An interactive program to visualize the influence of various parameters on the course of the pandemic / herd immunity.
The basic underlying mathematical model of the program is the SIR-model. Up to 4 models can be plotted simultaneously. The UI was made using Python and the Pygame-library. The source code is freely available on GitHub. An executable file can be downloaded via the following link (due to GitHub's file size limit).
https://drive.google.com/drive/folders/1dJDuhwRIn6pqgS-xkgCtONLGiq71wt_M?usp=sharing
The program shows 2 plots at all times. The upper plot shows the percentages of Infected people over time. For the plot below, the user can select to either plot the percentage of immune people (I+R), or the reproduction rate R over time. The reproduction rate R was calculated according to the following equation, with R0=beta/gamma, pC = reduction in transmission rates from safety measures, pI = reduction of transmission rates from virus immunities.
| User input | Description |
|---|---|
| Left-/Right-/Up-/Down-keys | Navigating through the parameters |
| Left-/Right-keys | If a parameter was selected: Increasing/decreasing its value |
| X-key | Selecting/ Deselecting a parameter |
| R-key | Switching the plot below to either: • Show percentage of immune people • Show R-value over time |
| C-/E-/M-/P-keys | Set preset variables for COVID-19, Ebola, Measles or Polio respectively |
By changing the parameters, the user can adjust/extend the classical SIR-model.
| Parameter name | Description |
|---|---|
| Show plot? | Boolean parameter that switches the visualization of plots on/off. |
| beta | Average number of infections that one person causes per timestep. |
| gamma | Inverse of the average time that one person stays infected. |
| Lockdown? | Boolean parameter that controls whether or not a lockdown is present.1,2 |
| L_start | Start date of lockdown (unit: days) |
| L_duration | Duration of lockdown (unit: days) |
| L_redRate | Reduction rate of contacts during lockdown (1: No reduction, 0: Strict lockdown with zero contacts) |
| P_Type? | Describes the type of population.1 Can have 3 different values: • Homogen: Homogeneous population • Gamma2: Heterogeneous population according to Britton et al. (2020) • Matrix2: Heterogeneous population according to Neipel et al. (2020) |
| P_H_var | Additional parameter that are used for the population heterogeneity models. • “Gamma”: Alpha-parameter • “Matrix”: Parameter used by SEIR-model. (Inverse of the incubation period of the disease) |
1 Lockdown?=False and P_Type?=”Homogen” correspond to the classical SIR-model.
2 See "extensions of SIR-model" below
Lockdown restrictions influence the parameter beta, as it describes the average number of infections that one person causes per timestep. We denote the new average number of infections that one person causes per timestep during lockdown as Li, with Li=Lockdown intensity. It is therefore:
- Li= 1: no lockdown,
- Li= 0: Strict lockdown during which every person has zero contacts
We can then split the course of time t=[1:200] into 3 time regions ("before lockdown", "during lockdown", "after lockdown") and model each region by its own SIR-model. If without restrictions, one person infects b other people per day, the 3 SIR-models then correspond to: 1st SIR model: Course of the pandemic until the lockdown (with beta=b) 2nd SIR model: Course of the pandemic during the lockdown (with beta=b*Li) 3rd SIR model: Course of the pandemic after lockdown (with beta=b)
This extension is based on the paper: Britton T., Ball F., Trapman P., A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2. Science 369, 846–849 (2020) https://science.sciencemag.org/content/sci/369/6505/846.full.pdf
Here a SEIR Model is used, which considers an additional exposed population. Exposed individuals are not infectious during the incubation period of the disease. We consider six different age groups with different transmission rates with respect to each of these six groups. This builds a 6x6 contact matrix beta. The P_H_var and gamma values, which are the inverse of the incubation and infectious period respectively, are assumed to be constant for these six age groups.
We start with one individual being exposed at time t=0 and the rest of the population being susceptible. The removed and infected population at time t=0 is zero. The S,E,I and R values are solved from the differential equations for each of these 6 age groups, which yields 4x6=24 values. The S, E, I and R values of the total population are calculated by adding the array of values corresponding to each age group.
The initial behaviour of the infected population depends on which of the six age groups is exposed first. The curve of the infected population might show a small initial peak. This behaviour of the model is because the first individual gets infected after the incubation period of the disease and this same individual joins the group of the removed population R while the next set of individuals, who got exposed, are still in their incubation period.
This extension is based on the paper: Neipel J., Bauermann J., Bo S., Harmon T., Jülicher F., Power-law population heterogeneity governs epidemic waves. PloS One 15, e0239678 (2020). https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0239678
It was implemented by introducing the mentioned additional gamma-term to the SIR-model.
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Up to 4 models can be plotted at once (m0, m1, m2, m3)
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R0-value gets printed automatically (R0=beta/gamma)
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The model contains N=1000 individuals in the population. The initial values of S, I and R at t=1 are S0=999, I0=1, and R=0.
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"classes"-folder and "data"-folder have to be in the same folder as "HerdImmunity.exe"/ "HerdImmunity.py"
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"HerdImmunity.exe" might take between 30-60 seconds to open
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"HerdImmunity.py" is the source code of "HerdImmunity.exe"
- Gamma-extension: Moritz Schramm
- Basic SIR-model, Matrix-extension: Tom Richard Vargis
- Description of the program in README-file, Pygame-UI-interface, implementation of lockown-measurements, icon.png: Marie Langer
- Font: TYPEW.ttf by Johan Holmdahl
DISCLAIMER REGARDING FONT USAGE:
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Type wheel
This truetype typewriter font was borned at Free Typewriter Fonts http://www.free-typewriter-fonts.com
You may distribute, sell, resell, change or modify the font - and you are allowed to upload the font onto any site or CD if this disclaimer is included in the download (text.txt).
Any other means prohibited.
Johan Holmdahl info@free-typewriter-fonts.com
April 2001
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If you encounter any bugs, please send a mail to marie.langer@mailbox.tu-dresden.de