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<a href="./plots.html" rel="bookmark" title="Permalink to Explanation of plots">Explanation of plots</a>
</h1>
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</div>
<div class="row">
<div class=" col-md-9 " id="content" role="main">
<div>
<div class="contents topic" id="contents">
<p class="topic-title">Contents</p>
<ul class="simple">
<li><a class="reference internal" href="#plot-1-7-day-incidence-rate" id="toc-entry-1">Plot 1: 7-day incidence rate</a></li>
<li><a class="reference internal" href="#plot-2-daily-changes-in-cases" id="toc-entry-2">Plot 2: daily changes in cases</a></li>
<li><a class="reference internal" href="#plot-3-daily-changes-in-deaths" id="toc-entry-3">Plot 3: daily changes in deaths</a></li>
<li><a class="reference internal" href="#plot-4-r-growth-factor-based-on-cases" id="toc-entry-4">Plot 4: R & growth factor based on cases</a><ul>
<li><a class="reference internal" href="#reproduction-number-r" id="toc-entry-5">Reproduction number R</a></li>
<li><a class="reference internal" href="#daily-growth-factor" id="toc-entry-6">Daily growth factor</a></li>
<li><a class="reference internal" href="#assessing-success-for-containment-measures" id="toc-entry-7">Assessing success for containment measures</a></li>
</ul>
</li>
<li><a class="reference internal" href="#plot-5-r-growth-factor-based-on-deaths" id="toc-entry-8">Plot 5: R & growth factor based on deaths</a></li>
<li><a class="reference internal" href="#plot-6-doubling-times" id="toc-entry-9">Plot 6: Doubling times</a></li>
<li><a class="reference internal" href="#plot-7-comparison-of-daily-new-cases-with-other-countries" id="toc-entry-10">Plot 7: Comparison of daily new cases with other countries</a></li>
<li><a class="reference internal" href="#plot-8-comparison-of-daily-new-deaths-with-other-countries" id="toc-entry-11">Plot 8: Comparison of daily new deaths with other countries</a></li>
<li><a class="reference internal" href="#the-role-of-dashed-lines" id="toc-entry-12">The role of dashed lines</a></li>
</ul>
</div>
<p>We discuss the plots from the top (number 1) to the bottom (number 8)
that are available for <a class="reference external" href="all-regions.html">all regions</a>.</p>
<div class="section" id="plot-1-7-day-incidence-rate">
<h2><a class="toc-backref" href="#toc-entry-1">Plot 1: 7-day incidence rate</a></h2>
<img src="./fig-uk-incidence.svg" alt="United Kingdom data"><ul class="simple">
<li>Shows how many people have been confirmed to be infected
during the last seven days prior to a date, per 100 thousand of inhabitants.</li>
<li>This is a cumulative normalised metric, and it is possible to use it for
comparison of different regions.</li>
<li>For low rates (around zero) the graph is green, and with the grows of the rate
it gradually becomes yellow, then red, then maroon.</li>
<li>The number near the cross on the right edge of the graph shows the latest value.</li>
<li>This number is generally reported in the media as <em>incidence rate</em>.</li>
<li>The interesting question is: how fast do they grow?</li>
</ul>
</div>
<div class="section" id="plot-2-daily-changes-in-cases">
<h2><a class="toc-backref" href="#toc-entry-2">Plot 2: daily changes in cases</a></h2>
<img src="./fig-south-korea2.svg" alt="South Korea data"><ul class="simple">
<li>Shows how many new confirmed cases are reported per day, shown as
blue bars.</li>
<li>Blue lines show a seven day (rolling) average over the bar data
(using a Gaussian window with a standard deviation of 3 days) - this
produces smoother data. It also helps to remove the effect of the
weekend (in some countries reported numbers drop during and just
after the weekend)</li>
<li>We would like to see these numbers of daily changes decrease from day
to day. The faster they go down the better. Ideally their reach 0.</li>
<li>For the data of South Korea, we can see that the peak of new
infections was around 1st March 2020, and following that the number
of new infections decreased to around 100 per day for the second half
of March, before the numbers started to decrease further.</li>
</ul>
</div>
<div class="section" id="plot-3-daily-changes-in-deaths">
<h2><a class="toc-backref" href="#toc-entry-3">Plot 3: daily changes in deaths</a></h2>
<img src="./fig-south-korea3.svg" alt="South Korea data"><ul class="simple">
<li>Red bars shows how many new deaths were reported for each day.</li>
<li>Red line shows a seven day (rolling) average over the bar data.</li>
<li>The number of deaths is expected to follow the number of infections
with some time delay, and reduced by a fraction (the <a class="reference external" href="https://en.wikipedia.org/wiki/Case_fatality_rate">case fatality
rate</a>).</li>
</ul>
</div>
<div class="section" id="plot-4-r-growth-factor-based-on-cases">
<h2><a class="toc-backref" href="#toc-entry-4">Plot 4: R & growth factor based on cases</a></h2>
<img src="./fig-south-korea4.svg" alt="South Korea data"><div class="section" id="reproduction-number-r">
<h3><a class="toc-backref" href="#toc-entry-5">Reproduction number R</a></h3>
<ul class="simple">
<li>The <a class="reference external" href="https://en.wikipedia.org/wiki/Basic_reproduction_number">Reproduction number R (Wikipedia)</a> expresses how
many people are infected from one person with COVID19.</li>
<li>If we assume an average infectious period, we can estimate R from the reported
new cases. This estimate of R is shown as the green line. The estimate of R
will be affected by inaccuracies in the reported numbers.</li>
<li>Following the Robert Koch Institute, we assume an average infectious period of
4 days. This means, for example, for a reproduction number R=2, that on
average each infected person will infect 2 other people after 4 days.</li>
<li>R needs to be below 1.0 for the total number of infections to
shrink. A value of R above 1 will lead to exponential growth.</li>
<li>See <a class="reference external" href="r-value.html">this page</a> for more details.</li>
</ul>
</div>
<div class="section" id="daily-growth-factor">
<h3><a class="toc-backref" href="#toc-entry-6">Daily growth factor</a></h3>
<ul class="simple">
<li>The growth factor is the ratio of new cases today
relative to new cases yesterday</li>
<li>Blue dots show these ratios for cases, and are
computed based on the smoothed data shown in plot 2 and 3.</li>
<li>The solid line is a 7-day rolling mean over these points to provide
smoother data.</li>
<li>As long as the growth value is greater than 1.0, the number of new
infections is increasing.</li>
<li>If the growth value is exactly 1.0, then we have the same number of
new infections every day.</li>
<li>The growth factor needs to be below 1.0 for the spread to slow down.</li>
</ul>
</div>
<div class="section" id="assessing-success-for-containment-measures">
<h3><a class="toc-backref" href="#toc-entry-7">Assessing success for containment measures</a></h3>
<ul>
<li><p class="first"><strong>The reproduction number R and the growth factors are measures of success for
the virus containment</strong>: as long as they are below 1.0, the spread
is slowing down. If the growth factor is greater than one, the number of newly
infected people is increasing from day to day; this must be avoided.</p>
</li>
<li><p class="first">What is the difference between R and the growth factor? They are computed
differently. The growth factor is simpler and more direct: it describes
changes in new infections from day to day.</p>
<p>The reproduction number R is used in epidemiology, and by politicians. It is
<a class="reference external" href="r-value.html">harder to estimate</a> than the daily growth factor.
The daily growth factor and
R cannot be compared directly, but both are qualitatively similar: they need
to stay under 1.0 for the infections to become fewer over time.</p>
</li>
<li><p class="first">Growth factor and R do not depend on the total number of new cases (but only
on the relative change over time). For that reason we can get high
fluctuations where the numbers of new cases is small (imagine there
was 1 case yesterday, and 7 cases today, this would give a growth factor of
7). In short: where the number of daily new cases is small, the growth
factor can appear to change quickly. For the same reason, R and the growth
factor are good indicators to assess changes in spread even with small case
numbers.</p>
</li>
</ul>
</div>
</div>
<div class="section" id="plot-5-r-growth-factor-based-on-deaths">
<h2><a class="toc-backref" href="#toc-entry-8">Plot 5: R & growth factor based on deaths</a></h2>
<img src="./fig-south-korea5.svg" alt="South Korea data"><p>Conventionally, the R number is computed on infections (here shown in plot 4 and
referred to as "R based on cases"). Assuming a fixed mortality rate, we can also
use the reported deaths to estimate R (as R depends not on the total numbers but
their change over time). This is done in plot 5, and the estimated R is shown in
yellow. See discussion of plot 4 above for details.</p>
<p>This computation of R based on deaths can be of interest, for example, for the
case of the <a class="reference external" href="html/United-Kingdom.html">United Kingdom</a>, where the daily new
infections remained approximately constant from about 10 April to about 10 May,
but at the same time the daily number of deaths shows a slow but persistent
reduction from day to day. A possible explanation for the situation would be
that the nearly constant number of daily infections results from a saturation of
the testing capacity, and that actually the spread is reducing (which is
reflected in the deaths, which may be more accurately captured and reported). In
that case, the R based on deaths may provide a more accurate estimate.</p>
<p>If deaths and infections are similarly captured and reported, both entities
should point to the same value, although the R based on deaths could be trailing
the estimate of R based on infections by several weeks; reflecting the duration
of the illness.</p>
</div>
<div class="section" id="plot-6-doubling-times">
<h2><a class="toc-backref" href="#toc-entry-9">Plot 6: Doubling times</a></h2>
<img src="./fig-south-korea6.svg" alt="South Korea data"><ul>
<li><p class="first">This plot computes the doubling time of the cases (blue) and deaths
(red), assuming that the growth of cases and deaths as shown in plot
1 is exponential.</p>
<ul>
<li><p class="first">In more detail, we compute the doubling period from one day to the
next and show this as transparent dots.</p>
<p>If a value of 3 is shown, this means that at that point in time,
it took 3 days for the numbers of cases (or deaths) to double.</p>
</li>
<li><p class="first">The solid line is a 7-day rolling mean over these data points and
provides more robust guidance.</p>
</li>
</ul>
</li>
<li><p class="first">Looking at the doubling time shown by the solid lines, we get another
indicator to assess the spread of the virus.</p>
<ul class="simple">
<li>For many countries, in the early stages of the outbreak, this
doubling time for confirmed infections (blue line) is somewhere
around 2 to 3 days.</li>
<li>As the spread of infections is reduced (by schools closing, people
exercising social distancing, staying at home, etc), the growth
rate of the exponential function becomes smaller, and
correspondingly the doubling time increases.</li>
<li><dl class="first docutils">
<dt>We would like to see the doubling time to get larger, as this an indication</dt>
<dd>that the growth of infections and deaths is decreasing (ideally infinitely
large, when no new infections occur).</dd>
</dl>
</li>
</ul>
</li>
<li><p class="first">The example data for South Korea shows that</p>
<ul class="simple">
<li>the doubling time for cases grows from around 3 days in the early stages</li>
<li>to reach a plateau with a doubling time of around 60 days from 15 March to 1
April 2020. During this time, the number of daily new infections (plot 2)
stabilises around 100, and the growth factor (blue line in plot 4) is approximately 1.0.</li>
<li>In April, the doubling time increases further, which corresonds to a
further reduction of daily new infections (plot 2), and a growth factor
(blue line in plot 4) below 1.0.</li>
</ul>
</li>
<li><p class="first">Occasionally, the solid lines are omitted if there are not enough data points.</p>
</li>
</ul>
</div>
<div class="section" id="plot-7-comparison-of-daily-new-cases-with-other-countries">
<h2><a class="toc-backref" href="#toc-entry-10">Plot 7: Comparison of daily new cases with other countries</a></h2>
<img src="./fig-south-korea7.svg" alt="South Korea data"><ul class="simple">
<li>The x-axis shows the number of days since a particular number of new
cases per day (such as 10) have occurred in that country, and the
y-axis the number of new cases for that day.</li>
<li>We see that countries follow similar paths, with the common
properties that the curve increases until the number of cases per day
peaks, and then the curve comes down again.</li>
<li>The y-axis is logarithmic.</li>
<li>Due to the logarithmic y-axis, this visualisation can help to
understand at what stage in the outbreak an area is (despite the
different size of the countries and numbers of cases).</li>
<li>The curve shows as 7-day rolling mean to provide a smoother line than
the noisier individual data points (in particular with fewer cases
per day). This leads to non-integer values.</li>
<li>[The curve for China starts at a higher value than 10, as the first recorded
value was 548. We could manually correct this.]</li>
</ul>
</div>
<div class="section" id="plot-8-comparison-of-daily-new-deaths-with-other-countries">
<h2><a class="toc-backref" href="#toc-entry-11">Plot 8: Comparison of daily new deaths with other countries</a></h2>
<img src="./fig-south-korea8.svg" alt="South Korea data"><ul class="simple">
<li>As Plot 7, but for deaths not cases.</li>
</ul>
</div>
<div class="section" id="the-role-of-dashed-lines">
<h2><a class="toc-backref" href="#toc-entry-12">The role of dashed lines</a></h2>
<p>We use a dashed line to indicate where the data is inaccurate (for some reason).</p>
<ol class="arabic">
<li><p class="first">Example 1: Last days in incidence, deaths and cases:</p>
<a href="https://oscovida.github.io/html/United-Kingdom.html"> United Kingdom
<img src="./plots-dashed-line-example-uk.png" alt="Plot from UK, JHU data">
</a><p>The most recent 7 days in incidence, and new cases and new deaths are shown as a dashed line.</p>
<p>See the bitmap above as an example. Here the dashed line is chosen because we
need a 7-day rolling average over the raw data to average out weekly
fluctuations. (Some additional Gaussian smoothing is applied to help the eye
which needs another 3 days). See function <tt class="docutils literal">compute_daily_change</tt> in
<a class="reference external" href="https://github.com/oscovida/oscovida/blob/master/oscovida/oscovida.py">https://github.com/oscovida/oscovida/blob/master/oscovida/oscovida.py</a> if you
care for the details).</p>
<p>In more detail: For the 7-day average for today, we need to sum the reported new cases from
the last 3 days, today, and the next 3 days and divide this by 7 (and we need
another 3 days in either direction for visual smoothing). For the most recent
7 days, we cannot do this (because we don't know the data from the future
yet). Instead, we compute a rolling average using the 7 previous(!) days.
Using this method, we can compute the averaged data up to day with the last
known data point.</p>
<p>However, the computation is different (and at times it might be possible to
note a step in the data as the line [and/or its slope] as it changes from
solid to dashed). This is highlighted through using a dashed line.</p>
</li>
<li><p class="first">Example 2: Deaths in German districts</p>
<a href="https://oscovida.github.io/html/Germany-Nordrhein-Westfalen-SK-Köln.html">
<img src="./plots-dashed-line-example-german-district.png" alt="Plot from Germany, Hamburg, RKI data">
</a><p>For the data on deaths for districts within Germany that is reported by the
RKI, we dash the most recent 6 weeks. The reason here is that the RKI reports
deaths with the date at which the infection took place, not the date on which
the person died. Before vaccinations were available, it was reasonable to
assume that COVID-related deaths would die within 6 weeks of the infection
taking place.</p>
<p>We have a detailed investigation of this phenomenon at <a class="reference external" href="https://oscovida.github.io/2020-germany-reporting-delay-meldeverzug.html">https://oscovida.github.io/2020-germany-reporting-delay-meldeverzug.html</a> .</p>
<p>We thought it is important to understand this when interpreting the data, and
have thus dashed the line in the plots for deaths in Germany for the most
recent 6 weeks.</p>
<p>[This six week period should be reviewed as the larger fraction of vaccinated
and older people changes this: younger and healthier patients may stay
significantly longer in medical and intensive care before they die from
COVID.]</p>
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