dt 0", + "postprocess_score": 0.9998369217, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 771, + 1465, + 1416, + 1546 + ], + "detect_score": -8.4292593002, + "content": "Let us call this sensitivity matrix M1. Consider now a larger set of measured variables m = (m1, m2, \u2026, mnm). The quantity \u2202mT/\u2202pi\u2202m/\u2202pj in Eq. 6 is given by", + "postprocess_score": 0.9994894266, + "detect_cls": "Equation label", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 851, + 1580, + 1414, + 1637 + ], + "detect_score": -3.8546807766, + "content": "\u2202 m T \u2500 (9) \u2202 p i \u2202m\u202f\u2500 \u2202 p j = \u2202 m 1 \u2500 \u2202 p i \u2202 m 1 \u2500 \u2202 p j + \u2202 m 2 \u2500 \u2202 p i \u2202 m 2 \u2500 \u2202 p j + \u2026 + \u2202 m n m \u2500 \u2202 p i \u2202 m n m \u2500 \u2202 p j", + "postprocess_score": 0.9539873004, + "detect_cls": "Equation", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 769, + 1660, + 1419, + 1789 + ], + "detect_score": -5.1834306717, + "content": "Therefore, integrating over time in the interval [0, \u221e ] and given the linearity property of the integrals, we find that the sensitivity matrix M of the set of the measured variables is given by the sum of the sensitivity matrices of the single measured variables. Formally, we have that", + "postprocess_score": 0.9997871518, + "detect_cls": "Equation label", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 1361, + 1826, + 1416, + 1840 + ], + "detect_score": -6.032348156, + "content": "8 of 13", + "postprocess_score": 0.9993976355, + "detect_cls": "Page Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -2.5295937061, + "content": "", + "postprocess_score": 0.9914755821, + "detect_cls": "Equation label", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 293, + 170, + 736, + 193 + ], + "detect_score": 1.7205687761, + "content": "(10) M = M 1 + M 2 + \u2026 + M n m", + "postprocess_score": 0.9992069602, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 91, + 236, + 738, + 424 + ], + "detect_score": -5.1577444077, + "content": "This property of the sensitivity matrices is useful to demonstrat- ing how measuring a further variable affects the sensitivity mea- sures \uf073 and \uf06d, as discussed in the following subsection and in Results. Last, because matrices M and H are positive semidefinite, their eigenvalues are nonnegative. For any positive semidefinite matrix A of order m, we shall denote its eigenvalues as 0 \u2264 \uf06c1(A) \u2264 \uf06c2(A) \u2264 \u2026 \u2264 \uf06cm(A).", + "postprocess_score": 0.9989339709, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 91, + 459, + 739, + 902 + ], + "detect_score": 0.7800967693, + "content": "Sensitivity measures and their properties Here, we discuss in more detail the sensitivity measures introduced in Results. First, we want to propose a measure to quantify the practical identifiability of the model parameters given the measured variables. To do this, we need to evaluate the sensitivity of the trajectories of the measured variables to a variation of the model parameters. If this sensitivity is small, then different sets of parame- ters will produce very similar trajectories of the measured variables, meaning that the parameters themselves are poorly identifiable. In particular, as a measure of the parameters identifiability, we can consider the worst scenario, namely, the case in which the perturba- tion of the parameters minimizes the change in the measured vari- ables. This happens when the variation of the model parameters \uf064p is aligned along the eigenvector v1 of M corresponding to the minimum eigenvalue \uf06c1(M). Given the definition of M, we have that _ \uf06c 1 (M) \u2225 \uf064p \u2225 ; hence, we can consider the quantity \u2225 \uf064m \u2225 = \u221a", + "postprocess_score": 0.999956131, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 361, + 935, + 736, + 961 + ], + "detect_score": -5.6010570526, + "content": "_ (11) \uf06c 1 (M) \uf073 = \u221a", + "postprocess_score": 0.9994494319, + "detect_cls": "Reference text", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 91, + 991, + 741, + 1321 + ], + "detect_score": -1.7148185968, + "content": "as an estimate of the sensitivity of the measured variables to the parameters. Note that, here and in the rest of the paper, \u2225v\u2225 denotes the Euclidean norm of a finite dimensional vector v, \u2225v\u22252 = v \u00b7 v, while for a function u(t), \u2225u\u2225 denotes the L2 norm of u in [0, \u221e ], i.e., \u2225 u \u2225 2 = \u222b0 \u221e u \u00b7 u\u00a0dt . Let us now focus on the hidden variables h. In general, as the hidden variables are not directly associated to empirical data, the largest uncertainty on the hidden variables is obtained in corre- spondence of a variation of the parameters along the eigenvector of H associated to the largest eigenvalue, namely, \uf06c n p (H). Hence, to quantify the sensitivity of the hidden variables to the parameters, one may consider", + "postprocess_score": 0.9999169111, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 332, + 1364, + 736, + 1395 + ], + "detect_score": -1.6192967892, + "content": "_ (12) \uf06c n p (H) \uf068 MAX = \u221a", + "postprocess_score": 0.9999527931, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 91, + 1437, + 739, + 1683 + ], + "detect_score": 0.9530110955, + "content": "However, it is crucial to note that the hidden variables ultimately depend on the parameters of the model, which are estimated by fitting data that are available for the measured variables only. As a consequence, it is reasonable to consider a quantity that evaluates how the uncertainty on the model parameters (determined by the uncertainty of the measured variables and by their sensitivity to the parameters) affects the identifiability of the hidden variables. There- fore, as a measure of the sensitivity of the hidden variables to the parameters, we consider", + "postprocess_score": 0.9998743534, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 340, + 1722, + 736, + 1788 + ], + "detect_score": -2.3205866814, + "content": "\uf064 p 1 T H\uf064 p 1 \u2500 (13) \uf068 2 = \uf064 p 1 T \uf064 p 1", + "postprocess_score": 0.9999281168, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -0.9199565649, + "content": "", + "postprocess_score": 0.9967074394, + "detect_cls": "Equation label", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 769, + 152, + 1417, + 566 + ], + "detect_score": -2.8748714924, + "content": "where \uf064p1 is a perturbation of the parameters along the eigenvector v1 of M corresponding to the minimum eigenvalue \uf06c1(M). Note that, when v1 and the eigenvector of H corresponding to the largest eigenvalue \uf06c n p (H) are aligned, by definition, we have \uf068 = \uf068MAX. Last, we want to define a quantity to estimate how much the hidden variables are perturbed given a variation of the measured ones. In particular, as a measure of the sensitivity of the hidden variables to the measured variables, we consider the maximum perturbation of the hidden variables given the minimum variation of the measured ones, which is \uf064 p T H\uf064p\u202f\u2500 \uf06d 2 = max (14) \u2225\uf064p\u2225=1 \uf064 p T M\uf064p Note that \uf06d2 can be computed considering the following gener- alized eigenvalue problem", + "postprocess_score": 0.9986617565, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 769, + 655, + 1451, + 1212 + ], + "detect_score": -5.4707622528, + "content": "where H and M are the sensitivity matrices for the hidden and the observed variables respectively, and \uf06ck = \uf06ck(M, H) denotes the k-th generalized eigenvalue of matrices M and H. We will denote by \uf06c n p the largest generalized eigenvalue and u the corresponding general- ized eigenvector. Note that, since both matrices are symmetric, if u is a right eigenvector, then uT is a left eigenvector. Multiplying each member of the equation by uT and dividing by uTMu, we obtain = max (16) \uf06c n p = u T Hu\u202f\u2500 \u2225v\u2225=1 v T Hv\u202f\u2500 u T Mu v T Mv where one can recognize the definition of \uf06d2 provided in Eq. 14. In other words, \uf06d2 represents the largest eigenvalue of the matrix M\u22121H. It is worth noting two aspects about the sensitivity measure \uf06d. First, given the definitions in Eqs. 11 and 12, for any \uf064p with \u2225\uf064p\u2225 = 1, we 2 and \uf064pTM\uf064p \u2265 \uf0732. As a consequence, have that \uf064 p T H\uf064p \u2264 \uf068 MAX we have that 2 (17) \u2500 \uf06d 2 \u2264 \uf068 MAX \uf073 2", + "postprocess_score": 0.8834430575, + "detect_cls": "Figure", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 770, + 1241, + 1416, + 1370 + ], + "detect_score": -0.6133924723, + "content": "Second, when the vector \uf064p that determines \uf06d is aligned with the eigenvector v1 of M, it is possible to express \uf06d in terms of the sensi- tivity measures \uf073 and \uf068. When \uf064p = \u2225 \uf064p \u2225 v1 = v1, recalling defini- tions in Eqs. 11 and 13, one obtains v 1 T M v 1 = \uf073 2 , while v 1 T H v 1 = \uf068 2 , from which it follows", + "postprocess_score": 0.9997585416, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 769, + 1464, + 1417, + 1795 + ], + "detect_score": -2.0133686066, + "content": "In addition, we note that if v1 and the eigenvector of H correspond- ing to its largest eigenvalue are aligned, one obtains that \uf06d = \uf068MAX/\uf073, which is the maximum value for the sensitivity measure \uf06d. We now demonstrate that the sensitivity of the hidden variables to the measured ones, \uf06d2, decreases as we measure one further vari- able. Let us assume now that we are able to measure one further variable, thus increasing the size of the set of measured variables to nm\u2032 = nm + 1 and, correspondingly, reducing that of the unmeasured variables to nh\u2032 = nh \u2212 1. Given the property in Eq. 10, the new sensitivity matrices can be written as M\u2032 = M + M1 and H\u2032 = H \u2212 M1, where by M1, we denote the sensitivity matrix for the newly mea- sured variable. The new generalized eigenvalue problem is", + "postprocess_score": 0.99999547, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 1026, + 600, + 1414, + 621 + ], + "detect_score": -3.6076159477, + "content": "(15) H u k = \uf06c k M u k", + "postprocess_score": 0.9901427031, + "detect_cls": "Page Header", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 1065, + 1402, + 1414, + 1437 + ], + "detect_score": -4.5055198669, + "content": "(18) \uf06d = \uf068\u202f\u2500 \uf073", + "postprocess_score": 0.9997122884, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 1361, + 1826, + 1416, + 1840 + ], + "detect_score": -6.7015337944, + "content": "9 of 13", + "postprocess_score": 0.9990872145, + "detect_cls": "Section Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -1.9497345686, + "content": "", + "postprocess_score": 0.998301208, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 172, + 152, + 718, + 173 + ], + "detect_score": -4.2676558495, + "content": "(19) H\u2032u\u2032= \uf06c\u2032M\u2032u\u2032 \u21d4 (H \u2212 M 1 ) u\u2032= \uf06c\u2032(M + M 1 ) u\u2032", + "postprocess_score": 0.5849023461, + "detect_cls": "Other", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 73, + 205, + 720, + 285 + ], + "detect_score": -2.0582268238, + "content": "where, for simplicity, we have denoted by \uf06c\u2032 the largest generalized eigenvalue of matrices M\u2032 and H\u2032. Left multiplying by u\u2032T and dividing by u\u2032TMu\u2032, we obtain", + "postprocess_score": 0.9705278277, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 146, + 308, + 718, + 432 + ], + "detect_score": -1.4274843931, + "content": "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u2265 u\u2032 T Hu\u2032 \u2500 = u\u2032 T H'u\u2032+ u\u2032 T M 1 u\u2032 \u2265 \u03bb n p = u T Hu\u202f\u2500 u T Mu u\u2032 T Mu\u2032 u\u2032 T Mu\u2032\u2212 u\u2032T M 1 u\u2032 (20) u\u2032 T H'u\u2032 \u2500 = \u03bb\u2032 u\u2032 T M \u1fbd u\u2032", + "postprocess_score": 0.9908812046, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 73, + 456, + 720, + 585 + ], + "detect_score": -0.9986514449, + "content": "where the first inequality comes from the definition of \uf06c n p , while the second comes from the fact that H, M, H\u2032, M\u2032, and M1 are positive semidefinite. In short, we find that \uf06c n p \u2265 \uf06c\u2032, meaning that, by mea- suring one variable, the sensitivity of the hidden variables to the measured ones decreases.", + "postprocess_score": 0.9998852015, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 74, + 623, + 698, + 675 + ], + "detect_score": -4.1667199135, + "content": "SIAR model and setup for numerical analysis The SIAR model of Fig.\u00a01 is described by the following equations", + "postprocess_score": 0.7995238304, + "detect_cls": "Reference text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 150, + 712, + 719, + 828 + ], + "detect_score": -0.5955280066, + "content": "\u23a7 s \u0307 = \u2212 s( \uf062 I \uf069 + \uf062 A a) \u23aa (21) \uf069\u202f\u0307 = (1 \u2212 \uf067 ) s( \uf062 I \uf069 + \uf062 A a ) + \uf061 AI a \u2212 \uf061 IR \uf069 \u23a8 \u23aa a \u0307 = \uf067s( \uf062 I \uf069 + \uf062 A a ) \u2212 ( \uf061 AI + \uf061 AR ) a \u23a9 r \u0307 = \uf061 IR \uf069 + \uf061 AR a", + "postprocess_score": 0.8899844289, + "detect_cls": "Equation label", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 73, + 878, + 723, + 1795 + ], + "detect_score": -0.3493070006, + "content": "where s(t), \uf069(t), a(t), and r(t) represent population densities, i.e., s(t) = S(t)/N, \uf069(t) = I(t)/N, a(t) = A(t)/N, and r(t) = R(t)/N, where S(t), I(t), A(t), and R(t) represent the number of susceptible, infec- tious, asymptomatic, and recovered individuals, and N is the size of the population, so that s(t) + \uf069(t) + a(t) + r(t) = 1. Here, \uf062I and \uf062A are the transmission rates for the symptomatic and the asymptomatic individuals, respectively, \uf067 is the probability for newly infected indi- viduals to show no symptoms, \uf061AI is the rate at which asymptomatic individuals become symptomatic, and \uf061IR and \uf061AR are the recovery rates for the two infectious populations. Note that all these parame- ters are positive quantities. Asymptomatic individuals are difficult to trace as the individuals themselves could be unaware about their state. As a consequence, we assume that the density of asymptomatic individuals is not measurable, while the densities of symptomatic and recovered indi- viduals are measured variables. According to the notation intro- duced in Eq. 1, we therefore have that m \u2261 [\uf069, r] and h \u2261 [s, a]. Note that, as a first approximation, here, we assume to be able to trace the asymptomatic individuals once they recover. The results presented in Fig.\u00a02 have been obtained considering the following setup. As the number of symptomatic infectious and recovered individuals are considered measurable, we have assumed that the initial conditions \uf069(0), r(0), and the rate of recovery \uf061IR are known parameters. Second, we have supposed to be able to measure, for instance, through backward contact tracing the rate at which asymptomatic individuals develop symptoms, i.e., \uf061AI. Hence, the vector of parameters to determine by calibrating the model is given by p = [a(0), \uf062I, \uf062A, \uf067, \uf061AR]. Table\u00a02 displays the value of the model parameters used to obtain the results shown in Fig.\u00a02. For the analysis of the four scenarios considered in Fig.\u00a03, the values of the model parameters have been set as given in Table\u00a03. Furthermore, to better contrast the results arising in the different case studies, in (A) and (C), we have considered p = [\uf069(0), a(0),", + "postprocess_score": 0.9999856949, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -3.3277609348, + "content": "", + "postprocess_score": 0.9999229908, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 769, + 150, + 1417, + 323 + ], + "detect_score": -0.5080314279, + "content": "Table 2. Values of the model parameters used for the case study in Fig. 2. \uf0690 a0 r0 \uf062I \uf062A \uf061IR \uf061AR \uf061AI 0.05 0.1 0 0.6 0.3 0.1 0.2 0.03", + "postprocess_score": 0.9989734888, + "detect_cls": "Table", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 769, + 356, + 1417, + 630 + ], + "detect_score": -4.5122990608, + "content": "Table 3. Values of the model parameters used for the case study in Fig. 3. \uf067 \uf0690 a0 r0 \uf062I \uf062A \uf061IR \uf061AR \uf061AI Case A 0.1 0.2 0.05 0.3 0.4 0.26 0.1 0.2 0.03 Case B 0.05 0.1 0 0.6 0.3 0.51 0.1 0.2 0.03 Case C 0.1 0.2 0.05 0.6 0.8 0.77 0.1 0.2 0.1 Case D 0.1 0.2 0.05 0.3 0.4 0.53 0.1 0.2 0.03", + "postprocess_score": 0.9844682813, + "detect_cls": "Reference text", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 752, + 656, + 1399, + 706 + ], + "detect_score": -5.0340023041, + "content": "r(0), \uf062I, \uf062A, \uf067, \uf061IR, \uf061AR, \uf061AI], while in (B) and (D), we have set p = [a(0), \uf062I, \uf062A, \uf067, \uf061AR].", + "postprocess_score": 0.9994002581, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 751, + 739, + 1400, + 1796 + ], + "detect_score": -0.3517702222, + "content": "Estimating the parameters of the SIAR model The SIAR model parameters have been estimated from data by adopting two different approaches, i.e., a nonlinear least square error minimization and a Bayesian inference. To generate a synthetic dataset, we have integrated the deterministic model in Eq. 21. To mimic measurement errors, we have adopted the following proce- dure. First, we compute r \u0304 (t) by adding to r(t) a uniform noise in the interval (\u2212 \uf064t/2, \uf064t/2), where \uf064t = \u2223r(t + 1) \u2212 r(t)\u2223, checking that the synthetic time series remains monotonically nondecreasing. We have then generated the data s \u0304 (t) in a similar fashion, this time controlling that the synthetic time series remains monotonically nonincreasing. To generate the data \uf069\u202f\u0304 (t) , we have added a Gaussian noise with zero mean and standard deviation (SD) equals to 3% to the time series, making sure that s \u0304 (t ) + \uf069\u202f\u0304 (t ) + r \u0304 (t ) \u2264 1 . Last, the data a \u0304 (t) have been evaluated using the fact that s \u0304 (t ) + \uf069\u202f\u0304 (t ) + a \u0304 (t ) + r \u0304 (t ) = 1 . The integration of Eq. 21 has been carried out by using the lsoda ordinary differential equation (ODE) solver (53,\u00a054) and then resa- mpling the data with a sampling period of one time unit. We assumed that the density of asymptomatic individuals a is not measurable, while the densities of symptomatic and recovered individuals, i.e., \uf069 and r, are measured variables. As regard to the least square error minimization approach, the model parameters have been estimated using a nonlinear optimiza- tion procedure (implemented via the function fmincon in MATLAB) with the following objective function to minimize ______________________________ \uf074 (22) 1 \u2500 2\uf074\u202f \u2211 ( (\uf069(k ) \u2212 _ \uf069 (k ) ) 2 + (r(k ) \u2212 _ r (k ) ) 2 ) k=1 d = \u221a where \uf069\u202f\u0304 (k) and r \u0304 (k) with k = 1, \u2026, \uf074 (with \uf074 = 50) represent the noisy synthetic time series of the densities of infectious and recovered individuals, respectively, while \uf069(k) and r(k) are the values of the corresponding variables obtained from the integration of Eq. 21. The core idea of Bayesian inference is to provide an a posteriori probability distribution for the model parameter vector, p, given an a priori probability distribution on the value of p and a likelihood function, which quantifies the goodness of a model in reproducing empirical data D. The relationship between these is given by the Bayes' theorem, which reads", + "postprocess_score": 0.9999866486, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 1433, + 712, + 1451, + 1206 + ], + "detect_score": -3.2835578918, + "content": "", + "postprocess_score": 0.5085400939, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 1351, + 1826, + 1416, + 1840 + ], + "detect_score": -5.7406949997, + "content": "10 of 13", + "postprocess_score": 0.9996981621, + "detect_cls": "Reference text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": 0.4311456382, + "content": "\uf070(p \u2223 D ) = \u2112(D \u2223 p ) \uf070(p) \u222b p \u2112(D \u2223 p ) \uf070(p ) dp", + "postprocess_score": 0.9634349346, + "detect_cls": "Equation label", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 267, + 149, + 736, + 213 + ], + "detect_score": -1.3325997591, + "content": "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 (23) \uf070(p \u2223 D ) = \u2112(D \u2223 p ) \uf070(p) \u222b p \u2112(D \u2223 p ) \uf070(p ) dp", + "postprocess_score": 0.9999341965, + "detect_cls": "Equation", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 91, + 236, + 741, + 983 + ], + "detect_score": -3.5776379108, + "content": "\uf070(p \u2223 D ) = \u2112(D \u2223 p ) \uf070(p) \u222b p \u2112(D \u2223 p ) \uf070(p ) dp where \uf070(p) indicates the prior distribution, \u2112(D\u2223p) the likelihood, and \uf070(p\u2223D) the posterior distribution. Usually, it is not possible to evaluate analytically the integral appearing in the denominator, es- pecially when a large number of parameters are considered. There- fore, one relies on MCMC algorithms, which allow one to approximate of the posterior distribution. The MCMC algorithm we used to implement the Bayesian inference is the DRAM (44). As the likeli- hood function \u2112(D\u2223p), we have considered the root mean square error, evaluated on the measurable variables only, i.e. \uf069(t) and r(t), which corresponds to Eq. 22, namely, to the objective function of the nonlinear optimization procedure. As we have assumed to have no a priori knowledge of the values of the model parameters, for the Bayesian inference, we have considered uniform prior probability distributions, which are the simplest and least informative choice (55,\u00a056). Flat priors do not require any additional information apart from setting the interval of possible parameter values. These inter- vals have been defined taking into account the following consider- ations. On the one hand, we have the initial conditions of the dynamical variables. As in the SIAR model, these represent popula- tion densities, we can assume the uniform prior distribution for their initial conditions to be defined in the interval [0,1]. Similarly, the parameter \uf067, which indicates the fraction of newly infected indi- viduals not developing symptoms, can be assumed to be defined in the same interval. As regard the remaining parameters, since we have assumed to not have any other information except for the fact that they are positive quantities, we can consider the uniform distri- bution to be extended in the interval [0, \u221e ].", + "postprocess_score": 0.9999178648, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 91, + 1017, + 741, + 1376 + ], + "detect_score": -6.3548154831, + "content": "Nine-compartment model for COVID-19 The nine-compartment model of Fig.\u00a05 can be considered as a variant of the SIDARTHE model (16). It is characterized by the presence of an incubation state, in which the individuals have been exposed to the virus (E) but are not yet infectious, and by infectious individuals, that, in addition to being symptomatic or asymptomatic, can be either detected or undetected. The model, therefore, includes four classes of infectious individuals: undetected asymptomatic (IA), undetected symptomatic and pauci-symptomatic (IS), home isolated (H, corre- sponding to detected asymptomatic and pauci-symptomatic), and treated in hospital (T, corresponding to detected symptomatic). Last, removed individuals can be undetected (Ru), detected (Rd), or deceased (D). The model dynamics is described by the following equations \u23aa", + "postprocess_score": 0.9704425335, + "detect_cls": "Reference text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 221, + 1412, + 736, + 1796 + ], + "detect_score": -7.0450839996, + "content": "\u23a7 S \u0307 \u00a0=\u00a0\u2212 S( \uf062 I A I A + \uf062 I S I S + \uf062 H H + \uf062 T T ) / N E \u0307 = S( \uf062 I A I A + \uf062 I S I S + \uf062 H H + \uf062 T T ) / N \u2212 ( \uf061 EI A + \uf061 EI S ) E I \u0307 A = \uf061 EI A E \u2212 ( \uf061 I A I S + \uf061 I A R u ) I A \u2212 \uf063 I A I S \u02d9 = \uf061 EI S E + \uf061 I A I S I A \u23aa \u2212 ( \uf061 I S H + \uf061 I S T + \uf061 I S R u + \uf061 I S D ) I S (24) H \u0307 = \uf061 I S H I S + \uf063 I A \u2212 ( \uf061 HT + \uf061 HR d ) H T \u0307 = \uf061 I S T I S + \uf061 HT H \u2212 ( \uf061 T R d + \uf061 TD ) T R \u0307 u = \uf061 I A R u I A + \uf061 I S R u I S R \u0307 d = \uf061 HR d H + \uf061 TR d T \u23a8 \u23aa \u23a9 D \u0307 = \uf061 I S D I S + \uf061 TD T", + "postprocess_score": 0.8992506266, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 92, + 1826, + 546, + 1844 + ], + "detect_score": -1.2907778025, + "content": "", + "postprocess_score": 0.9989314675, + "detect_cls": "Body Text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 769, + 152, + 1451, + 1796 + ], + "detect_score": -0.2476394176, + "content": "where the state variables represent the number of individuals in each compartment, N = 60 \u00b7 106 and S + E + IA + IS + H + T + Ru + Rd + D = N. The official data on the spreading of COVID-19\u00a0in Italy made available by the Civil Protection Department [Dipartimento della Protezione Civile, (49)] provide information only on four of the nine compartments of the model, namely, the home isolated (H), hospitalized (T), detected recovered (Rd), and deceased indi- viduals (D). These compartments constitute the set of the measured variables, while the other variables have to be considered as hidden, i.e., m \u2261 [H, T, Rd, D] and h \u2261 [S, E, IA, IS, Ru]. All the parameters appearing in (24) are considered unknown; thus, they need to be determined through fitting the model to the available data. It should also be noted that, as many nonpharmaceu- tical interventions have been issued/lifted, and the testing strategy has been changed several times over the course of the epidemics (47,\u00a048), not all parameters can be considered constant in the whole period used for the fitting. Hence, similarly to (16), we have divided the whole period of investigation (which in our case ranges from 24 February to 06 July 2020) into different windows, within each of which the parameters are assumed to be constant. In each time window, one allows only some parameters to vary according to what is reasonable to assume will be influenced by the government intervention during that time window. We distinguish two kinds of events that may require an adapta- tion of the model parameters. On the one hand, there are the non- pharmaceutical containment policies aimed at reducing the disease transmission. When these interventions are issued, the value of the parameters \uf062 may vary. On the other hand, the testing strategy, which affects the probability of detecting infected individuals, was also not uniform in the investigated period. When the testing policy changes, the value of the parameters \uf061 I S H , \uf061HT, and \uf061 HR d may vary. Here, we notice two important points. First, the value of \uf061 I S T is assumed to be constant in the whole period, as we suppose that there are no changes in how the symptomatic individuals requiring hospitalization are detected. Second, as a change in the sole param- eter \uf061 I S H would affect too much the average time an individual remains infected, then \uf061HT and \uf061 HR d also have to be included in the set of parameters that may change. On the basis of these consider- ations, the intervals in which each parameter remains constant or may change are identified. This defines the specific piece-wise waveform assumed for each of the parameters appearing in the model and, consequently, the effective number of values that need to be estimated for each parameter. Hereafter, we summarize the events defining the different windows in which the whole period of investigation is partitioned: 1) On 02 March, a policy limiting screening only to symptomatic individuals is introduced. 2) On 12 March, a partial lockdown is issued. 3) On 18 March, a stricter lockdown, which further limits nonessential activities, is imposed. 4) On 29 March, a wider testing campaign is launched. Starting from this date, as the number of tests has constantly increased while the number of new infections was decreasing, the parameters are allowed to change every 14 or 28 days, namely, on 11 April, 25 April, and 23 May. 5) On 04 May, a partial lockdown lift is proclaimed. 6) On 18 May, further restrictions are relaxed. 7) On 03 June, interregional mobility is allowed. This is the last time the model parameters are changed.", + "postprocess_score": 0.9999527931, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 1351, + 1826, + 1416, + 1840 + ], + "detect_score": -6.5557370186, + "content": "11 of 13", + "postprocess_score": 0.9987276196, + "detect_cls": "Section Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 91, + 75, + 1417, + 116 + ], + "detect_score": -0.1501361281, + "content": "", + "postprocess_score": 0.9991431236, + "detect_cls": "Reference text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 92, + 152, + 738, + 313 + ], + "detect_score": -6.259206295, + "content": "Note that, for the time period until 5 April, we have followed the same time partition used in (16). The model parameters have been estimated using a nonlinear optimization procedure (implemented via the function fmincon in MATLAB) with the following objective function to minimize e = ______________________________________________________________ _ ( (H(k) \u2212 _ H (k ) ) 2 + (T(k ) \u2212 T (k ) ) 2 + ( R d (k ) \u2212 _ R d (k ) ) 2 + (D(k ) \u2212 _ D (k ) ) 2 )", + "postprocess_score": 0.9980720282, + "detect_cls": "Equation label", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 91, + 329, + 741, + 622 + ], + "detect_score": -2.5377037525, + "content": "e = ______________________________________________________________ \u03c4 _ ( (H(k) \u2212 _ H (k ) ) 2 + (T(k ) \u2212 T (k ) ) 2 + ( R d (k ) \u2212 _ R d (k ) ) 2 + (D(k ) \u2212 _ D (k ) ) 2 ) 1 \u2500 4\u03c4\u202f \u2211 k=1 (25) \u221a where H \u0304 (k) , T \u0304 (k) , R \u0304 d (k) , and D \u0304 (k) with k = 1, \u2026, \uf074 (\uf074 = 134 days) represent the time series of daily data for isolated, hospitalized, detected recovered, and deceased individuals provided by the Civil Protection Department (49), and H(k), T(k), Rd(k), and D(k) are the values of the corresponding variables obtained from the integration of Eq. 24. The integration of Eq. 24 has been carried out by using a suitable ODE solver with maximum integration step size equal to 10\u22122 days and then resampling the data with a sampling period of 1 day.", + "postprocess_score": 0.9774880409, + "detect_cls": "Reference text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 93, + 661, + 738, + 1793 + ], + "detect_score": -1.4617025852, + "content": "REFERENCES AND NOTES 1. R. M. Anderson, H. Heesterbeek, D. Klinkenberg, T. D. Hollingsworth, How will country-based mitigation measures influence the course of the covid-19 epidemic? Lancet 395, 931\u2013934 (2020). 2. World Health Organization (WHO), Coronavirus disease (covid-19): Weekly epidemiological update (2020); https://who.int/emergencies/diseases/novel-coronavirus-2019/ situation-reports[accessed 15 October 2020]. 3. E. Dong, H. Du, L. Gardner, An interactive web-based dashboard to track covid-19 in real time. Lancet Infect. 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Aalto, L. Mombaerts, A. Fouquier d'H\u00e9rou\u00ebl, A. Husch, C. Ley, J. Gon\u00e7alves, A. Skupin, S. Magni, Modelling COVID-19 dynamics and potential for herd immunity by vaccination in Austria, Luxembourg and Sweden. J. Theor. Biol. 530, 110874 (2021). 56. G. E. P. Box, G. C. Tiao, Bayesian Inference in Statistical Analysis, vol. 40 (John Wiley & Sons, 2011).", + "postprocess_score": 0.9784602523, + "detect_cls": "Reference text", + "postprocess_cls": "Reference text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 769, + 391, + 1416, + 860 + ], + "detect_score": -2.8751766682, + "content": "Acknowledgments: We would like to thank V. Simoncini for pointing out the relation between the sensitivity \uf06d and the generalized eigenvalue and D. Proverbio for the useful discussion on the Bayesian approach for parameter estimation. V.L. acknowledges support from the Leverhulme Trust Research Fellowship 278 \"CREATE: The network components of creativity and success.\" V.L. and G.R. acknowledge support from University of Catania project \"Piano della Ricerca 2020/2022, Linea d'intervento 2, MOSCOVID.\" G.R. acknowledges support from Italian Ministry of Instruction, University and Research (MIUR) through PRIN project 2017, no. 2017KKJP4X. G.R. is a member of the \"Istituto Nazionale di Alta Matematica Francesco Severi (INdAM) and \"Gruppo Nazionale per il Calcolo Scientifico (GNCS). Author contributions: L.G., M.F., V.L., and G.R. conceived the research and developed the theory. L.G. carried out the numerical analysis. All authors wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: Epidemiological data displayed in Fig. 6 are publicly available data at the Italian Civil Protection repository (https://github.com/pcm-dpc/COVID-19/tree/master/dati-andamento- nazionale). Information about governmental containment policies in Italy are available at the Presidency of the Council of Ministers website (http://governo.it/it/coronavirus-misure-del- governo and http://governo.it/it/coronavirus-normativa). All the codes to perform the analyses discussed in the study and to produce Figs. 4 and 6 are made publicy available in the Zenodo repository (DOI: https://doi.org/10.5281/zenodo.5639320). All remaining data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.", + "postprocess_score": 0.9999506474, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 770, + 894, + 966, + 978 + ], + "detect_score": -5.0833125114, + "content": "Submitted 11 January 2021 Accepted 24 November 2021 Published 19 January 2022 10.1126/sciadv.abg5234", + "postprocess_score": 0.5784535408, + "detect_cls": "Page Footer", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 1433, + 712, + 1451, + 1206 + ], + "detect_score": -5.493847847, + "content": "", + "postprocess_score": 0.7526928186, + "detect_cls": "Equation", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 1351, + 1826, + 1416, + 1840 + ], + "detect_score": -1.5780289173, + "content": "13 of 13", + "postprocess_score": 0.9998119473, + "detect_cls": "Reference text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 201, + 124, + 817, + 200 + ], + "detect_score": -0.4084015489, + "content": "", + "postprocess_score": 0.9964281917, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 201, + 313, + 1206, + 405 + ], + "detect_score": -2.8792304993, + "content": "Lack of practical identifiability may hamper reliable predictions in COVID-19 epidemic models Luca Gallo, Mattia Frasca, Vito Latora, and Giovanni Russo", + "postprocess_score": 0.6030353308, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 202, + 485, + 469, + 528 + ], + "detect_score": -2.3431417942, + "content": "Sci. 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The title Science Advances is a registered trademark of AAAS. Copyright \u00a9 2022 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. 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"Problem annotating with SKEMA: Calling http://hopper.sista.arizona.edu/textFileToMentions failed with HTTP code 502" + } + ] + }, + "tds_status_code": 200, + "error": null + }, + "job_error": null + }, + "time": 166.79583835601807, + "accuracy": null, + "success": true + }, + "code_to_amr": { + "id": "extraction-a6629f9a-9a14-4e1b-b934-d8eb4ad81964", + "status": "failed", + "result": { + "created_at": "2023-10-17T19:38:34.180570", + "enqueued_at": "2023-10-17T19:38:34.181258", + "started_at": "2023-10-17T19:38:34.187614", + "job_result": null, + "job_error": "Traceback (most recent call last):\n File \"/usr/local/lib/python3.10/site-packages/rq/worker.py\", line 1428, in perform_job\n rv = job.perform()\n File \"/usr/local/lib/python3.10/site-packages/rq/job.py\", line 1278, in perform\n self._result = self._execute()\n File \"/usr/local/lib/python3.10/site-packages/rq/job.py\", line 1315, in _execute\n result = self.func(*self.args, **self.kwargs)\n File \"/./worker/operations.py\", line 739, in code_to_amr\n raise Exception(f\"Code extraction failure: {amr_response.text}\")\nException: Code extraction failure: {\"error\":\"MORAE PUT /models/PN failed to process payload\",\"payload\":{\"schema\":\"FN\",\"schema_version\":\"0.1.8\",\"name\":\"zip_file\",\"modules\":[{\"schema\":\"FN\",\"schema_version\":\"0.1.8\",\"name\":\"code\",\"fn\":{\"name\":null,\"b\":[{\"function_type\":\"MODULE\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module\",\"metadata\":3}],\"opi\":null,\"opo\":null,\"wopio\":null,\"bf\":[{\"function_type\":\"IMPORTED\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"odeint\",\"metadata\":null},{\"function_type\":\"EXPRESSION\",\"body\":72,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":219},{\"function_type\":\"EXPRESSION\",\"body\":73,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":222},{\"function_type\":\"EXPRESSION\",\"body\":74,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":225},{\"function_type\":\"EXPRESSION\",\"body\":75,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":228},{\"function_type\":\"EXPRESSION\",\"body\":76,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":231},{\"function_type\":\"EXPRESSION\",\"body\":77,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":233},{\"function_type\":\"EXPRESSION\",\"body\":78,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":235},{\"function_type\":\"EXPRESSION\",\"body\":79,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":237},{\"function_type\":\"EXPRESSION\",\"body\":80,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":246},{\"function_type\":\"ABSTRACT\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"pack\",\"metadata\":null},{\"function_type\":\"EXPRESSION\",\"body\":81,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":248},{\"function_type\":\"EXPRESSION\",\"body\":82,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":250},{\"function_type\":\"EXPRESSION\",\"body\":83,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":252},{\"function_type\":\"IMPORTED\",\"body\":null,\"import_type\":\"OTHER\",\"import_version\":null,\"import_source\":null,\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"numpy.arange\",\"metadata\":253},{\"function_type\":\"IMPORTED\",\"body\":84,\"import_type\":\"OTHER\",\"import_version\":null,\"import_source\":null,\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.arange_0\",\"metadata\":254},{\"function_type\":\"IMPORTED_METHOD\",\"body\":85,\"import_type\":\"OTHER\",\"import_version\":null,\"import_source\":null,\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"scipy.integrate.odeint_id83\",\"metadata\":255},{\"function_type\":\"ABSTRACT\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"pack\",\"metadata\":null},{\"function_type\":\"EXPRESSION\",\"body\":86,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":256},{\"function_type\":\"IMPORTED\",\"body\":null,\"import_type\":\"OTHER\",\"import_version\":null,\"import_source\":null,\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"matplotlib.pyplot.subplots\",\"metadata\":257},{\"function_type\":\"IMPORTED\",\"body\":87,\"import_type\":\"OTHER\",\"import_version\":null,\"import_source\":null,\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.subplots_0\",\"metadata\":258},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"Integer\",\"value\":1,\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":259},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"Integer\",\"value\":1,\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":260},{\"function_type\":\"EXPRESSION\",\"body\":88,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":262},{\"function_type\":\"EXPRESSION\",\"body\":89,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":264},{\"function_type\":\"ABSTRACT\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"pack\",\"metadata\":null},{\"function_type\":\"ABSTRACT\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"unpack\",\"metadata\":null},{\"function_type\":null,\"body\":90,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_0\",\"metadata\":267},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"b\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":268},{\"function_type\":\"EXPRESSION\",\"body\":91,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":270},{\"function_type\":\"EXPRESSION\",\"body\":92,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":272},{\"function_type\":\"EXPRESSION\",\"body\":93,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":274},{\"function_type\":null,\"body\":94,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_1\",\"metadata\":275},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"r\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":276},{\"function_type\":\"EXPRESSION\",\"body\":95,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":278},{\"function_type\":\"EXPRESSION\",\"body\":96,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":280},{\"function_type\":\"EXPRESSION\",\"body\":97,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":282},{\"function_type\":null,\"body\":98,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_2\",\"metadata\":283},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"r.\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":284},{\"function_type\":\"EXPRESSION\",\"body\":99,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":286},{\"function_type\":\"EXPRESSION\",\"body\":100,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":288},{\"function_type\":\"EXPRESSION\",\"body\":101,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":290},{\"function_type\":null,\"body\":102,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_3\",\"metadata\":291},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"r:\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":292},{\"function_type\":\"EXPRESSION\",\"body\":103,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":294},{\"function_type\":\"EXPRESSION\",\"body\":104,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":296},{\"function_type\":\"EXPRESSION\",\"body\":105,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":298},{\"function_type\":null,\"body\":106,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_4\",\"metadata\":299},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"r--\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":300},{\"function_type\":\"EXPRESSION\",\"body\":107,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":302},{\"function_type\":\"EXPRESSION\",\"body\":108,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":304},{\"function_type\":\"EXPRESSION\",\"body\":109,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":306},{\"function_type\":null,\"body\":110,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_5\",\"metadata\":307},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"r-.\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":308},{\"function_type\":\"EXPRESSION\",\"body\":111,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":310},{\"function_type\":\"EXPRESSION\",\"body\":112,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":312},{\"function_type\":\"EXPRESSION\",\"body\":113,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":314},{\"function_type\":null,\"body\":114,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_6\",\"metadata\":315},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"g\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":316},{\"function_type\":\"EXPRESSION\",\"body\":115,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":318},{\"function_type\":\"EXPRESSION\",\"body\":116,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":320},{\"function_type\":\"EXPRESSION\",\"body\":117,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":322},{\"function_type\":null,\"body\":118,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":\"module.plot_7\",\"metadata\":323},{\"function_type\":\"LITERAL\",\"body\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":{\"value_type\":\"List\",\"value\":\"k\",\"source_fn\":null,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"gromet_type\":\"LiteralValue\"},\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":324},{\"function_type\":\"EXPRESSION\",\"body\":119,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":326},{\"function_type\":\"EXPRESSION\",\"body\":120,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"source_language_version\":null,\"value\":null,\"gromet_type\":\"GrometBoxFunction\",\"name\":null,\"metadata\":328},{\"function_type\":\"EXPRESSION\",\"body\":121,\"import_type\":null,\"import_version\":null,\"import_source\":null,\"source_language\":null,\"s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lists available at ScienceDirect Science of the Total Environment j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s c i t o t e n v", + "postprocess_score": 0.9971317053, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 406, + 1278, + 480 + ], + "detect_score": -8.6862258911, + "content": "A simple SEIR-V model to estimate COVID-19 prevalence and predict SARS- CoV-2 transmission using wastewater-based surveillance data", + "postprocess_score": 0.7014417052, + "detect_cls": "Equation label", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 91, + 502, + 1106, + 680 + ], + "detect_score": -8.3644170761, + "content": "Tin Phan a,1, Samantha Brozak b,1, Bruce Pell c, Anna Gitter d, Amy Xiao e, Kristina D. Mena d, , Fuqing Wu d,\u204e Yang Kuang b,\u204e a Theoretical Biology and Biophysics Group, Los Alamos National Laboratory, NM, USA b School of Mathematical and Statistical Sciences, Arizona State University, AZ, USA c Department of Mathematics and Computer Science, Lawrence Technological University, MI, USA d The University of Texas Health Science Center at Houston, School of Public Health, Houston, TX, USA 77030 e Center for Microbiome Informatics and Therapeutics; Department of Biological Engineering, Massachusetts Institute of Technology", + "postprocess_score": 0.3689925373, + "detect_cls": "Other", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 495, + 736, + 808, + 748 + ], + "detect_score": -4.1669778824, + "content": "G R A P H I C A L A B S T R A C T", + "postprocess_score": 0.5772188902, + "detect_cls": "Page Footer", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 91, + 736, + 262, + 748 + ], + "detect_score": -1.3129056692, + "content": "H I G H L I G H T S", + "postprocess_score": 0.9908440709, + "detect_cls": "Table", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 777, + 416, + 1010 + ], + "detect_score": -6.0076951981, + "content": "\u2022 A simple and effective framework is devel- oped to bridge WBS and epidemic model. \u2022 SEIR-V model recapitulates the temporal dynamics of viral load in wastewater. \u2022 Model predicts the number of COVID-19 case peaked earlier and higher than re- ported data. \u2022 Incorporating viral decay in wastewater improves model performance and robust- ness.", + "postprocess_score": 0.5214386582, + "detect_cls": "Table Caption", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 488, + 777, + 1349, + 1163 + ], + "detect_score": 1.1858966351, + "content": "", + "postprocess_score": 0.9994689822, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 1208, + 287, + 1221 + ], + "detect_score": -3.7703084946, + "content": "A R T I C L E I N F O", + "postprocess_score": 0.9959999323, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 488, + 1208, + 624, + 1221 + ], + "detect_score": -0.7579526305, + "content": "A B S T R A C T", + "postprocess_score": 0.9888945818, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 1245, + 413, + 1265 + ], + "detect_score": -1.4069392681, + "content": "Editor: Dami\u00e0 Barcel\u00f3", + "postprocess_score": 0.4798469841, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 488, + 1245, + 1349, + 1549 + ], + "detect_score": -0.4412895739, + "content": "Wastewater-based surveillance (WBS) has been widely used as a public health tool to monitor SARS-CoV-2 transmis- sion. However, epidemiological inference from WBS data remains understudied and limits its application. In this study, we have established a quantitative framework to estimate COVID-19 prevalence and predict SARS-CoV-2 transmission through integrating WBS data into an SEIR-V model. We conceptually divide the individual-level viral shedding course into exposed, infectious, and recovery phases as an analogy to the compartments in a population-level SEIR model. We demonstrated that the effect of temperature on viral losses in the sewer can be straightforwardly incorporated in our framework. Using WBS data from the second wave of the pandemic (Oct 02, 2020\u2013Jan 25, 2021) in the Greater Boston area, we showed that the SEIR-V model successfully recapitulates the temporal dynamics of viral load in wastewater and predicts the true number of cases peaked earlier and higher than the number of reported cases by 6\u201316 days and 8.3\u201310.2 folds (R = 0.93). This work showcases a simple yet effective method to bridge WBS and quantitative ep- idemiological modeling to estimate the prevalence and transmission of SARS-CoV-2 in the sewershed, which could fa- cilitate the application of wastewater surveillance of infectious diseases for epidemiological inference and inform public health actions.", + "postprocess_score": 0.999281466, + "detect_cls": "Abstract", + "postprocess_cls": "Abstract" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 1286, + 413, + 1415 + ], + "detect_score": -3.5110452175, + "content": "Keywords: SEIR-V model Wastewater-based epidemiology Epidemic model SARS-CoV-2 Temperature", + "postprocess_score": 0.846640408, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 1615, + 1348, + 1679 + ], + "detect_score": -1.8872317076, + "content": "\u204e Corresponding authors. E-mail addresses: kuang@asu.edu (Y. Kuang), fuqing.wu@uth.tmc.edu (F. Wu). 1 These authors contribute equally", + "postprocess_score": 0.8801718354, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 90, + 1743, + 799, + 1830 + ], + "detect_score": -2.4200100899, + "content": "http://dx.doi.org/10.1016/j.scitotenv.2022.159326 Received 19 July 2022; Received in revised form 15 September 2022; Accepted 5 October 2022 Available online 8 October 2022 0048-9697/\u00a9 2022 Elsevier B.V. All rights reserved.", + "postprocess_score": 0.8235943913, + "detect_cls": "Other", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 91, + 88, + 172, + 99 + ], + "detect_score": -3.3805782795, + "content": "T. Phan et al.", + "postprocess_score": 0.9866392016, + "detect_cls": "Reference text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 92, + 132, + 224, + 146 + ], + "detect_score": -2.9703912735, + "content": "1. Introduction", + "postprocess_score": 0.6770319939, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 90, + 182, + 699, + 1567 + ], + "detect_score": -4.1110239029, + "content": "Wastewater-based surveillance (WBS) has been used as a public health tool to monitor SARS-CoV-2 infection in the population since the beginning of the COVID-19 pandemic. So far, WBS has been widely implemented in over 67 countries (Naughton et al., 2021). The Centers for Disease Control and Prevention (CDC) also launched the National Wastewater Surveillance System in late 2020 to monitor the spread of COVID-19 in the United States (CDC, 2020). Wastewater pools SARS-CoV-2 particles excreted by infected individuals irrespective of clinical symptoms or presentation, which pro- vides an opportunity to capture the viral shedding prior to symptoms and estimate the true magnitude of viral infections in communities (Bivins et al., 2020; Hart and Halden, 2020; Peccia et al., 2020; Randazzo et al., 2020; Saguti et al., 2021; Wu et al., 2022b). Previous work has shown that SARS-CoV-2 concentrations in wastewater were much higher than ex- pected from clinically reported cases and preceded clinically reported data by 4\u201310 days (Wu et al., 2020, 2022b; Peccia et al., 2020), and up to 14 days (Krivo\u0148\u00e1kov\u00e1 et al., 2021; Karthikeyan et al., 2022). Furthermore, the fast turnaround time of wastewater and flexible sampling strategy en- able WBS to provide a near real-time monitoring of viral transmission in the sewershed. Finally, WBS is less resource intensive than the large-scale, individual-based clinical testing and thus can be used as a cost-efficient tool for monitor the trend of viral infection in the population and new var- iants when combined with next-generation sequencing (Bivins et al., 2020; Safford et al., 2022; Wu et al., 2022a). These properties make WBS a feasi- ble public health tool to monitor SARS-CoV-2 in an endemic, which can also be customized for future pandemics. WBS has enabled researchers to estimate the total viral load in a sewershed; however, there are still limitations regarding quantifying and predicting viral transmission in a community. Few recent studies have tried to build classical susceptible-infected-removed (SIR)-type models to bridge the measured viral concentration and reported case number. For ex- ample, Proverbio et al. (2022) added a variable that keeps track of actively shedding individuals in a stochastic susceptible-exposed-infectious- recovered (SEIR) model and used a constant viral shedding rate to connect the number of infected cases to viral concentration in wastewater (Proverbio et al., 2022). Conversely, Brouwer et al. (2022) accounted for time dependent viral shedding rates by incorporating multiple subclasses with different shedding rates within each infected stage of the model to bet- ter predict viral concentrations and reported cases (Brouwer et al., 2022). A similar approach is conducted by Nourbakhsh et al. (2022), but with more sub-classification of the infected class (Nourbakhsh et al., 2022). These modeling approaches allow the modelers to connect viral concentrations in wastewater with the reported cases and predict the course of the pan- demic. Dynamical models in epidemiology often overlook the opportunity to utilize biologically interpretable and experimentally measurable parame- ters in the link between infected people and the shed viral RNA in wastewa- ter. The model structure is usually complicated with many parameters, so it is difficult to fully parametrize the models without running into issues such as model identifiability. Hence, our primary objective in this work is to le- verage our understanding of the biology of SARS-CoV-2 shedding to con- struct a simple, mechanistic, dynamic model that connects viral load in wastewater with the total number of infected cases in the sewershed. Our secondary objective is to introduce the effect of wastewater temperature into the modeling framework due to its significant impact on the viral loss (or decay) rate in the sewer (Hart and Halden, 2020).", + "postprocess_score": 0.9993200302, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 91, + 1599, + 315, + 1614 + ], + "detect_score": -3.3043739796, + "content": "2. Materials and methods", + "postprocess_score": 0.9991096854, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 90, + 1650, + 355, + 1668 + ], + "detect_score": 3.5132510662, + "content": "2.1. Samples and wastewater data", + "postprocess_score": 0.9999574423, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 90, + 1701, + 699, + 1795 + ], + "detect_score": -2.5398480892, + "content": "Raw, 24-h composite wastewater samples were collected from the Deer Island wastewater treatment plant in Massachusetts from October 02, 2020 to January 25, 2021. The Massachusetts wastewater treatment plant where we obtained samples has two major influent streams, which are referred to", + "postprocess_score": 0.9999319315, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 132, + 1349, + 783 + ], + "detect_score": -3.5681116581, + "content": "as the \"northern\" and \"southern\" influents. The daily flow rates during the sampling period for the northern and southern influents are 4.54e5\u20132.3e6 m3/day, and 2.16e5\u20131.19e6m3/day, respectively. Together the two catch- ments represent approximately 2.3 million wastewater customers in Mid- dlesex, Norfolk, and Suffolk counties, primarily in urban and suburban neighborhoods. There are 5100 miles of local sewers transporting wastewa- ter into 227 miles of interceptor pipes to the wastewater treatment plant (www.mwra.com), and the typical turnaround time for the plant to treat wastewater is 24 h. Samples were processed as they were received. Experi- mental methods and data were reported in our previous work (Wu et al., 2022b; Xiao et al., 2022). Briefly, the samples were pasteurized at 60 \u00b0C for 1 h for disinfection, and then filtered with 0.2 \u03bcm hydrophilic polyether- sulfone membrane (Millipore Sigma) to remove bacterial cells and debris. Then, 15-ml filtrate was concentrated to ~200 ul with Amicon Ultra Cen- trifugal Filter (30-kDa cutoff, Millipore Sigma), and lysed with Qiagen AVL buffer followed by RNA extraction with Qiagen RNeasy kit. SARS- CoV-2 concentrations were quantified by one-step reverse transcription- polymerase chain reaction (RT-PCR) with the Taqman Fast Virus 1-Step Master Mix (Thermofisher) and CDC N1 and N2 primers/probes. Ct values were transformed to copies per ml of wastewater using standard curves for N1 and N2 targets established with synthetic SARS-CoV-2 RNA (Twist Bio- science) as the template. To compute the total viral load in the sewershed, we first averaged the viral concentration in the northern and southern influ- ents by the sampling date, which is then multiplied by the total influent flow rates (i.e., sum of flow rates of northern and southern influents) on the same day.", + "postprocess_score": 0.9999728203, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 815, + 928, + 830 + ], + "detect_score": -3.5608215332, + "content": "2.2. Clinical data source", + "postprocess_score": 0.9998873472, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 865, + 1348, + 1137 + ], + "detect_score": 0.8875368237, + "content": "The clinical COVID-19 case data for Norfolk, Suffolk, and Middlesex Counties served by the Massachusetts wastewater treatment plant were downloaded from Massachusetts government website (www.mass.gov). The plant covers about 71.9 % of the total population in the three counties, including almost all of Suffolk County (99.8 %), 59.8 % of Middlesex County, and 68.7 % of Norfolk County, based on the 2020 Census popula- tion data. For simplicity, we summed the number of clinical cases from each county to represent the total cases in the catchment of the wastewater treatment plant, which is used to compare with the modeling results. Tem- poral fecal viral shedding data from COVID-19 patients were kindly pro- vided by (W\u00f6lfel et al., 2020).", + "postprocess_score": 0.9999845028, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 1169, + 1340, + 1188 + ], + "detect_score": -5.0583562851, + "content": "2.3. Relationship between wastewater viral concentrations and infectious cases", + "postprocess_score": 0.6296675205, + "detect_cls": "Body Text", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 741, + 1220, + 1349, + 1335 + ], + "detect_score": -2.0624237061, + "content": "Assuming we can obtain the fecal viral shedding distribution function over time, we can approximate a constant rate of fecal viral shedding over the duration of infectiousness. In this way, the viral RNA production is proportional to the number of people in the infectious compartment I of the SEIR model. That is:", + "postprocess_score": 0.9997256398, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 1377, + 1347, + 1397 + ], + "detect_score": -4.1548461914, + "content": "\u00f0 total viral production in wastewater\u2248\u03b1 \u03b2 1 \u03b3 \u00de I, (1)", + "postprocess_score": 0.8794711828, + "detect_cls": "Other", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 1437, + 1349, + 1528 + ], + "detect_score": -4.0802359581, + "content": "where the proportional constant is defined based on biological parameters similar to (Saththasivam et al., 2021): \u03b1 is the fecal load with unit g/day/ person, \u03b2 is the viral shedding rate in stool with unit viral copies/g, and \u03b3 is the fraction of viral loss in the sewer.", + "postprocess_score": 0.9998830557, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 1564, + 1119, + 1583 + ], + "detect_score": -2.8297317028, + "content": "2.4. Approximation of fecal viral shedding profile", + "postprocess_score": 0.9981726408, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 740, + 1615, + 1351, + 1794 + ], + "detect_score": 3.3602070808, + "content": "A key component of this approach is the generation of fecal viral shed- ding profile. Let f(t) be the function that describes the temporal fecal viral shedding profile. Upon infection, the shedding of virus in stool should be very small, then reaches a peak before decreasing to 0. Mathematically, this means f(0) = 0, lim t!\u221e f t\u00f0 \u00de \u00bc 0 and f(t) has a unique maximum for some t > 0. While beta and gamma functions are often used to represent f (t) (Wu et al., 2022a; Ferretti et al., 2020; He et al., 2020), we introduce a", + "postprocess_score": 0.9999153614, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 716, + 1830, + 723, + 1841 + ], + "detect_score": -2.4126019478, + "content": "2", + "postprocess_score": 0.9997413754, + "detect_cls": "Body Text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 1015, + 88, + 1348, + 103 + ], + "detect_score": -5.6758308411, + "content": "Science of the Total Environment 857 (2023) 159326", + "postprocess_score": 0.999943018, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 88, + 88, + 1349, + 1841 + ], + "detect_score": -2.3572018147, + "content": "T. Phan et al. Science of the Total Environment 857 (2023) 159326 phenomenological function f(t) that is more tractable than the standard duration to 8 days, which is equivalent to fixing \u03b4 \u00bc 1 8 per day. Thus, in beta and gamma functions: our model, parameters \u03bb, \u03b1, \u03b2, and \u03b3 need to be estimated. By fitting the model to wastewater data covering the second wave of the pandemic, specifically, from Oct 2 to Dec 16, 2020, we can approximate the : f t\u00f0 \u00de \u00bc \u03c91t (2) \u03c92 2 \u00fe t2 susceptible (to an emerging variant) to be the entire population served by the wastewater treatment plant. For simplification, we assume that there are no infectious individuals initially (I(0) = 0), only infected individuals In this form, \u03c91 is a magnitude modifier parameter (log10 viral RNA copy (E(0) > 0) due to the assumed lack of immunity to new circulating variants. per g per day) and \u03c92 (day) represents the timing for peak viral shedding and The initial value for the virus concentration in wastewater can be taken influences the timing and the magnitude of the peak of the viral shedding from the first data point. Thus, E(0) is the only unknown initial condition. profile. Specifically, f(t) peaks at \u03c91 when t = \u03c92. Thus, if the peak timing 2\u03c92 The parameters and initial conditions that remain to be estimated are: \u03bb, and magnitude of the viral shedding profile are known, then f(t) can be \u03b1, \u03b2, \u03b3, and E(0). Since the viral production rate is \u03b1\u03b2(1 \u2212 \u03b3), and we only uniquely defined. It is necessary to mention that f(t) is the overall viral shed- have viral concentration (or total viral load) data, it is impossible to esti- ding into the wastewater from infected individuals; however, it mostly mate a unique set of values, or specific values, for \u03b1, \u03b2, and \u03b3. For example, means fecal shedding in this work. We did not include the viral shedding the product of \u03b1 = 1, \u03b2 = 2, \u03b3 = 0.5 is the same as when \u03b1 = 10, \u03b2 = 1, \u03b3 = from urine or other sources (sputum or saliva) because previous studies 0.9. This reflects the pertinent issue of model identifiability in mathemati- showed that no or low level of virus was detected in urine samples of typical cal models in biology and epidemiology (Tuncer et al., 2022; Eisenberg patients despite high viral load (W\u00f6lfel et al., 2020; Jones et al., 2020), and et al., 2013; Wu et al., 2019; Ciupe and Tuncer, 2022). Thus, an important the total amount of virus in sputum or saliva are likely to be insignificant step in our approach is the direct estimations of \u03b2 and \u03b3, which would allow compared to stool due to the huge difference in volume. us to identify \u03b1 uniquely. 2.5. Simple wastewater epidemiological model 2.6. Incorporating the effect of temperature on the viral degradation rate S0 \u00bc \u03bbIS In order to account for temperature variation over time, a sine curve E0 \u00bc \u03bbIS kE was fit to the average of temperatures at the northern and southern influ- (3) I0 \u00bc kE \u03b4I ents (Brozak et al., 2022). The curve describing the temperature in degrees \u00f0 \u00deI V 0 \u00bc \u03b1\u03b2 1 \u03b3 Celsius at time t (Fig. S1) is given by \u00f0 T t\u00f0 \u00de \u00bc 3:6249 sin 0:0202t 4:4665 \u00de \u00fe 16:2298: In this model, S denotes the susceptible population, E is the infected but yet to be infectious population (or the exposed class), I is the infectious The temperature-adjusted half-life is described by class, and V is the cumulative viral load in wastewater. The R compartment (recovered individuals) does not contribute to the transmission dynamics in \u00f0 \u2212 T t\u00f0 \u00de\u2212T0 \u00de=10 \u00b0C; \u00f04\u00de \u03b7 T\u00f0 \u00de \u00bc \u03b70Q10 the SEIR model, hence omitted here. Susceptible people are infected by the infectious class at a rate \u03bbI. Exposed individuals become infectious at a rate k. Infectious individuals recover at a rate \u03b4 and shed virus at a rate \u03b1 \u00d7 \u03b2, where \u03b70 is the half-life in hours at ambient temperature T0 and Q10 is the temperature-dependent rate of change (McMahan et al., 2021; Hart and where \u03b1 is the fecal load and \u03b2 is the average viral shedding rate in Eq. (1). The time spent in the E and I classes are exponentially distributed Halden, 2020). Q10 is typically between 2 and 3 for biological systems, and assumed here to be 2.5 (B\u02c7ehr\u00e1dek, 1930; Reyes et al., 2008). with average duration of 1/k and 1/\u03b4, respectively. \u03b3 is the viral degrada- The temperature-adjusted first-order decay rate \u03be(T) (per hour) is then tion and loss rate in the sewer pipes, so only a fraction (1 \u2212 \u03b3) of virus is detected in the wastewater sample. The expression for V follows directly from Eq. (1). \u03be T\u00f0 \u00de \u00bc ln 2 \u03b7 T\u00f0 \u00de : Several studies note that infectious virus is detectable in nose and throat swabs only when the total viral load is above 105\u22126 copies/mL (Killingley et al., 2022; Ke et al., 2021; W\u00f6lfel et al., 2020; Kampen et al., 2021). We used the simple exponential decay equation V\u2032 \u00bc b\u03b7 T\u00f0 \u00deV to esti- Since a certain level of infectious viruses is required for disease transmis- mate the losses in the sewer \u03b3. Then, sion, this implies that the infectious period does not start until the viral load (within host) reaches above 105\u22126 virus copies/mL. The shedding of \u00f05\u00de V t\u00f0 \u00de \u00bc V 0e\u2212\u03be T\u00f0 \u00det ; infectious virus that links to transmission happens early and rapidly dimin- ishes within 10 days after symptom onset; however, significant heterogene- where V0 is the amount of viral RNA in the sewers at time t = 0. Thus, the ity exists (Ke et al., 2022; Heitzman-Breen and Ciupe, 2022; Boucau et al., amount of virus that arrives to the wastewater treatment plant is 2022). This agrees with previous observations that viral loads above 106 copies/mL are associated with a high probability of transmission (Ke \u00f0 \u00de \u00bc V 0e\u2212\u03be T\u00f0 \u00detarrive ; \u00f06\u00de V tarrive et al., 2021). Together, these observations suggest that in this SEIR epi- demic model, we can separate the exposed class (E) based on the duration before viral load reaches 105\u22126 copies/mL, and the infectious class (I) where tarrive is the time it takes the viral RNA to travel to the wastewater based on the duration that viral load stays above 105\u22126 copies/mL. This re- treatment plant after excretion. The amount of virus lost is given by V0 \u2212 sults in an incubation period of about 3 days and an infectious period of 8 V(tarrive). Thus, the proportion of viral RNA lost in the sewer is given by days based on the viral dynamics profile in the SARS-CoV-2 Human Chal- \u00f0 \u00de \u00f0 \u00de lenge experiment in healthy young adults (Killingley et al., 2022). These es- \u00bc 1\u2212 V tarrive \u00bc 1\u2212e\u2212\u03be T\u00f0 \u00detarrive ; \u00f07\u00de \u03b3 T\u00f0 \u00de \u00bc V0\u2212V tarrive timates are within previous estimated ranges of 2\u20137 days for incubation V 0 V 0 periods (Li et al., 2020; Lauer et al., 2020; Guan et al., 2020) and consistent with the updated guideline from CDC where the average infectious dura- where the last equality follows from Eq. (6). We provide estimates of the tion is about 2 days before and 8 days after symptom onset (CDC, 2022a). die-off fraction under various scenarios in Table S3 (Supplementary Mate- Thus, we fix the average exposed duration to 3 days, which is equivalent rial). Note that \u03b3(T) varies with temperature over the course of fitting and to fixing k \u00bc 1 forecasting. 3 per day (Fig. 1A). Similarly, we fix the average infectious 3", + "postprocess_score": 0.9296919107, + "detect_cls": "Reference text", + "postprocess_cls": "Reference text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 88, + 1348, + 566 + ], + "detect_score": 6.1171679497, + "content": "T. Phan et al. Science of the Total Environment 857 (2023) 159326 A B ) ) g g r r Average viral shedding rate 8 8 e e p p S E I R s s e e i i p p 6 6 o o c c , , 0 0 1 1 I g g 4 4 o o l l ( ( g g 8 d n n i i d d 2 2 d d e e E R h h Fecal shedding s s l l 3 d Model shedding a a r r 0 0 i i V V 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Days post infection Days post infection", + "postprocess_score": 0.9999575615, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 603, + 1348, + 736 + ], + "detect_score": 4.9207391739, + "content": "Fig. 1. Illustration and fitting fecal viral shedding dynamics. (A) Illustration of the fecal viral shedding dynamics based on the infection progression. The viral shedding profile is divided into three periods shaded: Exposed (E), Infectious (I), and Recovered (R). The red-shaded region is the period of infectiousness I, which is corresponding to the compartment I in the SEIR model. (B) Fitting of the proposed viral shedding function to viral shedding in hospitalized patients' stool data from (Wolfel et al. 2020). While aggregated data seem to show a viral shedding peak at around day 13\u201314, declining trends were found in the 9 individual cases. The average viral shedding rate in stool during the infectious period (from day 3 to day 11) is 4.49 \u00d7 107 viral RNA per g. The horizontal dashed line is the average fecal viral shedding rate for infectious individuals inferred from the model. The viral shedding peak is set at the 4th day post infection.", + "postprocess_score": 0.9999700785, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 90, + 790, + 215, + 808 + ], + "detect_score": 1.4086709023, + "content": "2.7. Data fitting", + "postprocess_score": 0.9999818802, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 90, + 840, + 700, + 1036 + ], + "detect_score": -0.0020545702, + "content": "Our goal is to fit the SEIR-V model to viral concentration in wastewater data to infer the true number of cases. Then, we compare the predicted number of cases with the daily reported case data. In our model, the vari- able V is the cumulative viral load in wastewater. Thus, the difference of V in every 24-hour period reflects the daily measurement data of total virus concentration in wastewater. To reflect this observation, we aim to minimize the sum of square error (SSEV) between these two quantities in our fitting. Hence, our minimization objective is: Z 2 ", + "postprocess_score": 0.9999901056, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 90, + 1072, + 697, + 1124 + ], + "detect_score": -1.1369520426, + "content": " Z 2 td 0 \u00de : log V s\u00f0 \u00deds \u00f08\u00de \u2212 log ^V td\u00f0 SSEV \u00bc \u2211td td\u22121 Here,bV td\u00f0", + "postprocess_score": 0.9986017346, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 1163, + 699, + 1287 + ], + "detect_score": -3.2176587582, + "content": "\u00de is the total virus concentration experimentally measured on day Here,bV td\u00f0 td, which equals to viral RNA concentration in wastewater (CRNA) multi- tdV\u2032(s)ds is the corresponding quantity plied by the total flow (F) data. \u222btd\u22121 in our model. Once we obtain a reasonable fit to the data, the inferred num- ber of true cases is given by:", + "postprocess_score": 0.9998463392, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 90, + 1323, + 698, + 1342 + ], + "detect_score": -1.6057292223, + "content": "\u00f0 \u00de, (9) Daily case number \u00bc Ccumulative cases td\u00f0 \u00de Ccumulative cases td 1", + "postprocess_score": 0.8665263653, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 1381, + 699, + 1503 + ], + "detect_score": -2.8548688889, + "content": "where Ccumulative cases(t) is a variable that keeps track of the cumulative in- fected cases, e.g., Ccumulative cases\u2032 = \u03bbIS. For the minimization algorithm, we use MATLAB function fmincon and multistart. Similarly, the fecal viral shedding function is fitted by minimiz- ing the objective function SSEf: 2 tn\u00f0 \u00de bf", + "postprocess_score": 0.9999229908, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 90, + 1537, + 698, + 1575 + ], + "detect_score": -5.2137589455, + "content": " 2 f tn\u00f0 \u00de bf tn\u00f0 \u00de , (10) SSEf \u00bc \u2211tn wherebf tn\u00f0", + "postprocess_score": 0.9907886386, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 91, + 1614, + 699, + 1795 + ], + "detect_score": 0.1121105999, + "content": "wherebf tn\u00f0 \u00de is the fecal shedding data at day tn. Note that, we assume re- ported data represents a single time point, which is equivalent to assuming the viral shedding is approximately constant over the course of one day. A more technical approach would be to integrate f(t) similar to Eq. (8), then \u00de. Instead, here we pass the integration to average it to compare withbf tn\u00f0 the average stool shed per day (\u03b1), and the average viral shedding over one day is given by (\u03b1 \u00d7 \u03b2) during the infectious period.", + "postprocess_score": 0.9999229908, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 741, + 789, + 818, + 804 + ], + "detect_score": -4.2351064682, + "content": "3. Result", + "postprocess_score": 0.9965554476, + "detect_cls": "Equation label", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 740, + 840, + 1300, + 859 + ], + "detect_score": -2.98132658, + "content": "3.1. Determining the average fecal viral shedding rate in infectious period", + "postprocess_score": 0.9609001279, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 738, + 891, + 1350, + 1591 + ], + "detect_score": -1.2788931131, + "content": "We observed that there is a striking similarity in the viral load profiles for the nose, throat, and stool for infected individuals from the time of infec- tion to recovery qualitatively (W\u00f6lfel et al., 2020, Killingley et al., 2022, Van Kampen et al., 2021). In all three cases, high viral load/shedding is as- sociated with the infectious duration of the infection. This observation sug- gests that in the classical SEIR epidemic model, we can make the simplifying assumption that the infectious individuals contribute substan- tially to the viral pools in wastewater. As illustrated in Fig. 1A, the viral shedding profile is divided into three periods shaded: Exposed (E), Infec- tious (I), and Recovered (R). With this framework, we can approximate the viral load in wastewater using the viral shedding from the infectious population. Furthermore, we can estimate the average viral shedding rate based on the viral shedding function f(t) and the fixed average duration of infectiousness (see Materials and Methods). Fig. 1B shows the best fit of the model to the fecal viral shedding rate data in W\u00f6lfel et al. (W\u00f6lfel et al., 2020). We assumed five days from infec- tion to symptom onset in the fecal viral shedding data, which is in range of 2\u201314 days estimated for the general population (CDC, 2022b; Lauer et al., 2020). Furthermore, we fixed the viral peak at day four (\u03c92 = 4 day). There is no well-established timing of the peak fecal viral shedding rate; however, the peak time for viral load in nose and throat is around 4.7 and 6.2 days after inoculation, respectively (Killingley et al., 2022), and maybe even earlier in stool (Wu et al., 2022a). The best fit parameter is \u03c91 = 71.97 log10 viral RNA copy per g day. Using the best fit, we estimate the average fecal viral shedding rate for an infectious individual to be: Z 11 Z 11 71:97t f \u03b2 \u00bc 1 t\u00f0 \u00dedt \u00bc 1 11 3 8 16 \u00fe t2 dt\u22487:65 log 10viral RNA per g (11) 3 3", + "postprocess_score": 0.9999816418, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 738, + 1625, + 1347, + 1691 + ], + "detect_score": -5.7005910873, + "content": "A conversion gives: (12) \u03b2 \u00bc 4:49 107 viral RNA per g:", + "postprocess_score": 0.9223031402, + "detect_cls": "Other", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 741, + 1722, + 1350, + 1795 + ], + "detect_score": -1.796698451, + "content": "This number is close to the measured median viral RNA load 107.68 (ranging from 104.1 to 1010.27) copies/ml in infected individuals in South Korea (Han et al., 2020), and the extrapolated fecal shedding rate of", + "postprocess_score": 0.9999842644, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 716, + 1830, + 723, + 1841 + ], + "detect_score": -4.4765014648, + "content": "4", + "postprocess_score": 0.999961853, + "detect_cls": "Section Header", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 91, + 88, + 172, + 99 + ], + "detect_score": -2.0568401814, + "content": "T. Phan et al.", + "postprocess_score": 0.9883300066, + "detect_cls": "Reference text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 91, + 128, + 698, + 201 + ], + "detect_score": -4.4418640137, + "content": "107.30 (ranging from 105.74 \u2212 108.28) copies/g of 711 infected individuals in the dormitories at University of Arizona (Schmitz et al., 2021). Thus, we fixed fecal viral shedding rate \u03b2 in our SEIR-V model to this value.", + "postprocess_score": 0.9962953925, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 90, + 233, + 693, + 252 + ], + "detect_score": -0.2949813008, + "content": "3.2. SEIR-V model captures the temporal dynamics of clinical COVID-19 cases", + "postprocess_score": 0.9982734919, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 90, + 283, + 699, + 1010 + ], + "detect_score": 3.6132590771, + "content": "We developed an SEIR-V model to understand SARS-CoV-2 transmis- sion using WBS data in the second wave of the pandemic and the computed average fecal viral shedding rate during the period of infectiousness. Fig. 2 shows the best fit and its inference with parameter values and possible ranges summarized in Table 1. We fitted the model to total viral RNA copies in wastewater data up to the grey dashed line (December 18, 2020), then simulated the model out to January 25, 2021, see Fig. 2A. The fitting region was chosen before the peak in the viral RNA data, so that we could test the model's prediction of the peak against the data. Additionally, the fitting re- gion from October 02, 2020 to December 18, 2020 potentially limits the in- fluence from vaccination and the emergence of the alpha variant, which began near the end of 2020. Using the best fit parameters, we computed the number of new cases and compared it to the reported cases. As shown in Fig. 2B, the model sim- ulation recapitulates the trend of clinically reported daily new cases and predicts an earlier and higher peak than reported case data by 16 days and 10.2-fold, respectively. We made a correlation plot between the model simulated cases and the reported case data (Fig. 2C). The higher pre- dicted number of cases and the high correlation coefficient (R = 0.93, R2 = 0.87) imply that the model accurately captures the trend of the reported case data, while accounting for the underreported rate. This indicates that the method preserves both key properties of WBS data, which is that the trend of viral concentration in wastewater leads the trend of reported cases and can be used to estimate the true prevalence without being im- pacted by the underreporting rate. In the next step, we demonstrate how the effect of temporal variation in temperature on viral loss can be incorporated in our framework. Further- more, by incorporating the temporal effect of temperature, we can directly estimate the variation in \u03b3(T), the fraction of viral loss in the sewershed.", + "postprocess_score": 0.9998784065, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 137, + 1056, + 511, + 1612 + ], + "detect_score": -4.3547387123, + "content": "A WW data Model ) 15.2 s e i p o c e m o 14.8 n e g , 0 1 g o l 14.4 ( d a o l l a r i v 14.0 l a t fitting prediction o T SSE = 2.19 13.6 Nov Dec Jan Oct Date (2020-2021)", + "postprocess_score": 0.8509461284, + "detect_cls": "Body Text", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 286, + 1630, + 444, + 1650 + ], + "detect_score": -2.6574094296, + "content": "Date (2020-2021)", + "postprocess_score": 0.5573527217, + "detect_cls": "Body Text", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 1015, + 88, + 1348, + 103 + ], + "detect_score": -4.5569787025, + "content": "Science of the Total Environment 857 (2023) 159326", + "postprocess_score": 0.9996827841, + "detect_cls": "Section Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 983, + 1630, + 1272, + 1649 + ], + "detect_score": -3.472931385, + "content": "Reported daily COVID-19 cases", + "postprocess_score": 0.6684665084, + "detect_cls": "Equation label", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 538, + 1056, + 896, + 1650 + ], + "detect_score": -3.2347295284, + "content": "B 4.4 Model Reported ) 0 1 4.0 g o l ( 9 1 - D I 3.6 V O C f o e c n 3.2 e d i . c n i y l i a 2.8 D \u0394Tlead = 16 days 2.4 Nov Dec Jan Oct Date (2020-2021)", + "postprocess_score": 0.999953866, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 922, + 1055, + 1302, + 1613 + ], + "detect_score": -3.5226433277, + "content": "C x104 2.5 s e s a c 2 9 1 - D I V O C 1.5 y l i a d d e t c i 1 d e r p - l e d o 0.5 M r = 0.93 0 0 500 1500 2500 Reported daily COVID-19 cases", + "postprocess_score": 0.9999471903, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 740, + 182, + 1350, + 1011 + ], + "detect_score": 2.806425333, + "content": "SARS-CoV-2 RNA in wastewater is subject to degradation which is af- fected by many factors such as temperature and travel time (Bivins et al., 2020; McCall et al., 2022). We accounted for these two factors to determine the fraction of viral decay \u03b3(T) in the model. Sensitivity analyses were per- formed to investigate how the two parameters impact model fitting. First, we tested SARS-CoV-2 degradation rates at T0 = 20 \u00b0C wastewater at high titers (\u03b70 = 0.99 days or 23.76 hours) and low titers (\u03b70 = 7.9 days or 189.6 hours). Results showed a consistently better fit using viral degra- dation rate at low titers (Table S1). Precise estimation of the travel time is challenging given the varied flow rates and geographical distances to the wastewater treatment plant. Here, we assumed the average travel time is 18 h based on the professional experience from the treatment plant where we sampled. We also tested the sensitivity of this value by assuming 24-, 30- and 36-h travelling time and found little differences in the modeling fitting (Table S1). Viral concentration in wastewater is typically low, so we used the degradation rate at low titers and compared the die-off fraction for different temperature and travel time. Results in Table S3 showed that the viral die-off fraction differs about 5\u20136 fold from 10 to 30 \u00b0C. Next, we incorporated the temporal-varying temperature data (Fig. S1) into the model framework with a travel time 18-hour and assessed model performance. By incorporating the effect of the temporal variations in tem- perature, the viral degradation rate also varies with temperature and time (see Materials and Methods). This temporal variation allows the model to capture the trend of clinical data with a smaller SSE compared to a constant degradation rate. Thus, it demonstrates the importance of incorporating temperature in our modeling framework. Additionally, the difference is sta- tistically significant, due to one less fitting parameter, based on the corrected Akaike information criterion (Fig. 3A, B and S2) (Burnham and Anderson, 2004). We observe that the model simulation predicts an earlier peak than reported case data by 6 days, which is 10 days shorter compared to the prediction of the model without the temporal temperature effect (Fig. 3B and S2A). Additionally, the model predicts the true number of cases to be about 8.3 times higher than the reported number of cases as", + "postprocess_score": 0.9999933243, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 740, + 132, + 1294, + 150 + ], + "detect_score": 2.8490505219, + "content": "3.3. Incorporation of wastewater temperature improves model prediction", + "postprocess_score": 0.9982050657, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 90, + 1685, + 1349, + 1795 + ], + "detect_score": 3.755320549, + "content": "Fig. 2. Model fit and prediction to wastewater data covering the second wave of pandemic. (A) Best fit to virus concentration data in wastewater from October 2 to December 18, 2020 (dashed grey line), and model prediction to January 25, 2021. Red dots are the measured viral load in wastewater and blue curve is the modeling result. (B) Model estimation of the true number of COVID-19 cases (blue curve) and clinically reported cases (red curve). The blue and red dashed lines are dates when the two curves peak, and \u0394Tlead is the time difference between the two peaks. (C) Correlation between simulation cases and reported cases. Best fit parameters: \u03bb = 9.66 \u00d7 10\u22128 day\u22121 person\u22121, \u03b1 = 249 g, \u03b3 = 0.08, and E(0) = 11 people.", + "postprocess_score": 0.9999878407, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 716, + 1831, + 723, + 1841 + ], + "detect_score": -4.3222708702, + "content": "5", + "postprocess_score": 0.9999787807, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 88, + 172, + 99 + ], + "detect_score": -5.912352562, + "content": "T. Phan et al.", + "postprocess_score": 0.9993900061, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 1015, + 88, + 1348, + 103 + ], + "detect_score": -4.8945999146, + "content": "Science of the Total Environment 857 (2023) 159326", + "postprocess_score": 0.9999288321, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 90, + 133, + 1349, + 476 + ], + "detect_score": -1.2362197638, + "content": "Table 1 Parameters in the model. Definition Unit Value References S Susceptible population People S(0) = 2.3 \u00d7 106 - fixed (Wu et al., 2022b) E Exposed population People E(0)\u2212fitting I Infectious population People I(0) = 0 - fixed \u03bb Transmission rate Per day per person Fitting 1/k Exposed duration Day 3 days W\u00f6lfel et al., 2020; Killingley et al., 2022; Wu et al., 2022a; Van Kampen et al., 2021; 1/\u03b4 Infectious duration Day 8 days W\u00f6lfel et al., 2020, Killingley et al., 2022, Wu et al., 2022a; Van Kampen et al., 2021 \u03b1 Fecal load Gram 51\u2013796 g - fitting Rose et al., 2015 \u03b2 Viral shedding in stool Viral RNA copies per gram Fitting \u03b3 Fraction of viral loss in sewer Per day Fitting and estimated Fitting \u03c91 Magnitude modifier log10 viral RNA per g day Peak timing for viral shedding Day 4 day - fixed Killingley et al., 2022; Wu et al., 2022a. \u03c92 Note that \u03b2 and \u03c91 are obtained from fitting to viral shedding data in stool (W\u00f6lfel et al., 2020).", + "postprocess_score": 0.9998477697, + "detect_cls": "Table", + "postprocess_cls": "Table" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 511, + 699, + 705 + ], + "detect_score": -1.8251239061, + "content": "compared to a predicted factor of 10.2 without the temporal effect of tem- perature (Fig. 3B and S2A). The predicted initial exposed population is 1182 people, which is a more reasonable estimate compared to the 11 ex- posed individuals predicted without the temporal effect of temperature (Fig. 3B). Those results have shown that incorporating the travel time and the temporal variation in temperature reduces the possibility of model unidentifiability and significantly improve the model performance (Table 1).", + "postprocess_score": 0.9999850988, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 739, + 206, + 753 + ], + "detect_score": -3.3792192936, + "content": "4. Discussion", + "postprocess_score": 0.9998637438, + "detect_cls": "Page Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 790, + 698, + 909 + ], + "detect_score": -1.666852355, + "content": "Wastewater pools viral signals excreted by infected individuals across the whole spectrum of disease symptoms from asymptomatic and subclinical-symptomatic to symptomatic (Lee et al., 2020). This inclusive- ness of all virus-shedding individuals offers an opportunity to better esti- mate the magnitude of viral infections in communities (Hart and Halden,", + "postprocess_score": 0.9999761581, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 174, + 939, + 201, + 968 + ], + "detect_score": -5.3732018471, + "content": "A", + "postprocess_score": 0.3679983318, + "detect_cls": "Page Footer", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 174, + 987, + 568, + 1573 + ], + "detect_score": -3.0665364265, + "content": "15.5 ) s e i p o c e m o 15.0 n e g , 0 1 g o l ( d a 14.5 o l l a r i v l a t o T fitting prediction 14.0 SSE = 2.00 Nov Dec Jan Date (2020-2021)", + "postprocess_score": 0.9999325275, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 609, + 1262, + 1267, + 1541 + ], + "detect_score": -2.6036937237, + "content": ") 18 s y Predicted n \u0394Tlead a 10 o k i d t ( a Reported a e k 12 p m a i t t e s a p e t r d 9 l a e o 6 f d e n m i U t g 8 0 a L Without temperature With temperature", + "postprocess_score": 0.9996494055, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 609, + 988, + 1266, + 1235 + ], + "detect_score": -3.7522037029, + "content": "-276 AICC ) 3 E(0) o t 0 C I 1 e r g -272 A u o l d 2 s ( e o t 9 p c 1 x e - -268 r e r D l 1 I o a i V t C i o n I C -264 0", + "postprocess_score": 0.9982543588, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 609, + 939, + 631, + 968 + ], + "detect_score": -3.213070631, + "content": "B", + "postprocess_score": 0.4231648743, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 740, + 511, + 1348, + 909 + ], + "detect_score": -3.4156730175, + "content": "2020; Sanju\u00e1n and Domingo-Calap, 2021; Wu et al., 2020). However, it is challenging to convert viral concentrations in wastewater to the number of infected cases. Our group and peers previously reported methods to esti- mate the infection prevalence by wastewater viral load (McMahan et al., 2021; Nourbakhsh et al., 2022; Wu et al., 2020). These efforts, however, are limited because of inconsideration of dynamic viral shedding rates dur- ing the disease course and viral degradation in wastewater. In this study, we established a quantitative framework to estimate the number of infectious COVID-19 cases and predict SARS-CoV-2 transmission through integrating wastewater surveillance data and development of an SEIR-V model. As an analogy to the four compartments of the SEIR model to simulate infectious disease dynamics at the population level, the individual-level fecal viral shedding course was divided into three periods including exposed (incubation), infectious, and recovery (Fig. 1A). The di- vision is based on the observation that the temporal viral profiles in the nose, mouth, and stool are strikingly similar qualitatively with high viral", + "postprocess_score": 0.9995493293, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 91, + 1610, + 1350, + 1795 + ], + "detect_score": 2.3391828537, + "content": "Fig. 3. Incorporating the effect of temporal variation of wastewater temperature in the SEIR-V model. (A) Best fit to viral concentration data in wastewater from October 2 to December 18, 2020 (dashed grey line), and model prediction to January 25, 2021. Red dots are the measured viral load in wastewater and blue curve is the modeling result. (B) Comparison of the SEIR-V models with and without incorporating temperature effect. Top left: corrected Akaike information criterion (AICc) values, the statistically significant AICc difference is 9.2; Top right: initial populations exposed to SARS-CoV-2; Bottom left: wastewater lead time difference at peak; Bottom right: fold of difference between the number of predicted cases and clinically reported cases. The AIC/AICc are calculated assuming normal distribution of residuals with mean zero \u00f0 \u00de using the formulas AIC \u00bc n log sse \u00fe 2k and AICc \u00bc AIC \u00fe 2k k\u00fe1 n n k 1, where n is the number of data used for fitting, k is the number of fitting parameter and the SSE is calculated based on the data used for fitting. Light blue represents the model without including temperate effect, while blue represents the model with temperature effect. Best fit parameters when incorporating temperature: \u03bb = 9.06 \u00d7 10\u22128 day\u22121 person\u22121, \u03b1 = 360 g,and E(0) = 1182 people.", + "postprocess_score": 0.9999763966, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 716, + 1830, + 723, + 1841 + ], + "detect_score": -4.9381217957, + "content": "6", + "postprocess_score": 0.9999630451, + "detect_cls": "Body Text", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 91, + 88, + 172, + 99 + ], + "detect_score": -4.3614678383, + "content": "T. Phan et al.", + "postprocess_score": 0.9997631907, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 90, + 132, + 701, + 1791 + ], + "detect_score": 2.9589669704, + "content": "load associated with infectiousness (Killingley et al., 2022; W\u00f6lfel et al., 2020). With this concept, we estimated the population-level average viral shedding rate during the infectious phase using clinically reported SARS- CoV-2 concentrations in hospitalized patients' stool samples (Fig. 1B). This estimated viral shedding rate is an average of infected individuals in the population and does not consider the heterogenous viral shedding dy- namics among infected individuals (W\u00f6lfel et al., 2020; Killingley et al., 2022; Stanca and Tuncer et al., 2022). Thus, our model can be improved by including viral shedding data during the early phase of the infection and large-scale individual-level shedding dynamics data. It is noteworthy to mention that the \"I\" in the SEIR model is the \"infec- tious\" class, not the \"infected\" class. This contrasts with the conventional ap- proaches that use mean or median viral shedding rate in a group of tested samples regardless of the phase of the infection (Saththasivam et al., 2021; Petala et al., 2022; Schmitz et al., 2021). By focusing on the infectious popula- tion, which is also the main contributor of viral shedding in wastewater, we greatly simplify the typical complex structure of the SEIR-type models that im- plement WBS (Fig. S3) and reduce the likelihood of model unidentifiability. By fitting an SEIR-V model to wastewater data within our framework, we show that the method retains key advantages of using wastewater. Spe- cifically, the inferred case data from the best fit parameters leads the re- ported case data by 6\u201316 days and implies a large ratio (8.3\u201310.2) of true prevalence to clinically reported cases, which are consistent with previous results (Wu et al., 2020; Wu et al., 2022a; Eikenberry et al., 2020; Angulo et al., 2021). We also incorporate the important effects of temperature with temporal variations and travel time on the viral degradation rate in a simple manner that is applicable to a larger time scale. We note that exten- sion to incorporate time-dependent variations of the fecal viral shedding rate within this framework is straightforward but will require careful con- sideration for the convergence of the numerical method. Together, our work shows the potential and flexibility of the framework to incorporate WBS in epidemic models. The foundation of our framework is independent of the epidemic model formulation, yet its application depends greatly on the epidemic models for specific situations. For example, if we want to apply the framework to cap- ture a period with significant changes to social behavior, perhaps due to the effect of a social intervention, then an appropriate change to the structure of the SEIR model to reflect these structures is necessary (Johnston and Pell, 2020; Fenichel et al., 2011; Pell et al., 2018). However, if multiple variants are of interest, then the SEIR model itself needs to be extended to a multi- variant version and incorporate known biological properties of different variants (Dyson et al., 2021; Gonzalez-Parra et al., 2021). Similarly, inter- ventions (such as vaccination) and the impact of social gatherings must first be included in the epidemic model prior to its integration within our framework (Saad-Roy et al., 2021; Giordano et al., 2021; Buckner et al., 2021; Makhoul et al., 2020). Our modeling framework represents a simplified picture that describes the connection between viral transmission in the human population and viral concentration in wastewater. For this demonstrative purpose, we make various simplifying assumptions that would need to be adjusted for application of the framework to specific situations. Firstly, the SEIR model assumes a completely susceptible, homogenous, and well-mixed population. In practice, specific contacting/population structure of the re- gion being studied must be taken into account to provide accurate estimates of relevant epidemiological quantities, such as the basic or effective repro- duction number. Secondly, we assume that the viral concentration in waste- water is contributed mainly by the infectious group. A direct calculation of the respective viral contribution from each group, assuming a cut-off threshold of 2 log using the best estimate of the viral shedding function f (t) gives 12 % , 52% and 36% relative contribution from E, I and R classes, respectively. Taking this into account would roughly reduce the fold- difference between predicted and reported cases by about 50 %. A simple approach would be to consider the contribution from the E and R classes similar to that of the I class (e.g., by finding the average viral shedding rate \u03b2 for each class). Alternatively, a convolution of f(t) and everyone who has ever been infected can also be done to account for the ever-", + "postprocess_score": 0.9999821186, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 740, + 132, + 1349, + 732 + ], + "detect_score": -1.6178803444, + "content": "changing viral shedding rate. However, such a model would need to recon- cile the differences between the temporal variations in the fecal viral shed- ding and infectiousness. This can be done by studying the relationship between viral dynamics in the stool and the nose or throat. Finally, while we only consider the effects of temperature and travel time on viral decay rate, other in-sewer factors, such as organic matter, particles, pH, solvents, detergents, and microbes could also affect the viral die-off fractions and should be considered when appropriate (Bertels et al., 2022; Chahal et al., 2016; Gundy et al., 2009). Dynamical epidemic models are useful tools to track pandemic progres- sion and to assess the potential impact of hypothetical situations such as stay-at-home orders or the emergence of a resistant viral strain. However, sparsely reported case data with high uncertainty, due partially to the high underreporting rate, can compromise the ability of epidemic models to provide an accurate forecast of the pandemic and limit their application to retrospective studies. Hence, WBS, which bypasses both the tremendous difficulty in data collection faced by the standard clinical reporting practice and the high underreporting rate, represents a potential solution to address this challenge faced by the modeling community. WBS data also provides a leading indicator of the pandemic progression and is not limited to SARS- CoV-2, thus it can further enhance the prediction and applicability of epi- demic models for public health purposes. Together, this aspect of our framework highlights the importance of interdisciplinary collaboration to better address public health concerns.", + "postprocess_score": 0.9999877214, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 741, + 783, + 869, + 797 + ], + "detect_score": -4.6811404228, + "content": "5. Conclusions", + "postprocess_score": 0.9999186993, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 740, + 834, + 1348, + 903 + ], + "detect_score": -4.1127729416, + "content": "In this study, we have established a quantitative framework to estimate COVID-19 prevalence and predict SARS-CoV-2 transmission by incorporating WBS data in a simple epidemic SEIR-V model. The main conclusions are:", + "postprocess_score": 0.9998942614, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 741, + 923, + 1348, + 1220 + ], + "detect_score": -2.6458160877, + "content": "\u2022 We constructed a simple and effective framework to incorporate WBS data to epidemic models. The developed SEIR-V model captures the tem- poral dynamics of clinical COVID-19 cases and preserves key advantages of WBS data over reported case data. \u2022 We illustrated how the effect of travel time and temperature on viral decay can be incorporated within our framework to improve model per- formance and robustness, which is an important component to model dis- ease transmission in real world applications. \u2022 The modeling framework is a valuable platform to integrate WBS with ep- idemic models to provide accurate and robust estimates of the pandemic progression and examine the potential impact of interventions to inform public health decision making.", + "postprocess_score": 0.9989010096, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 741, + 1292, + 1115, + 1311 + ], + "detect_score": -1.5060588121, + "content": "CRediT authorship contribution statement", + "postprocess_score": 0.9968529344, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 740, + 1343, + 1348, + 1690 + ], + "detect_score": -3.6606755257, + "content": "Tin Phan: Conceptualization; Methodology; Formal analysis; Writing - Original Draft; Visualization; Writing - review & editing; Samantha Brozak: Conceptualization; Methodology; Formal analysis; Writing - review & editing; Bruce Pell: Conceptualization; Methodology; Formal analysis; Writing - review & editing; Anna Gitter: Discussion; Writing - review & editing. Amy Xiao: Discussion; Writing - review & editing. Kristina D. Mena: Discussion; Writing - review & editing. Yang Kuang: Conceptualization; Methodology; Funding acquisition; Formal analysis; Writing - review & editing; Supervision. Fuqing Wu: Conceptualization; Methodology; Funding acquisition; For- mal analysis; Visualization; Writing - Original Draft; Writing - review & editing; Supervision.", + "postprocess_score": 0.9992858768, + "detect_cls": "Section Header", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 740, + 1726, + 887, + 1745 + ], + "detect_score": -4.266954422, + "content": "Data availability", + "postprocess_score": 0.953463614, + "detect_cls": "Figure", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 769, + 1776, + 1289, + 1795 + ], + "detect_score": -3.1315531731, + "content": "The data and code are provided in the supplementary materials", + "postprocess_score": 0.6431524158, + "detect_cls": "Page Header", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 716, + 1831, + 723, + 1841 + ], + "detect_score": -3.8575510979, + "content": "7", + "postprocess_score": 0.9998402596, + "detect_cls": "Figure", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 1015, + 88, + 1348, + 103 + ], + "detect_score": -4.3236327171, + "content": "Science of the Total Environment 857 (2023) 159326", + "postprocess_score": 0.9973967075, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 88, + 172, + 99 + ], + "detect_score": -4.5096955299, + "content": "T. Phan et al.", + "postprocess_score": 0.9962707758, + "detect_cls": "Reference text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 132, + 387, + 150 + ], + "detect_score": -2.1095561981, + "content": "Declaration of competing interest", + "postprocess_score": 0.9763538837, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 90, + 182, + 698, + 251 + ], + "detect_score": -1.8652979136, + "content": "The authors declare that they have no known competing financial inter- ests or personal relationships that could have appeared to influence the work reported in this paper.", + "postprocess_score": 0.9999423027, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 90, + 283, + 255, + 302 + ], + "detect_score": -1.675317049, + "content": "Acknowledgement", + "postprocess_score": 0.9998850822, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 90, + 334, + 700, + 504 + ], + "detect_score": 0.3789759576, + "content": "This work is supported by Faculty Startup funding from the Center of Infectious Diseases at UTHealth, the UT system Rising STARs award, and the Texas Epidemic Public Health Institute (TEPHI) to F.W. This work was also supported by Director's postdoctoral fellowship at Los Alamos National Laboratory to T.P.; Y.K. and S.B. are partially supported by the US National Science Foundation Rules of Life program DEB -1930728 and the NIH grant 5R01GM131405-02.", + "postprocess_score": 0.9999879599, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 90, + 537, + 382, + 555 + ], + "detect_score": 0.3091455996, + "content": "Appendix A. Supplementary data", + "postprocess_score": 0.999886632, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 587, + 697, + 631 + ], + "detect_score": -1.4228743315, + "content": "Supplementary data to this article can be found online at https://doi. org/10.1016/j.scitotenv.2022.159326.", + "postprocess_score": 0.999725163, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 91, + 677, + 186, + 692 + ], + "detect_score": -3.0792274475, + "content": "References", + "postprocess_score": 0.9992426634, + "detect_cls": "Page Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 90, + 731, + 700, + 1787 + ], + "detect_score": 5.5025992393, + "content": "Angulo, Frederick J., Finelli, Lyn, Swerdlow, David L., 2021. 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Identifiability and estimation of multiple transmission path- ways in cholera and waterborne disease. J. Theor. Biol. 324, 84\u2013102. Fenichel, Eli P., et al., 2011. Adaptive human behavior in epidemiological models. Proc. Natl. Acad. Sci. 108 (15), 6306\u20136311. Ferretti, Luca, et al., 2020. Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing. Science 368 (6491), eabb6936. Giordano, Giulia, et al., 2021. Modeling vaccination rollouts, SARS-CoV-2 variants and the re- quirement for non-pharmaceutical interventions in Italy. Nat. Med. 27 (6), 993\u2013998. Gonzalez-Parra, Gilberto, Mart\u00ednez-Rodr\u00edguez, David, Villanueva-Mic\u00f3, Rafael J., 2021. Im- pact of a new SARS-CoV-2 variant on the population: a mathematical modeling approach. Math.Comput.Appl. 26 (2), 25. Guan, Wei-Jie, et al., 2020. Clinical characteristics of coronavirus disease 2019 in China. N. Engl. J. Med. 382 (18), 1708\u20131720. 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Water Res. 118070.", + "postprocess_score": 0.9999731779, + "detect_cls": "Reference text", + "postprocess_cls": "Reference text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 716, + 1830, + 723, + 1841 + ], + "detect_score": -5.099729538, + "content": "8", + "postprocess_score": 0.9980364442, + "detect_cls": "Figure Caption", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 1015, + 88, + 1348, + 103 + ], + "detect_score": -5.3176832199, + "content": "Science of the Total Environment 857 (2023) 159326", + "postprocess_score": 0.9999252558, + "detect_cls": "Reference text", + "postprocess_cls": "Page Header" + } + ], + "tds_status_code": 200 + }, + "job_error": null + }, + "time": 4.052867650985718, + "accuracy": null, + "success": true + }, + "variable_extraction": { + "id": "extraction-01703257-02ee-463c-b1cc-148d75480fcf", + "status": "finished", + "result": { + "created_at": 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"enqueued_at": "2023-10-17T19:40:29.498784", + "started_at": "2023-10-17T19:40:29.505565", + "job_result": null, + "job_error": "Traceback (most recent call last):\n File \"/usr/local/lib/python3.10/site-packages/rq/worker.py\", line 1428, in perform_job\n rv = job.perform()\n File \"/usr/local/lib/python3.10/site-packages/rq/job.py\", line 1278, in perform\n self._result = self._execute()\n File \"/usr/local/lib/python3.10/site-packages/rq/job.py\", line 1315, in _execute\n result = self.func(*self.args, **self.kwargs)\n File \"/./worker/operations.py\", line 739, in code_to_amr\n raise Exception(f\"Code extraction failure: {amr_response.text}\")\nException: Code extraction failure: {\"error\":\"MORAE PUT /models/PN failed to process 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19:40:30.463587\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#hours\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":315,\"line_end\":315,\"col_begin\":4,\"col_end\":9,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":315,\"line_end\":315,\"col_begin\":11,\"col_end\":16,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.463631\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#hours\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":315,\"line_end\":315,\"col_begin\":11,\"col_end\":16,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":315,\"line_end\":315,\"col_begin\":18,\"col_end\":22,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.463672\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#hours\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":315,\"line_end\":315,\"col_begin\":18,\"col_end\":22,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":315,\"line_end\":315,\"col_begin\":24,\"col_end\":26,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.463711\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#hours\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":315,\"line_end\":315,\"col_begin\":24,\"col_end\":26,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.463836\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":316,\"line_end\":316,\"col_begin\":17,\"col_end\":19,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.463818\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#use first time point\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":316,\"line_end\":316,\"col_begin\":17,\"col_end\":19,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":316,\"line_end\":316,\"col_begin\":4,\"col_end\":19,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.463767\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#use first time 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19:40:30.467835\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":327,\"line_end\":327,\"col_begin\":4,\"col_end\":28,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.467494\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468420\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":329,\"line_end\":329,\"col_begin\":11,\"col_end\":12,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468402\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":329,\"line_end\":329,\"col_begin\":4,\"col_end\":12,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468348\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"List\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468638\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":330,\"line_end\":330,\"col_begin\":11,\"col_end\":37,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468618\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#print(f'ICs: {ICs}')\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":330,\"line_end\":330,\"col_begin\":11,\"col_end\":37,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":330,\"line_end\":330,\"col_begin\":4,\"col_end\":37,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468555\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#print(f'ICs: {ICs}')\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":330,\"line_end\":330,\"col_begin\":4,\"col_end\":37,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":334,\"line_end\":334,\"col_begin\":14,\"col_end\":66,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.468779\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#err = np.sum(np.log10(results[:, 4]) - np.log10(data**2))\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":334,\"line_end\":334,\"col_begin\":14,\"col_end\":66,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":337,\"line_end\":337,\"col_begin\":17,\"col_end\":24,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.469723\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":337,\"line_end\":337,\"col_begin\":17,\"col_end\":34,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.469596\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":19,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470246\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":19,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470128\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470502\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"List\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470659\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":-1,\"line_end\":-1,\"col_begin\":-1,\"col_end\":-1,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470637\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470502\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":30,\"col_end\":38,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470925\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":30,\"col_end\":38,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":30,\"col_end\":58,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470822\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":30,\"col_end\":58,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":39,\"col_end\":45,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.471442\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":39,\"col_end\":45,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":39,\"col_end\":57,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.471113\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":39,\"col_end\":57,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":58,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.471594\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":11,\"col_end\":58,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":338,\"line_end\":338,\"col_begin\":4,\"col_end\":58,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.470027\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#remove NAs\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":338,\"line_end\":338,\"col_begin\":4,\"col_end\":58,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":340,\"line_end\":340,\"col_begin\":10,\"col_end\":16,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.472096\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":340,\"line_end\":340,\"col_begin\":10,\"col_end\":26,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.471982\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.472257\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":340,\"line_end\":340,\"col_begin\":24,\"col_end\":25,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.472237\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":340,\"line_end\":340,\"col_begin\":17,\"col_end\":25,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.472327\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":357,\"line_end\":357,\"col_begin\":7,\"col_end\":38,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.472715\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":357,\"line_end\":357,\"col_begin\":7,\"col_end\":40,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.472613\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.473252\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":359,\"line_end\":359,\"col_begin\":8,\"col_end\":10,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.473233\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":359,\"line_end\":359,\"col_begin\":0,\"col_end\":10,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.473181\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474198\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":365,\"line_end\":365,\"col_begin\":13,\"col_end\":20,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474180\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474293\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":365,\"line_end\":365,\"col_begin\":21,\"col_end\":23,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474274\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474380\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":365,\"line_end\":365,\"col_begin\":25,\"col_end\":26,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474364\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":365,\"line_end\":365,\"col_begin\":21,\"col_end\":26,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474425\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":365,\"line_end\":365,\"col_begin\":13,\"col_end\":26,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474500\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":365,\"line_end\":365,\"col_begin\":0,\"col_end\":26,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474114\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"List\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474741\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":366,\"line_end\":366,\"col_begin\":5,\"col_end\":25,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474721\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":366,\"line_end\":366,\"col_begin\":0,\"col_end\":25,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474658\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"List\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474942\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":367,\"line_end\":367,\"col_begin\":5,\"col_end\":33,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474924\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":367,\"line_end\":367,\"col_begin\":0,\"col_end\":33,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.474870\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"List\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.475127\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":368,\"line_end\":368,\"col_begin\":5,\"col_end\":40,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.475108\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":368,\"line_end\":368,\"col_begin\":0,\"col_end\":40,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.475060\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":373,\"line_end\":373,\"col_begin\":9,\"col_end\":73,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.475248\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":374,\"line_end\":374,\"col_begin\":0,\"col_end\":21,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.475986\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.476628\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":377,\"line_end\":377,\"col_begin\":13,\"col_end\":15,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.476609\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#use first time point\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":377,\"line_end\":377,\"col_begin\":13,\"col_end\":15,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":377,\"line_end\":377,\"col_begin\":0,\"col_end\":15,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.476557\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#use first time point\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":377,\"line_end\":377,\"col_begin\":0,\"col_end\":15,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":378,\"line_end\":378,\"col_begin\":4,\"col_end\":15,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.476762\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.476916\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":378,\"line_end\":378,\"col_begin\":13,\"col_end\":14,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.476893\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477165\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":6,\"col_end\":7,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477146\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":6,\"col_end\":7,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":10,\"col_end\":16,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477392\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":10,\"col_end\":16,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":10,\"col_end\":31,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477230\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":10,\"col_end\":31,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":17,\"col_end\":19,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477625\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":17,\"col_end\":19,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":17,\"col_end\":30,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477738\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":17,\"col_end\":30,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":6,\"col_end\":31,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477918\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":6,\"col_end\":31,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":379,\"line_end\":379,\"col_begin\":0,\"col_end\":31,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.477081\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"#total population served by DITP\",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":379,\"line_end\":379,\"col_begin\":0,\"col_end\":31,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478234\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":381,\"line_end\":381,\"col_begin\":5,\"col_end\":12,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478217\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":381,\"line_end\":381,\"col_begin\":0,\"col_end\":12,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478158\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":5,\"col_end\":9,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478815\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478954\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":7,\"col_end\":8,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478934\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":5,\"col_end\":9,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478815\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":11,\"col_end\":21,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479125\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479288\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":23,\"col_end\":24,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479268\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":23,\"col_end\":28,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479358\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":11,\"col_end\":29,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479462\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":5,\"col_end\":30,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479549\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":383,\"line_end\":383,\"col_begin\":0,\"col_end\":30,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.478690\"}}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479957\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":384,\"line_end\":384,\"col_begin\":5,\"col_end\":6,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479938\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":384,\"line_end\":384,\"col_begin\":0,\"col_end\":6,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.479880\"}}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":385,\"line_end\":385,\"col_begin\":11,\"col_end\":18,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.480195\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"# use first data point \",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":385,\"line_end\":385,\"col_begin\":11,\"col_end\":18,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":385,\"line_end\":385,\"col_begin\":11,\"col_end\":23,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.480327\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"# use first data point \",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":385,\"line_end\":385,\"col_begin\":11,\"col_end\":23,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":385,\"line_end\":385,\"col_begin\":5,\"col_end\":24,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.480431\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"# use first data point \",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":385,\"line_end\":385,\"col_begin\":5,\"col_end\":24,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":385,\"line_end\":385,\"col_begin\":0,\"col_end\":24,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.480087\"}},{\"gromet_type\":\"SourceCodeComment\",\"comment\":\"# use first data point \",\"comment_type\":\"OTHER\",\"context_function_name\":null,\"code_file_reference_uid\":null,\"line_begin\":385,\"line_end\":385,\"col_begin\":0,\"col_end\":24,\"is_metadatum\":true,\"provenance\":null}],[{\"gromet_type\":\"source_code_data_type\",\"source_language\":\"Python\",\"source_language_version\":\"3.8\",\"data_type\":\"\",\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 19:40:30.481107\"}},{\"gromet_type\":\"SourceCodeReference\",\"code_file_reference_uid\":\"a052b07b-07da-9285-51c8-fb0ebb85ea27\",\"line_begin\":387,\"line_end\":387,\"col_begin\":7,\"col_end\":8,\"is_metadatum\":true,\"provenance\":{\"gromet_type\":\"Provenance\",\"method\":\"skema_code2fn_program_analysis\",\"timestamp\":\"2023-10-17 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These permissions are", + "postprocess_score": 0.7646376491, + "detect_cls": "Page Footer", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 198, + 1261, + 1285, + 1294 + ], + "detect_score": -5.2244668007, + "content": "granted for free by Elsevier for as long as the COVID-19 resource centre", + "postprocess_score": 0.9178978801, + "detect_cls": "Page Footer", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 1, + "bounding_box": [ + 629, + 1320, + 853, + 1346 + ], + "detect_score": -6.2553391457, + "content": "remains active.", + "postprocess_score": 0.9822595119, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 562, + 90, + 902, + 105 + ], + "detect_score": -4.2609267235, + "content": "Chaos, Solitons and Fractals 139 (2020) 110057", + "postprocess_score": 0.9996607304, + "detect_cls": "Other", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 103, + 138, + 1360, + 334 + ], + "detect_score": -3.9039206505, + "content": "Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos", + "postprocess_score": 0.9984810948, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 103, + 406, + 1287, + 477 + ], + "detect_score": -6.4830160141, + "content": "A SIR model assumption for the spread of COVID-19 in different communities", + "postprocess_score": 0.474674046, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 104, + 508, + 750, + 537 + ], + "detect_score": -4.1851797104, + "content": "\u2217 b , b Ian Cooper a , Argha Mondal , Chris G. Antonopoulos", + "postprocess_score": 0.4634104669, + "detect_cls": "Section Header", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 103, + 556, + 657, + 592 + ], + "detect_score": -6.072933197, + "content": "a School of Physics, The University of Sydney, Sydney, Australia b Department of Mathematical Sciences, University of Essex, Wivenhoe Park, UK", + "postprocess_score": 0.5278699398, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 501, + 659, + 658, + 672 + ], + "detect_score": -1.0358011723, + "content": "a b s t r a c t", + "postprocess_score": 0.9985902905, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 103, + 659, + 335, + 672 + ], + "detect_score": -2.2189629078, + "content": "a r t i c l e i n f o", + "postprocess_score": 0.9996395111, + "detect_cls": "Other", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 499, + 694, + 1362, + 1148 + ], + "detect_score": -0.062396463, + "content": "In this paper, we study the effectiveness of the modelling approach on the pandemic due to the spreading of the novel COVID-19 disease and develop a susceptible-infected-removed (SIR) model that provides a theoretical framework to investigate its spread within a community. Here, the model is based upon the well-known susceptible-infected-removed (SIR) model with the difference that a total population is not defined or kept constant per se and the number of susceptible individuals does not decline monotonically. To the contrary, as we show herein, it can be increased in surge periods! In particular, we investigate the time evolution of different populations and monitor diverse significant parameters for the spread of the disease in various communities, represented by China, South Korea, India, Australia, USA, Italy and the state of Texas in the USA. The SIR model can provide us with insights and predictions of the spread of the virus in communities that the recorded data alone cannot. Our work shows the importance of modelling the spread of COVID-19 by the SIR model that we propose here, as it can help to assess the impact of the disease by offering valuable predictions. Our analysis takes into account data from January to June, 2020, the period that contains the data before and during the implementation of strict and control measures. We propose predictions on various parameters related to the spread of COVID-19 and on the number of susceptible, infected and removed populations until September 2020. By comparing the recorded data with the data from our modelling approaches, we deduce that the spread of COVID- 19 can be under control in all communities considered, if proper restrictions and strong policies are implemented to control the infection rates early from the spread of the disease. \u00a9 2020 Elsevier Ltd. All rights reserved.", + "postprocess_score": 0.9998980761, + "detect_cls": "Abstract", + "postprocess_cls": "Abstract" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 102, + 694, + 425, + 943 + ], + "detect_score": -4.4695625305, + "content": "Article history: Received 18 June 2020 Accepted 23 June 2020 Available online 28 June 2020 Keywords: COVID-19 pandemic infectious disease virus spreading SIR model forecasting", + "postprocess_score": 0.7268810868, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 104, + 1223, + 247, + 1238 + ], + "detect_score": -4.6796755791, + "content": "1. Introduction", + "postprocess_score": 0.9984081388, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 103, + 1273, + 710, + 1621 + ], + "detect_score": 1.5009797812, + "content": "In December 2019, a novel strand of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China causing a severe and potentially fatal respiratory syndrome, i.e., COVID-19. Since then, it has become a pandemic declared by World Health Orga- nization (WHO) on March 11, which has spread around the globe [1\u20135] . WHO published in its website preliminary guidelines with public health care for the countries to deal with the pandemic [6] . Since then, the infectious disease has become a public health threat. Italy and USA are severely affected by COVID-19 [7\u20139] . Mil- lions of people are forced by national governments to stay in self- isolation and in difficult conditions. The disease is growing fast in many countries around the world. In the absence of availability of a proper medicine or vaccine, currently social distancing, self- quarantine and wearing a face mask have been emerged as the", + "postprocess_score": 0.9999797344, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 103, + 1667, + 537, + 1716 + ], + "detect_score": -4.1203799248, + "content": "\u2217 Corresponding author. E-mail address: arghamondalb1@gmail.com (A. Mondal).", + "postprocess_score": 0.9313198328, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 104, + 1742, + 474, + 1778 + ], + "detect_score": -0.3083037436, + "content": "https://doi.org/10.1016/j.chaos.2020.110057 0960-0779/\u00a9 2020 Elsevier Ltd. All rights reserved.", + "postprocess_score": 0.9993738532, + "detect_cls": "Page Footer", + "postprocess_cls": "Page Footer" + }, + { + "pdf_name": "paper.pdf", + "page_num": 2, + "bounding_box": [ + 754, + 1223, + 1361, + 1672 + ], + "detect_score": 1.6423951387, + "content": "most widely-used strategy for the mitigation and control of the pandemic. In this context, mathematical models are required to estimate disease transmission, recovery, deaths and other significant param- eters separately for various countries, that is for different, spe- cific regions of high to low reported cases of COVID-19. Differ- ent countries have already taken precise and differentiated mea- sures that are important to control the spread of the disease. How- ever, still now, important factors such as population density, in- sufficient evidence for different sym ptoms, transmission mecha- nism and unavailability of a proper vaccine, makes it difficult to deal with such a highly infectious and deadly disease, especially in high population density countries such as India [10\u201312] . Recently, many research articles have adopted the modelling approach, using real incidence datasets from affected countries and, have investi- gated different characteristics as a function of various parameters of the outbreak and the effects of intervention strategies in differ- ent countries, respective to their current situations.", + "postprocess_score": 0.9999089241, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 80, + 89, + 1030, + 105 + ], + "detect_score": -5.5781354904, + "content": "2 I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057", + "postprocess_score": 0.9998384714, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 80, + 139, + 688, + 1647 + ], + "detect_score": 0.6064856052, + "content": "It is imperative that mathematical models are developed to provide insights and make predictions about the pandemic, to plan effective control strategies and policies [13\u201315] . Modelling ap- proaches [8,16\u201321] are helpful to understand and predict the pos- sibility and severity of the disease outbreak and, provide key in- formation to determine the intensity of COVID-19 disease inter- vention. The susceptible-infected-removed (SIR) model and its ex- tended modifications [22\u201325] , such as the extended-susceptible- infected-removed (eSIR) mathematical model in various forms have been used in previous studies [26\u201328] to model the spread of COVID-19 within communities. Here, we propose the use of a novel SIR model with different characteristics. One of the major assumptions of the classic SIR model is that there is a homogeneous mixing of the infected and susceptible populations and that the total population is constant in time. In the classic SIR model, the susceptible population de- creases monotonically towards zero. However, these assumptions are not valid in the case of the spread of the COVID-19 virus, since new epicentres spring up around the globe at different times. To account for this, the SIR model that we propose here does not con- sider the total population and takes the susceptible population as a variable that can be adjusted at various times to account for new infected individuals spreading throughout a community, resulting in an increase in the susceptible population, i.e., to the so-called surges. The SIR model we introduce here is given by the same sim- ple system of three ordinary differential equations (ODEs) with the classic SIR model and can be used to gain a better understand- ing of how the virus spreads within a community of variable pop- ulations in time, when surges occur. Importantly, it can be used to make predictions of the number of infections and deaths that may occur in the future and provide an estimate of the time scale for the duration of the virus within a community. It also provides us with insights on how we might lessen the impact of the virus, what is nearly impossible to discern from the recorded data alone! Consequently, our SIR model can provide a theoretical framework and predictions that can be used by government authorities to control the spread of COVID-19. In our study, we used COVID-19 datasets from [29] in the form of time-series, spanning January to June, 2020. In particular, the time series are composed of three columns which represent the d , total cases I active cases I d and Deaths D d in time (rows). These tot datasets were used to update parameters of the SIR model to un- derstand the effects and estimate the trend of the disease in var- ious communities, represented by China, South Korea, India, Aus- tralia, USA, Italy and the state of Texas in the USA. This allowed us to estimate the development of COVID-19 spread in these com- munities by obtaining estimates for the number of deaths D , sus- ceptible S , infected I and removed R m populations in time. Conse- quently, we have been able to estimate its characteristics for these communities and assess the effectiveness of modelling the disease. The paper is organised as following: In Sec. 2 , we introduce the SIR model and discuss its various aspects. In Sec. 3 , we ex- plain the approach we used to study the data in [29] and in Sec. 4 , we present the results of our analysis for China, South Ko- rea, India, Australia, USA, Italy and the state of Texas in the USA. Section 5 discusses the implications of our study to the \"flattening the curve\" approach. Finally, in Sec. 6 , we conclude our work and discuss the outcomes of our analysis and its connection to the ev- idence that has been already collected on the spread of COVID-19 worldwide.", + "postprocess_score": 0.99999547, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 80, + 1683, + 597, + 1728 + ], + "detect_score": 0.2470552921, + "content": "2. The SIR model that can accommodate surges in the susceptible population", + "postprocess_score": 0.999853611, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 80, + 1759, + 686, + 1804 + ], + "detect_score": -3.0758883953, + "content": "The world around us is highly complicated. For example, how a virus spreads, including the novel strand of Coronavirus (SARS-", + "postprocess_score": 0.9980859756, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 730, + 139, + 1337, + 943 + ], + "detect_score": 0.1737413555, + "content": "CoV-2) that was identified in Wuhan, Hubei Province, China, de- pends upon many factors, among which some of them are consid- ered by the classic SIR model, which is rather simplistic and cannot take into consideration surges in the number of susceptible indi- viduals. Here, we propose the use of a modified SIR model with characteristics, based upon the classic SIR model. In particular, one of the major assumptions of the classic SIR model is that there is a homogeneous mixing of the infected I and susceptible S popu- lations and that the total population N is constant in time. Also, in the SIR model, the susceptible population S decreases monoton- ically towards zero. These assumptions however are not valid in the case of the spread of the COVID-19 virus, since new epicen- tres spring up around the globe at different times. To account for this, we introduce here a SIR model that does not consider the to- tal population N , but rather, takes the susceptible population S as a variable that can be adjusted at various times to account for new infected individuals spreading throughout a community, resulting in its increase. Thus, our model is able to accommodate surges in the number of susceptible individuals in time, whenever these oc- cur and as evidenced by published data, such as those in [29] that we consider here. Our SIR model is given by the same, simple system of three or- dinary differential equations (ODEs) with the classic SIR model that can be easily implemented and used to gain a better understanding of how the COVID-19 virus spreads within communities of variable populations in time, including the possibility of surges in the sus- ceptible populations. Thus, the SIR model here is designed to re- move many of the complexities associated with the real-time evo- lution of the spread of the virus, in a way that is useful both quan- titatively and qualitatively. It is a dynamical system that is given by three coupled ODEs that describe the time evolution of the follow- ing three populations:", + "postprocess_score": 0.9999853373, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 735, + 968, + 1336, + 1316 + ], + "detect_score": -3.291107893, + "content": "1. Susceptible individuals, S ( t ): These are those individuals who are not infected, however, could become infected. A susceptible in- dividual may become infected or remain susceptible. As the virus spreads from its source or new sources occur, more in- dividuals will become infected, thus the susceptible population will increase for a period of time (surge period). 2. Infected individuals, I ( t ): These are those individuals who have already been infected by the virus and can transmit it to those individuals who are susceptible. An infected individual may re- main infected, and can be removed from the infected popula- tion to recover or die. 3. Removed individuals, R m ( t ): These are those individuals who have recovered from the virus and are assumed to be immune, R m ( t ) or have died, D ( t ).", + "postprocess_score": 0.9936820269, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 730, + 1343, + 1339, + 1539 + ], + "detect_score": -1.5980840921, + "content": "Furthermore, it is assumed that the time scale of the SIR model is short enough so that births and deaths (other than deaths caused by the virus) can be neglected and that the number of deaths from the virus is small compared with the living popula- tion. Based on these assumptions and concepts, the rates of change of the three populations are governed by the following system of ODEs, what constitutes the SIR model used in this study", + "postprocess_score": 0.9999789, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 732, + 1559, + 1336, + 1714 + ], + "detect_score": -2.6257579327, + "content": "dS(t ) = , \u2212aS(t ) I(t ) dt dI(t ) = , aS(t ) I(t ) \u2212 bI(t ) dt (t ) d R m = , bI(t ) (1) dt", + "postprocess_score": 0.6855009198, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 3, + "bounding_box": [ + 730, + 1734, + 1336, + 1800 + ], + "detect_score": -5.3550691605, + "content": "where a and b are real, positive, parameters of the initial exponen- tial growth and final exponential decay of the infected population I .", + "postprocess_score": 0.4789528251, + "detect_cls": "Other", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 411, + 89, + 1360, + 104 + ], + "detect_score": -2.9698488712, + "content": "I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057 3", + "postprocess_score": 0.9999964237, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 104, + 139, + 712, + 948 + ], + "detect_score": -0.4069412649, + "content": "It has been observed that in many communities, a spike in the number of infected individuals, I , may occur, which results in a surge in the susceptible population, S , recorded in the COVID-19 datasets [29] , what amounts to a secondary wave of infections. To account for such a possibility, S in the SIR model (1) , can be reset to S surge at any time t s that a surge occurs, and thus it can accom- modate multiple such surges if recorded in the published data in [29] , what distinguishes it from the classic SIR model. The evolution of the infected population I is governed by the second ODE in system 1 , where a is the transmission rate con- stant and b the removal rate constant. We can define the basic = aS(t ) /b, as the fate of the evolu- effective reproductive rate R e tion of the disease depends upon it. If R e is smaller than one, the infected population I will decrease monotonically to zero and if dI(t ) < \u21d2 < greater than one, it will increase, i.e., if 0 1 and if R e dt dI(t ) > \u21d2 > 0 R e 1 . Thus, the effective reproductive rate R e acts as dt a threshold that determines whether an infectious disease will die out quickly or will lead to an epidemic. > 1 and S \u2248 1, the rate At the start of an epidemic, when R e dI(t ) \u2248 of infected population is described by the approximation dt ( a \u2212 b ) I(t ) and thus, the infected population I will initially increase (a \u2212b) = exponentially according to I(t ) I(0) e t . The infected popula- tion will reach a peak when the rate of change of the infected pop- = , = ulation is zero, d I(t ) /d t 0 1 . After the and this occurs when R e peak, the infected population will start to decrease exponentially, \u221d \u2192 \u221e following I(t ) e \u2212bt . Thus, eventually (for t ), the system will \u2192 \u2192 approach S 0 and I 0. Interestingly, the existence of a thresh- old for infection is not obvious from the recorded data, however can be discerned from the model. This is crucial in identifying a possible second wave where a sudden increase in the suscepti- > 1, and to another exponential ble population S will result in R e growth of the number of infections I .", + "postprocess_score": 0.9999687672, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 103, + 977, + 252, + 997 + ], + "detect_score": -2.0502951145, + "content": "3. Methodology", + "postprocess_score": 0.9999917746, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 102, + 1028, + 712, + 1237 + ], + "detect_score": -0.3939710259, + "content": "The data in [29] for China, South Korea, India, Australia, USA, Italy and the state of Texas (communities) are organised in the form of time-series where the rows are recordings in time (from January to June, 2020), and the three columns are, the total cases d I (first column), number of infected individuals I d (second col- tot umn) and deaths D d (third column). Consequently, the number of removals can be estimated from the data by d d = d . R I \u2212 I d \u2212 D (2) m tot", + "postprocess_score": 0.9999479055, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 103, + 1253, + 711, + 1804 + ], + "detect_score": 1.2454432249, + "content": "Since we want to adjust the numerical solutions to our proposed SIR model (1) to the recorded data from [29] , for each dataset (community), we consider initial conditions in the interval [0,1] and scale them by a scaling factor f to fit the recorded data by visual inspection. In particular, the initial conditions for the three = populations are set such that S(0) 1 (i.e., all individuals are con- d d = = < , sidered susceptible initially), I(0) (0) I / f 1 where I R m max max is the maximum number of infected individuals I d . Consequently, d the parameters a, b, f and I are adjusted manually to fit the max recorded data as best as possible, based on a trial-and-error ap- proach and visual inspections. A preliminary analysis using non- linear fittings to fit the model to the published data [29] provided at best inferior results to those obtained in this paper using our trial-and-error approach with visual inspections, in the sense that the model solutions did not follow as close the published data, what justifies our approach in the paper. A prime reason for this is that the published data (including those in [29] we are using here) are data from different countries that follow different methodolo- gies to record them, with not all infected individuals or deaths ac- counted for. \u2265 0 at any t \u2265 0. System (1) can In this context, S, I and R m be solved numerically to find how the scaled (by f ) susceptible S ,", + "postprocess_score": 0.9999829531, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 752, + 139, + 1361, + 512 + ], + "detect_score": -2.9485728741, + "content": "infected I and removed R m populations (what we call model solu- tions) evolve with time, in good agreement with the recorded data. In particular, since this system is simple with well-behaved solu- tions, we used the first-order Euler integration method to solve = / = . it numerically, and a time step h 20 0 50 0 0 0 04 that corre- of 200 days since January, sponds to a final integration time t f 2020. This amounts to double the time interval in the recorded data in [29] and allows for predictions for up to 100 days after January, 2020. Obviously, what is important when studying the spread of a virus is the number of deaths D and recoveries R in time. As these numbers are not provided directly by the SIR model (1) , we estimated them by first, plotting the data for deaths D d vs the d d d , = d + d = removals R where R D R I \u2212 I d and then fitting the m m tot plotted data with the nonlinear function (cid:2) (cid:3) \u2212k R m = \u2212 e", + "postprocess_score": 0.9999756813, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 753, + 529, + 1361, + 745 + ], + "detect_score": 1.0110032558, + "content": "(cid:2) (cid:3) \u2212k R m = , D 1 \u2212 e (3) D 0 and k are constants estimated by the non-linear fitting. where D 0 The function is expressed in terms of only model values and is fitted to the curve of the data. Thus, having obtained D from the non-linear fitting, the number of recoveries R can be described in time by the simple observation that it is given by the scaled re- movals, R m from the SIR model (1) , less the number of deaths, D from Eq. (3) , = \u2212 D.", + "postprocess_score": 0.999980092, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 754, + 765, + 1360, + 785 + ], + "detect_score": -3.8874840736, + "content": "= R \u2212 D. (4) R m", + "postprocess_score": 0.9967126846, + "detect_cls": "Equation label", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 753, + 823, + 845, + 839 + ], + "detect_score": -2.3766357899, + "content": "4. Results", + "postprocess_score": 0.9980232716, + "detect_cls": "Equation label", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 4, + "bounding_box": [ + 753, + 874, + 1362, + 1804 + ], + "detect_score": 0.1522195786, + "content": "The rate of increase in the number of infections depends on the product of the number of infected and susceptible individuals. An understanding of the system of Eqs. (1) explains the staggering in- crease in the infection rate around the world. Infected people trav- eling around the world has led to the increase in infected numbers and this results in a further increase in the susceptible population [14] . This gives rise to a positive feedback loop leading to a very rapid rise in the number of active infected cases. Thus, during a surge period, the number of susceptible individuals increases and as a result, the number of infected individuals increases as well. For example, as of 1 March, 2020, there were 88,590 infected indi- viduals and by 3 April, 2020, this number had grown to a stagger- ing 1,015,877 [29] . Understanding the implications of what the system of Eqs. (1) tells us, the only conclusion to be drawn using scien- tific principles is that drastic action needs to be taken as early as possible, while the numbers are still low, before the exponen- tial increase in infections starts kicking in. For example, if we consider the results of policies introduced in the UK to mitigate the spread of the disease, there were 267,240 total infections and 37,460 deaths by 27 May and in the USA, 1,755,803 and 102,107, total infections and deaths, respectively. Thus, even if one starts with low numbers of infected individuals, the number of infec- tions will at first grow slowly and then, increase approximately exponentially, then taper off until a peak is reached. Comparing these results for the UK and USA with those for South Korea, where steps were taken immediately to reduce the susceptible population, there were 11,344 total infections and 269 deaths by 27 May. The number of infections in China reached a peak about 16 February, 2020. The government took extreme actions with clo- sures, confinement, social distancing, and people wearing masks. This type of action produces a decline in the number of infections and susceptible individuals. If the number of susceptible individu- als does not decrease, then the number of infections just gets in- creased rapidly. As at this moment, there is no effective vaccine developed, the only way to reduce the number of infections is to reduce the number of individuals that are susceptible to the dis-", + "postprocess_score": 0.9999960661, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 79, + 89, + 1030, + 105 + ], + "detect_score": -4.5869941711, + "content": "4 I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057", + "postprocess_score": 0.9999125004, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 167, + 137, + 488, + 512 + ], + "detect_score": -4.0467090607, + "content": "", + "postprocess_score": 0.9999753237, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 158, + 543, + 488, + 909 + ], + "detect_score": -4.0443725586, + "content": "", + "postprocess_score": 0.9999625683, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 905, + 783, + 1259, + 898 + ], + "detect_score": -3.2955610752, + "content": "", + "postprocess_score": 0.6019153595, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 905, + 623, + 1259, + 738 + ], + "detect_score": -3.1234717369, + "content": "", + "postprocess_score": 0.5252876878, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 905, + 592, + 1055, + 607 + ], + "detect_score": -4.7366247177, + "content": "", + "postprocess_score": 0.9935482144, + "detect_cls": "Figure", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 905, + 559, + 1061, + 574 + ], + "detect_score": -3.1159658432, + "content": "", + "postprocess_score": 0.9811373949, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 514, + 543, + 834, + 909 + ], + "detect_score": -3.177952528, + "content": "", + "postprocess_score": 0.9999858141, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 527, + 145, + 834, + 512 + ], + "detect_score": -2.3328213692, + "content": "", + "postprocess_score": 0.9999841452, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 865, + 144, + 1179, + 512 + ], + "detect_score": -2.7618238926, + "content": "", + "postprocess_score": 0.9999905825, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 80, + 932, + 1338, + 969 + ], + "detect_score": 4.9351878166, + "content": "Fig. 1. China: Model predictions for the period from 22 January to 9 August, 2020 with data from January to June, 2020. The data show a discrete jump in deaths D in mid-April.", + "postprocess_score": 0.9999896288, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 79, + 1018, + 687, + 1590 + ], + "detect_score": 1.2801407576, + "content": "ease. Consequently, the rate of infection tends to zero only if the susceptible population goes to zero. Here, we have applied the SIR model (1) considering data from various countries and the state of Texas in the USA provided in [29] . Assuming the published data are reliable, the SIR model (1) can be applied to assess the spread of the COVID-19 disease and predict the number of infected, removed and recovered pop- ulations and deaths in the communities, accommodating at the same time possible surges in the number of susceptible individ- uals. Figures 1\u201317 show the time evolution of the cumulative total infections I tot , current infected individuals, I , recovered individuals, R , dead individuals, D , and normalized susceptible populations, S for China, South Korea, India, Australia, USA, Italy and Texas in the USA, respectively. The crosses show the published data [29] and the smooth lines, solutions and predictions from the SIR model. The cumulative total infections plots also show a curve for the ini- tial exponential increase in the number of infections, where the number of infections doubles every five days. The figures also show predictions, and a summary of the SIR model parameters in (1) and published data in [29] for easy comparisons. We start by analysing the data from China and then move on to the study of the data from South Korea, India, Australia, USA, Italy and Texas.", + "postprocess_score": 0.9999921322, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 79, + 1633, + 164, + 1648 + ], + "detect_score": -3.4596066475, + "content": "4.1. China", + "postprocess_score": 0.9987799525, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 80, + 1683, + 687, + 1804 + ], + "detect_score": 2.7597084045, + "content": "The number of infections peaked in China about 16 February, 2020 and since then, it has slowly decreased. The decrease only occurs when the susceptible population numbers decrease and this decrease in susceptible numbers only occurred through the drastic actions taken by the Chinese government. China quarantined and", + "postprocess_score": 0.9998686314, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 730, + 1018, + 1339, + 1189 + ], + "detect_score": 0.3867059946, + "content": "confirmed potential patients, and restricted citizens' movements as well as international travel. Social distancing was widely practiced, and most of the people wore face masks. The actual numbers of infections have decreased at a greater rate than predicted by the SIR model (see Figs. 1 and 2 ). Our results in Figs. 1 and 2 provide evidence that the Chinese government has done well in limiting the impact of the spread of COVID-19.", + "postprocess_score": 0.9999861717, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 730, + 1224, + 873, + 1240 + ], + "detect_score": -5.2714262009, + "content": "4.2. South Korea", + "postprocess_score": 0.9997320771, + "detect_cls": "Figure", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 730, + 1275, + 1338, + 1543 + ], + "detect_score": 0.0972661227, + "content": "From the plots shown in Figs. 3 and 4 , it is obvious that the South Korean government has done a wonderful job in controlling the spread of the virus. The country has implemented an extensive virus testing program. There has also been a heavy use of surveil- lance technology: closed-circuit television (CCTV) and tracking of bank cards and mobile phone usage, to identify who to test in the first place. South Korea has achieved a low fatality rate (currently one percent) without resorting to such authoritarian measures as in China. The most conspicuous part of the South Korean strategy is simple enough: implementation of repeated cycles of test and contact trace measures.", + "postprocess_score": 0.9999654293, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 730, + 1582, + 812, + 1597 + ], + "detect_score": -2.9834554195, + "content": "4.3. India", + "postprocess_score": 0.9997512698, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 5, + "bounding_box": [ + 730, + 1633, + 1337, + 1804 + ], + "detect_score": 2.7195117474, + "content": "To match the recorded data from India with predictions from the SIR model (1) , it is necessary to include a number of surge pe- riods, as shown in Fig. 5 . This is because the SIR model cannot pre- dict accurately the peak number of infections, if the actual num- bers in the infected population have not peaked in time. It is most likely the spread of the virus as of early June, 2020 is not con- tained and there will be an increasing number of total infections.", + "postprocess_score": 0.9999297857, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 411, + 89, + 1360, + 104 + ], + "detect_score": -0.3649011552, + "content": "I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057 5", + "postprocess_score": 0.9999512434, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 297, + 356, + 728, + 635 + ], + "detect_score": -1.8897101879, + "content": "", + "postprocess_score": 0.9997722507, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 492, + 821, + 528, + 850 + ], + "detect_score": -5.4656028748, + "content": "", + "postprocess_score": 0.8075119257, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 984, + 828, + 1022, + 857 + ], + "detect_score": -1.5792958736, + "content": "", + "postprocess_score": 0.660996139, + "detect_cls": "Body Text", + "postprocess_cls": "Equation label" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 748, + 497, + 1160, + 791 + ], + "detect_score": -3.5085437298, + "content": "", + "postprocess_score": 0.9999063015, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 748, + 168, + 1167, + 471 + ], + "detect_score": -1.5734174252, + "content": "", + "postprocess_score": 0.9999638796, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 932, + 1612, + 1282, + 1721 + ], + "detect_score": -4.2674560547, + "content": "", + "postprocess_score": 0.7202917933, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 932, + 1459, + 1282, + 1568 + ], + "detect_score": -2.9999716282, + "content": "", + "postprocess_score": 0.4926264584, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 932, + 1428, + 1082, + 1442 + ], + "detect_score": -4.4816603661, + "content": "", + "postprocess_score": 0.9996041656, + "detect_cls": "Equation label", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 932, + 1398, + 1091, + 1412 + ], + "detect_score": -2.1646308899, + "content": "", + "postprocess_score": 0.9355637431, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 537, + 1004, + 1203, + 1354 + ], + "detect_score": -3.6628472805, + "content": "", + "postprocess_score": 0.9997650981, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 104, + 880, + 1362, + 916 + ], + "detect_score": 2.4908134937, + "content": "Fig. 2. China: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999934435, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 181, + 996, + 520, + 1354 + ], + "detect_score": -3.0143077374, + "content": "", + "postprocess_score": 0.9998668432, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 201, + 1381, + 862, + 1731 + ], + "detect_score": -3.1576080322, + "content": "", + "postprocess_score": 0.9998521805, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 6, + "bounding_box": [ + 240, + 1754, + 1224, + 1770 + ], + "detect_score": -1.6391944885, + "content": "Fig. 3. South Korea: Model predictions for the period from 26 February to 13 September, 2020 with data from February to June, 2020.", + "postprocess_score": 0.999976635, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 80, + 89, + 1030, + 105 + ], + "detect_score": -5.086025238, + "content": "6 I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057", + "postprocess_score": 0.9999493361, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 273, + 302, + 658, + 571 + ], + "detect_score": -1.0018720627, + "content": "", + "postprocess_score": 0.9999177456, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 481, + 770, + 516, + 798 + ], + "detect_score": -3.8868014812, + "content": "", + "postprocess_score": 0.8541102409, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 947, + 769, + 985, + 798 + ], + "detect_score": -1.9957865477, + "content": "", + "postprocess_score": 0.3257133663, + "detect_cls": "Body Text", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 676, + 460, + 1137, + 722 + ], + "detect_score": -2.7878081799, + "content": "", + "postprocess_score": 0.9993543029, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 676, + 137, + 1143, + 411 + ], + "detect_score": -4.144402504, + "content": "", + "postprocess_score": 0.9999591112, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 899, + 1362, + 1259, + 1583 + ], + "detect_score": -5.8744635582, + "content": "", + "postprocess_score": 0.44838956, + "detect_cls": "Reference text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 899, + 1332, + 1051, + 1345 + ], + "detect_score": -4.7969961166, + "content": "", + "postprocess_score": 0.9977968931, + "detect_cls": "Page Footer", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 898, + 1273, + 1070, + 1286 + ], + "detect_score": -2.9009726048, + "content": "", + "postprocess_score": 0.9779056907, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 515, + 1257, + 830, + 1595 + ], + "detect_score": -2.9700899124, + "content": "", + "postprocess_score": 0.9999526739, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 853, + 893, + 1168, + 1232 + ], + "detect_score": -4.8892045021, + "content": "", + "postprocess_score": 0.9999582767, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 516, + 893, + 830, + 1232 + ], + "detect_score": -3.9394185543, + "content": "", + "postprocess_score": 0.9984112978, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 80, + 821, + 1338, + 857 + ], + "detect_score": 4.9554133415, + "content": "Fig. 4. South Korea: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999959469, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 184, + 886, + 492, + 1232 + ], + "detect_score": -3.1495950222, + "content": "", + "postprocess_score": 0.9997445941, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 158, + 1257, + 492, + 1595 + ], + "detect_score": -2.5730977058, + "content": "", + "postprocess_score": 0.9999455214, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 261, + 1618, + 1156, + 1633 + ], + "detect_score": 0.6287079453, + "content": "Fig. 5. India: Model predictions for the period from 14 March to 30 September, 2020 with data from March to June, 2020.", + "postprocess_score": 0.999966979, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 80, + 1658, + 687, + 1779 + ], + "detect_score": -4.9946875572, + "content": "However, by adding new surge periods, a higher and delayed peak can be predicted and compared with future data. In Fig. 5 , a conse- quence of the surge periods is that the peak is delayed and higher than if no surge periods were applied. The model predictions for the 30 September, 2020 including the surges are: 330,0 0 0 total in-", + "postprocess_score": 0.9963923097, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 7, + "bounding_box": [ + 730, + 1658, + 1337, + 1779 + ], + "detect_score": 0.1459338814, + "content": "fections, 700 active infections and 7,500 deaths, whereas if there were no surge periods, there would be 130,0 0 0 total infections, 700 active infections and 6,300 deaths, with the peak of 60,000, which is about 40% of the current number of active cases occuring around 20 May 2020. Thus, the model can still give a rough esti-", + "postprocess_score": 0.9999347925, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 411, + 89, + 1360, + 104 + ], + "detect_score": -0.1532128155, + "content": "I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057 7", + "postprocess_score": 0.9999480247, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 297, + 357, + 682, + 855 + ], + "detect_score": -1.8648295403, + "content": "", + "postprocess_score": 0.9998959303, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 999, + 817, + 1038, + 848 + ], + "detect_score": -3.6528127193, + "content": "", + "postprocess_score": 0.5562535524, + "detect_cls": "Equation label", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 768, + 492, + 1166, + 785 + ], + "detect_score": -2.0164601803, + "content": "", + "postprocess_score": 0.9997652173, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 768, + 166, + 1167, + 465 + ], + "detect_score": -4.4959640503, + "content": "", + "postprocess_score": 0.9991463423, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 927, + 1612, + 1283, + 1723 + ], + "detect_score": -3.0358397961, + "content": "", + "postprocess_score": 0.9886011481, + "detect_cls": "Other", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 927, + 1458, + 1283, + 1568 + ], + "detect_score": -3.2949621677, + "content": "", + "postprocess_score": 0.9291894436, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 927, + 1427, + 1080, + 1442 + ], + "detect_score": -2.3839042187, + "content": "", + "postprocess_score": 0.9996833801, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 927, + 1396, + 1087, + 1411 + ], + "detect_score": -2.7310013771, + "content": "", + "postprocess_score": 0.9959367514, + "detect_cls": "Figure Caption", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 540, + 1380, + 857, + 1734 + ], + "detect_score": -3.544885397, + "content": "", + "postprocess_score": 0.999986887, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 104, + 877, + 1362, + 914 + ], + "detect_score": -0.416898787, + "content": "Fig. 6. India: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999952316, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 181, + 989, + 1199, + 1352 + ], + "detect_score": -4.988483429, + "content": "", + "postprocess_score": 0.9916501045, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 194, + 1380, + 515, + 1734 + ], + "detect_score": -3.7899298668, + "content": "", + "postprocess_score": 0.9999480247, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 8, + "bounding_box": [ + 280, + 1756, + 1185, + 1772 + ], + "detect_score": -4.0235710144, + "content": "Fig. 7. Australia: Model predictions for the period from 22 January to 9 August, 2020 with data from January to June, 2020.", + "postprocess_score": 0.9999250174, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 80, + 89, + 1030, + 105 + ], + "detect_score": -2.2837231159, + "content": "8 I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057", + "postprocess_score": 0.9999710321, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 274, + 337, + 655, + 618 + ], + "detect_score": -2.9510085583, + "content": "", + "postprocess_score": 0.9998918772, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 484, + 810, + 521, + 839 + ], + "detect_score": -2.033257246, + "content": "", + "postprocess_score": 0.4747511446, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 964, + 809, + 1003, + 839 + ], + "detect_score": -4.1995396614, + "content": "", + "postprocess_score": 0.3830924034, + "detect_cls": "Body Text", + "postprocess_cls": "Equation label" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 690, + 506, + 1143, + 774 + ], + "detect_score": -1.937095046, + "content": "", + "postprocess_score": 0.9998959303, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 690, + 170, + 1143, + 449 + ], + "detect_score": -5.7125692368, + "content": "", + "postprocess_score": 0.9952076077, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 890, + 1483, + 1259, + 1718 + ], + "detect_score": -6.6361227036, + "content": "", + "postprocess_score": 0.9060491323, + "detect_cls": "Reference text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 890, + 1452, + 1042, + 1466 + ], + "detect_score": -3.0463967323, + "content": "", + "postprocess_score": 0.9953071475, + "detect_cls": "Equation label", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 890, + 1390, + 1048, + 1404 + ], + "detect_score": -3.997625351, + "content": "", + "postprocess_score": 0.9251260161, + "detect_cls": "Figure Caption", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 504, + 1373, + 820, + 1728 + ], + "detect_score": -3.3029215336, + "content": "", + "postprocess_score": 0.9997759461, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 510, + 990, + 820, + 1346 + ], + "detect_score": -1.574930191, + "content": "", + "postprocess_score": 0.9998363256, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 850, + 989, + 1161, + 1346 + ], + "detect_score": -4.1359729767, + "content": "", + "postprocess_score": 0.9990087748, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 80, + 861, + 1338, + 898 + ], + "detect_score": -2.4086289406, + "content": "Fig. 8. Australia: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999665022, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 158, + 982, + 478, + 1346 + ], + "detect_score": -2.1802184582, + "content": "", + "postprocess_score": 0.9997614026, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 166, + 1372, + 478, + 1728 + ], + "detect_score": -4.044552803, + "content": "", + "postprocess_score": 0.9998310804, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 9, + "bounding_box": [ + 272, + 1751, + 1145, + 1767 + ], + "detect_score": -3.1573388577, + "content": "Fig. 9. USA: Model predictions for the period from 22 January to 9 August, 2020 with data from January to June, 2020.", + "postprocess_score": 0.9999227524, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 411, + 89, + 1361, + 104 + ], + "detect_score": -2.8498427868, + "content": "I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057 9", + "postprocess_score": 0.9999816418, + "detect_cls": "Body Text", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 297, + 308, + 662, + 859 + ], + "detect_score": 1.1488877535, + "content": "", + "postprocess_score": 0.9999864101, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 988, + 820, + 1028, + 851 + ], + "detect_score": -4.9271850586, + "content": "", + "postprocess_score": 0.5114684105, + "detect_cls": "Body Text", + "postprocess_cls": "Equation label" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 752, + 478, + 1167, + 782 + ], + "detect_score": -2.5091948509, + "content": "", + "postprocess_score": 0.9980486631, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 751, + 138, + 1167, + 449 + ], + "detect_score": -4.1455821991, + "content": "", + "postprocess_score": 0.9999665022, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 104, + 881, + 1362, + 918 + ], + "detect_score": 2.2922720909, + "content": "Fig. 10. USA: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999892712, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 104, + 967, + 711, + 1062 + ], + "detect_score": -2.2898876667, + "content": "mate of future infections and deaths, as well as the time it may take for the number of infections to drop to safer levels, at which time restrictions can be eased, even without an accurate prediction in the peak in active infections (see Figs. 5 and 6 ).", + "postprocess_score": 0.9999017715, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 103, + 1097, + 218, + 1113 + ], + "detect_score": -2.8522717953, + "content": "4.4. Australia", + "postprocess_score": 0.9997265935, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 103, + 1148, + 711, + 1648 + ], + "detect_score": 0.7121558785, + "content": "A surge in the susceptible population was applied in early March, 2020 in the country. The surge was caused by 2,700 pas- sengers disembarking from the Ruby Princes cruise ship in Sydney and then, returning to their homes around Australia. More than 750 passengers and crew have become infected and 26 died. Two government enquires have been established to investigate what went wrong. Also, at this time many infected overseas passengers arrived by air from Europe and the USA. The Australian govern- ment was too slow in quarantining arrivals from overseas. From mid-March, 2020 until mid-May, 2020, the Australian gov- ernments introduced measures of testing, contact tracing, social distancing, staying at home policy, closure of many businesses and encouraging people to work from home. From Figs. 7 and 8 , it can be observed that actions taken were successful as the actual num- ber of infections declined in accord with the model predictions. There have been no further surge periods. From end of May, 2020, these restrictions are being removed in stages. The SIR model can be used when future data becomes available to see if the number of susceptible individuals starts to increase. If so, the model can accommodate this by introducing surge factors.", + "postprocess_score": 0.9999815226, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 103, + 1684, + 176, + 1699 + ], + "detect_score": -3.4812300205, + "content": "4.5. USA", + "postprocess_score": 0.993467629, + "detect_cls": "Body Text", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 104, + 1734, + 711, + 1804 + ], + "detect_score": 0.3524923325, + "content": "As of early June, 2020, the peak number of infections has not been reached. When a peak in the data is not reached, it is more difficult to fit the model predictions to the data. In the model, it is", + "postprocess_score": 0.9998614788, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 753, + 967, + 1361, + 1543 + ], + "detect_score": 1.2409342527, + "content": "necessary to add a few surge periods. This is because new epicen- tres of the virus arose at different times. The virus started spread- ing in Washington State, followed by California, New York, Chicago and the southern states of the USA. The need to add surge periods shows clearly that the spread of the virus is not under control. In the USA, by the end of May, 2020, the number of active in- fected cases has not yet peaked and the cumulative total num- ber of infections keeps getting bigger. This can be accounted for in the SIR model by considering how the susceptible population changes with time in May. During that time, to match the data to the model predictions, surge periods were used where the nor- malized susceptible population S was reset to 0.2 every four days. What is currently happening in the USA is that as susceptible in- dividuals become infected, their population decreases, with these infected individuals mixing with the general population, leading to an increase in the susceptible population. This is shown in the model by the variable for the susceptible population, S , varying from about 0.06 to 0.20, repeatedly during May. Until this vicious cycle is broken, the cumulative total infected population will keep growing at a steady rate and not reach an almost steady-state. The fluctuating normalized susceptible variable provides clear evidence that government authorities do not have the spread of the virus under control (see Figs. 9 and 10 ).", + "postprocess_score": 0.9999960661, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 753, + 1583, + 839, + 1597 + ], + "detect_score": -2.1333229542, + "content": "4.6. Texas", + "postprocess_score": 0.9998719692, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 10, + "bounding_box": [ + 752, + 1633, + 1361, + 1802 + ], + "detect_score": 2.0024363995, + "content": "The plots in Figs. 11 and 12 show that the peak in the total cumulative number of infections has not been reached as early as June, however, the peak is probably not far away. If there are no surges in the susceptible population, then one could expect that by late September, 2020, the number of infections will have fallen to very small numbers and the virus will have been well under control with the total number of deaths in the order of 2,0 0 0. In", + "postprocess_score": 0.9999774694, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 81, + 89, + 1030, + 105 + ], + "detect_score": -3.4840359688, + "content": "10 I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057", + "postprocess_score": 0.9998757839, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 236, + 138, + 513, + 386 + ], + "detect_score": -2.6236674786, + "content": "", + "postprocess_score": 0.9999366999, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 228, + 405, + 513, + 646 + ], + "detect_score": -3.4855673313, + "content": "", + "postprocess_score": 0.9996958971, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 871, + 563, + 1181, + 638 + ], + "detect_score": -6.324213028, + "content": "", + "postprocess_score": 0.3460729718, + "detect_cls": "Table", + "postprocess_cls": "Other" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 871, + 416, + 1187, + 533 + ], + "detect_score": -3.6509039402, + "content": "", + "postprocess_score": 0.7445672154, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 951, + 1377, + 990, + 1407 + ], + "detect_score": -1.826046586, + "content": "", + "postprocess_score": 0.3218106627, + "detect_cls": "Body Text", + "postprocess_cls": "Equation label" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 724, + 1041, + 1134, + 1337 + ], + "detect_score": -1.4917943478, + "content": "", + "postprocess_score": 0.9987780452, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 724, + 713, + 1142, + 1014 + ], + "detect_score": -1.4294627905, + "content": "", + "postprocess_score": 0.9999728203, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 535, + 405, + 810, + 646 + ], + "detect_score": -2.7379980087, + "content": "", + "postprocess_score": 0.9997373223, + "detect_cls": "Page Header", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 836, + 144, + 1108, + 386 + ], + "detect_score": -2.4109332561, + "content": "", + "postprocess_score": 0.9999467134, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 535, + 144, + 810, + 386 + ], + "detect_score": -3.4479863644, + "content": "", + "postprocess_score": 0.9998819828, + "detect_cls": "Body Text", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 256, + 669, + 1161, + 684 + ], + "detect_score": -0.799395144, + "content": "Fig. 11. Texas: Model predictions for the period from 12 March to 28 September, 2020 with data from March to June, 2020.", + "postprocess_score": 0.9999872446, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 274, + 905, + 655, + 1404 + ], + "detect_score": 2.1702032089, + "content": "", + "postprocess_score": 0.9997959733, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 80, + 1430, + 1338, + 1466 + ], + "detect_score": 0.2603513002, + "content": "Fig. 12. Texas: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999912977, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 79, + 1491, + 688, + 1764 + ], + "detect_score": 1.2270652056, + "content": "mid-May, 2020, some restrictions have been lifted in the state of Texas. The SIR model can be used to model some of the possible scenarios if the early relaxation of restrictions leads to increasing number of susceptible populations. If there is a relatively small in- crease in the future number of susceptible individuals, no series impacts occur. However, if there is a large outbreak of the virus, then the impacts can be dramatic. For example, at the end of June, = . , 2020, if S was reset to 0.8 (S 0 8) a second wave of infections occurs with the peak number of infections occurring near the end of July, with the second wave peak being higher than the initial peak number of infections. Subsequently, the number of deaths", + "postprocess_score": 0.9999864101, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 730, + 1491, + 1336, + 1633 + ], + "detect_score": -3.3011353016, + "content": "will rise from about 2,0 0 0 to nearly 5,0 0 0, as shown in Figs. 13 and 14 . If governments start lifting their containment strategies too quickly, then it is probable there will be a second wave of infec- tions with a larger peak in active cases, resulting to many more deaths.", + "postprocess_score": 0.9998557568, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 730, + 1683, + 807, + 1702 + ], + "detect_score": -5.0556755066, + "content": "4.7. Italy", + "postprocess_score": 0.9787009358, + "detect_cls": "Figure", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 11, + "bounding_box": [ + 730, + 1734, + 1339, + 1804 + ], + "detect_score": 0.5494459271, + "content": "Figure 15 shows clearly that the peak of the pandemic has been reached in Italy and without further surge periods, the spread of the virus is contained and number of active cases is declining", + "postprocess_score": 1.0, + "detect_cls": "Body Text", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 411, + 89, + 1359, + 104 + ], + "detect_score": -0.1321257204, + "content": "I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057 11", + "postprocess_score": 0.9999877214, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 352, + 138, + 685, + 666 + ], + "detect_score": -0.3160360456, + "content": "", + "postprocess_score": 0.9999830723, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 406, + 707, + 1058, + 723 + ], + "detect_score": -2.3823757172, + "content": "Fig. 13. Texas: Model predictions with a surge period occurring at the end of June, 2020.", + "postprocess_score": 0.9999729395, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 146, + 809, + 667, + 1238 + ], + "detect_score": -0.147577703, + "content": "", + "postprocess_score": 0.999917984, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 104, + 1261, + 712, + 1296 + ], + "detect_score": 2.7339990139, + "content": "Fig. 14. Texas: If a second wave occurs, there could be increase in the number of deaths, D .", + "postprocess_score": 0.9999880791, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 103, + 1354, + 711, + 1424 + ], + "detect_score": -5.2247648239, + "content": "rapidly. The plots in panels (a), (b) in Fig. 16 are a check on how well the model can predict the time evolution of the virus. These plots also assist in selecting the model's input parameters.", + "postprocess_score": 0.9896364808, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 104, + 1481, + 320, + 1500 + ], + "detect_score": -7.8255968094, + "content": "5. Flattening the curve", + "postprocess_score": 0.9998935461, + "detect_cls": "Page Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 104, + 1531, + 710, + 1804 + ], + "detect_score": -2.319409132, + "content": "The term flattening the curve has rapidly become a rallying cry in the fight against COVID-19, popularised by the media and gov- ernment officials. Claims have been made that flattening the curve results in: (i) reduction in the peak number of cases, thereby help- ing to prevent the health system from being overwhelmed and (ii) in an increase in the duration of the pandemic with the total burden of cases remaining the same. This implies that social dis- tancing measures and management of cases, with their devastat- ing economic and social impacts, may need to continue for much longer. The picture which has been widely shown in the media is shown in Fig. 17 (a).", + "postprocess_score": 0.9999457598, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 704, + 143, + 1046, + 512 + ], + "detect_score": -1.1937574148, + "content": "", + "postprocess_score": 0.9999790192, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 732, + 533, + 1053, + 557 + ], + "detect_score": -2.7895007133, + "content": "", + "postprocess_score": 0.9044197202, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 732, + 576, + 1112, + 613 + ], + "detect_score": -5.0163025856, + "content": "", + "postprocess_score": 0.7597658634, + "detect_cls": "Page Footer", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 732, + 632, + 1103, + 684 + ], + "detect_score": -4.4615859985, + "content": "", + "postprocess_score": 0.7663061619, + "detect_cls": "Table", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 754, + 808, + 1361, + 1232 + ], + "detect_score": 2.5839173794, + "content": "The idea presented in the media as shown in Fig. 17 (a) is that by flattening the curve, the peak number of infections will de- crease, however, the total number of infections will be the same and the duration of the pandemic will be longer. Hence, they con- cluded that by flattening the curve , it will have a lesser impact upon the demands in hospitals. Figure 17 (b) gives the scientific meaning of flattening the curve . By governments imposing appropriate mea- sures, the number of susceptible individuals can be reduced and combined with isolating infected individuals, will reduce the peak number of infections. When this is done, it actually shortens the time the virus impacts the society. Thus, the second claim has no scientific basis and is incorrect. What is important is reducing the peak in the number of infections and when this is done, it shortens the duration in which drastic measures need to be taken and not lengthen the period as stated in the media and by government of- ficials. Figure 17 shows that the peak number of infections actually reduces the duration of the impact of the virus on a community.", + "postprocess_score": 0.9999802113, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 754, + 1278, + 891, + 1294 + ], + "detect_score": -3.5016644001, + "content": "6. Conclusions", + "postprocess_score": 0.999835968, + "detect_cls": "Figure", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 12, + "bounding_box": [ + 754, + 1329, + 1361, + 1804 + ], + "detect_score": 1.7761548758, + "content": "Mathematical modelling theories are effective tools to deal with the time evolution and patterns of disease outbreaks. They provide us with useful predictions in the context of the impact of inter- vention in decreasing the number of infected-susceptible incidence rates [30\u201332] . In this work, we have augmented the classic SIR model with the ability to accommodate surges in the number of susceptible individuals, supplemented by recorded data from China, South Ko- rea, India, Australia, USA, Italy and the state of Texas in the USA to provide insights into the spread of COVID-19 in communities. In all cases, the model predictions could be fitted to the published data reasonably well, with some fits better than others. For China, the actual number of infections fell more rapidly than the model prediction, which is an indication of the success of the measures implemented by the Chinese government. There was a jump in the number of deaths reported in mid-April in China, which results in a less robust estimate of the number of deaths predicted by the SIR model. The susceptible population dropped to zero very quickly in South Korea showing that the government was quick to act in con-", + "postprocess_score": 0.9999955893, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 81, + 89, + 1030, + 105 + ], + "detect_score": -3.6609020233, + "content": "12 I. Cooper, A. Mondal and C.G. 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Italy: Model predictions for the period from 26 February to 13 September, 2020 with data from February to June, 2020.", + "postprocess_score": 0.9999657869, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 274, + 902, + 628, + 1463 + ], + "detect_score": -0.4556277394, + "content": "", + "postprocess_score": 0.9999705553, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 80, + 1485, + 1338, + 1522 + ], + "detect_score": 2.6966998577, + "content": "Fig. 16. Italy: (a) Nonlinear fitting with Eq. (3) using a trial-and-error method to estimate the number of deaths, D from the removed population, R m (see text for the details). (b) Plots of the number of removals, R m against the cumulative total infections I tot and current active cases I .", + "postprocess_score": 0.9999936819, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 79, + 1571, + 688, + 1793 + ], + "detect_score": 0.6412225366, + "content": "trolling the spread of the virus. As of the beginning of June, 2020, the peak number of infections in India has not yet been reached. Therefore, the model predictions give only minimum estimates of the duration of the pandemic in the country, the total cumula- tive number of infections and deaths. The case study of the virus in Australia shows the importance of including a surge where the number of susceptible individuals can be increased. This surge can be linked to the arrival of infected individuals from overseas and infected people from the Ruby Princess cruise ship. The data from", + "postprocess_score": 0.9999789, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 13, + "bounding_box": [ + 730, + 1571, + 1337, + 1793 + ], + "detect_score": -4.3732595444, + "content": "USA is an interesting example, since there are multiple epicentres of the virus that arise at different times. This makes it more diffi- cult to select appropriate model parameters and surges where the susceptible population is adjusted. The results for Texas show that the model can be applied to communities other than countries. Italy provides an example where there is excellent agreement be- tween the published data and model predictions. Thus, our SIR model provides a theoretical framework to inves- tigate the spread of the COVID-19 virus within communities. The", + "postprocess_score": 0.9945401549, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 411, + 89, + 1360, + 104 + ], + "detect_score": -1.8371825218, + "content": "I. Cooper, A. Mondal and C.G. Antonopoulos / Chaos, Solitons and Fractals 139 (2020) 110057 13", + "postprocess_score": 0.9999974966, + "detect_cls": "Page Header", + "postprocess_cls": "Page Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 297, + 218, + 623, + 486 + ], + "detect_score": -3.0958020687, + "content": "", + "postprocess_score": 0.9999445677, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 477, + 519, + 524, + 547 + ], + "detect_score": -6.4340114594, + "content": "", + "postprocess_score": 0.9272556901, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 486, + 620, + 520, + 646 + ], + "detect_score": -3.5570061207, + "content": "", + "postprocess_score": 0.6571654081, + "detect_cls": "Figure", + "postprocess_cls": "Equation" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 879, + 625, + 915, + 653 + ], + "detect_score": -3.0718894005, + "content": "", + "postprocess_score": 0.5860510468, + "detect_cls": "Page Header", + "postprocess_cls": "Equation label" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 673, + 139, + 1167, + 588 + ], + "detect_score": 1.2988601923, + "content": "", + "postprocess_score": 0.9999545813, + "detect_cls": "Figure", + "postprocess_cls": "Figure" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 104, + 676, + 1362, + 730 + ], + "detect_score": -0.936245203, + "content": "Fig. 17. Flattening the curve: Panel (a): The flattening of the curve diagram used widely in the media to represent a means of reducing the impacts of COVID-19. Panel (b) If the number of susceptible individuals is reduced, then the peak number of infections will be less and the time for the number of infections to fall to low numbers is reduced.", + "postprocess_score": 0.9999760389, + "detect_cls": "Figure Caption", + "postprocess_cls": "Figure Caption" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 103, + 793, + 712, + 1698 + ], + "detect_score": 2.6646597385, + "content": "model can give insights into the time evolution of the spread of the virus that the data alone does not. In this context, it can be applied to communities, given reliable data are available. Its power also lies to the fact that, as new data are added to the model, it is easy to adjust its parameters and provide with best-fit curves between the data and the predictions from the model. It is in this context then, it can provide with estimates of the number of likely deaths in the future and time scales for decline in the number of infections in communities. Our results show that the SIR model is more suitable to predict the epidemic trend due to the spread of the disease as it can accommodate surges and be adjusted to the recorded data. By comparing the published data with predictions, it is possible to predict the success of government interventions. The considered data are taken in between January and June, 2020 that contains the datasets before and during the implementation of strict and control measures. Our analysis also confirms the success and failures in some countries in the control measures taken. Strict, adequate measures have to be implemented to further prevent and control the spread of COVID-19. Countries around the world have taken steps to decrease the number of infected citizens, such as lock-down measures, awareness programs promoted via media, hand sanitization campaigns, etc. to slow down the trans- mission of the disease. Additional measures, including early detec- tion approaches and isolation of susceptible individuals to avoid mixing them with no-symptoms and self-quarantine individuals, traffic restrictions, and medical treatment have shown they can help to prevent the increase in the number of infected individuals. Strong lockdown policies can be implemented, in different areas, if possible. In line with this, necessary public health policies have to be implemented in countries with high rates of COVID-19 cases as early as possible to control its spread. The SIR model used here is only a simple one and thus, the predictions that come out might not be accurate enough, something that also depends on the pub- lished data and their trustworthiness. However, as the model data show, one thing that is certain is that COVID-19 is not going to go way quickly or easily.", + "postprocess_score": 0.9999793768, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 754, + 793, + 1076, + 813 + ], + "detect_score": -5.0445175171, + "content": "Declaration of Competing Interest", + "postprocess_score": 0.9239397645, + "detect_cls": "Page Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 754, + 844, + 1360, + 914 + ], + "detect_score": 0.0377864242, + "content": "The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.", + "postprocess_score": 0.9999830723, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 753, + 945, + 935, + 964 + ], + "detect_score": -1.8052092791, + "content": "Acknowledgements", + "postprocess_score": 0.9999091625, + "detect_cls": "Page Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 754, + 996, + 1363, + 1062 + ], + "detect_score": 1.2173963785, + "content": "AM is thankful for the support provided by the Department of Mathematical Sciences, University of Essex, UK to complete this work.", + "postprocess_score": 0.9999659061, + "detect_cls": "Body Text", + "postprocess_cls": "Body Text" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 754, + 1091, + 853, + 1106 + ], + "detect_score": -3.2485990524, + "content": "References", + "postprocess_score": 0.9999169111, + "detect_cls": "Section Header", + "postprocess_cls": "Section Header" + }, + { + "pdf_name": "paper.pdf", + "page_num": 14, + "bounding_box": [ + 755, + 1138, + 1361, + 1732 + ], + "detect_score": -0.8136662841, + "content": "[1] World Health organization, coronavirus disease (COVID-19) outbreak. https:// www.who.int/emergencies/diseases/novel-coronavirus-2019 . 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The mathematics of infectious diseases. SIAM review 20 0 0;42(4):599\u2013653 . [24] Hethcote HW . The basic epidemiology models: models, expressions for r0, pa- rameter estimation, and applications. In Mathematical understanding of infec- tious disease dynamics 2009;16:1\u201361 . [25] Weiss HH. The SIR model and the foundations of public health. MATerials MATem\u00e0tics, 0 0 01-17 2013 . http://mat.uab.cat/matmat/PDFv2013/v2013n03. pdf [26] Amaro JE , Dudouet J , Orce JN . Global analysis of the COVID-19 pandemic using simple epidemiological models 2020 . ArXiv preprint arXiv:2005.06742 [27] Calafiore GC , Novara C , Possieri C . A modified sir model for the covid-19 con- tagion in Italy 2020 . ArXiv preprint arXiv:2003.14391 [28] Ndairou F , Area I , Nieto JJ , Torres DF . Mathematical modeling of covid-19 transmission dynamics with a case study of wuhan. Chaos, Solitons & Fractals 2020:109846 . [29] Coronavirus worldometer website. https://www.worldometers.info/ coronavirus/ . 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