Ganz einfach die Ausbreitung des Virus ausrechnen

Aus den Medien erfahren wir nicht nur die tägliche gemeldeten Coronavirus-Fallzahlen und die Todesfälle, sondern auch etwas über die Ausbreitungsgeschwindigkeit der Krankheit. In dem Zusammenhang ist meist von einer »Verdopplung der Fälle« die Rede. Anlässlich des Starts meiner Serie (Warum eine eigene Coronavirus-Statistik?) hatte ich bereits zwei statistische Eigenschaften der Epidemie erwähnt, die sich aus den Erfahrungen in China und Italien ergeben haben und die Verdopplung aufgreifen. Bei ungebremster Ausbreitung gilt:

  1. Die Zahl der gemeldete Fälle verdoppelt sich alle 3 Tage
  2. Die Zahl der Toten verdoppelt sich alle 2 Tage

Die Zeitspanne, in der sich die Zahl an Mikroben in einer Population verdoppelt, nennen Mikrobiologen Generationszeit. Der Anstieg der Individuenzahl wird in der exponentiellen Phase durch eine Gleichung beschrieben, die wie die Zinsenzins-Formel funktioniert. Als Faustformel wenden Banker und Sparer in diesem Zusammenhang gerne die 72er-Regel an. Mit ihr lässt sich näherungsweise hochrechnen, nach wie vielen Jahren sich eine verzinsliche Kapitalanlage im Nennwert verdoppelt, durch den Effekt des Zinseszins (was heute – bei Zinsen von 0,001 %  – eine utopische Kalkulation darstellt). Dazu teilt man 72 durch die Prozentzahl des jährlichen Zinssatzes des angelegten Betrages.

Und genau so lässt sich die Ausbreitungsgeschwindigkeit des Coronavirus ausrechnen:

Die Formel für die schnelle Berechnung einer Generationszeit, mit t=Zeit [gemessen in Tagen] und ∆= [Prozent Veränderung zum Vortag]

Hier ein aktuelles Rechenbeispiel mit den gemeldeten Fallzahlen aus Deutschland von gestern (71.108) und vorgestern (66.885):
Die prozentuale Steigerung aus den beiden Tagen ergibt sich, wenn man die größere Zahl durch die kleinere dividiert (71.108 : 66.885), machht 1,063… und entspricht einem Zuwachs von 6,3 %. Dann einfach die 72 durch 6,3 teilen, was eine Verdopplung der registrierten Coronavirus-Fälle in 11, 4 Tagen ergibt.

Angela Merkel hat in einem Audiopodcast am Samstag das Ziel ausgegeben, ab einem dauerhaften Wachstum > 10 Tage über eine Lockerung der Maßnahmen nachzudenken … wir sind auf dem besten Weg dahin.


Daily Covid-19 Stats Nº 21

New visua­li­sa­tion of the corona spread in Germany: the daily new cases (bars), sum of the regis­tered cases (dotted line), sum of the dead (black) … in between the poli­tical measures to curb the spread

End of Day Summary (21). Apart from my scien­tific streak, it’s mainly my inte­rest in design that has led me to build my daily coro­na­virus statis­tics. That’s why I have rethought and reor­ga­nized my style of visua­li­za­tion today. After more than three weeks with steeply rising expo­nen­tial curves the daily devia­tions in the lower part of the coor­di­nate systems are hardly noti­ce­able. My focus are now the daily changes, not the fore­casts, which can still be seen in the table, as well as the growth rates. The table is now at the end of the article.

The first figure (above) shows the increase of the regis­tered Covid19 cases in Germany, from March 9–31. The focus is now on the daily new cases (bars), where the flat­tening of the infec­tions is more appa­rent than in the dotted curve with the summed cases. The deaths (black bars) at the bottom of the chart seem a bit lost … I would like to change that in a revi­sion soon. Particularly when looking back over an entire month, it is helpful to fade in the poli­tical measures, some of which at the end of the month seem much further away than they really are.

The second picture (above) is dedi­cated to the coro­na­virus condi­tions in my home­town Berlin. At the sugges­tion of some Twitter follo­wers, I have now added up the total number of regis­tered cases against 5 subgroups: the cured (green), the people in domestic quaran­tine (blue), the Covid19 pati­ents (orange), the severe ICU cases (yellow) and the deaths (black with figures).

You can hardly see it in this presen­ta­tion, but in the table below: The growth rate in Berlin today decreased from ×1.4 to ×1.2 (related to 3 days), which corre­sponds to a doub­ling of regis­tered cases in 9.5 days. On Saturday, Angela Merkel set the goal “doub­ling in 10 days” … behold: We are slowly approa­ching this thres­hold.

Today’s Table – the complete picture:

Today, the number of regis­tered cases in Germany has risen by 4100, and thus the deve­lop­ment has slowed down again, what is also reflected in the growth rate, which falls by 0.1 for the fourth time: from ×1.5 (Sat), to ×1.4 (Sun), to ×1.3 (yesterday) to ×1.2; this corre­sponds to a doub­ling in 9.5 days, which can today really be consi­dered a success of the distan­cing measures.

The increase in the number of deaths has also slowed today, to ×1.4 (from ×1.5 yesterday, this value had changed over­night), which means a doub­ling in 5 days and is still unfor­tu­n­a­tely high. But all viro­lo­gists confirm that the number of deaths will lag behind the number of cases by at least a week.


Daily Covid-19 Stats Nº 20

Today’s coro­na­virus spreadsheet: regis­tered Cases and Deaths in Germany and Berlin today

End of Day Summary (20). Today, the number of regis­tered cases in Germany has risen by 1,834, and thus the deve­lop­ment has slowed down consi­der­ably, which is also reflected in the lower growth rate: after 6 days with ×1.5 and yesterday with ×1.4 we are down to ×1.3 – the lowest growth rate in over 4 weeks. In view of the fact that the amount of testing has increased signi­fi­cantly in recent days, this could perhaps already be seen as a clear reac­tion to the social distan­cing measures.

The increase in the number of deaths has also slowed today, to ×1.4, after having been ×1.5 in each of the last two days, due to the dramatic deaths in 2 nursing homes in nort­hern Germany. The Berlin figures follow the national trend, even a tenth more encou­ra­ging: ×1.2 (regis­tered cases) and ×1.4 (regis­tered deaths).

Today’s coro­na­virus curves for Germany, predicted cases vs. confirmed cases, March 30, 2020

The curve of confirmed cases (blue) is by far flatter than the statis­tical predic­tion (gray), with a growth rate of ×1.3.

Today’s coro­na­virus curves for Germany, predicted deaths vs. confirmed deaths, March 30, 2020

The number of deaths is still rising dyna­mi­cally, bur the growth lowered a bit, from ×1.5 to ×1.4

Author’s note: The above values are purely specu­la­tive esti­ma­tions using simple mathe­ma­tical model­ling (based on regis­tered cases/deaths) and are not confirmed by health autho­ri­ties nor any other national public autho­rity.


Attempt to understand the US coronavirus situation

Yesterday’s coro­na­virus curves for USA, predicted cases vs. confirmed cases, March 28, 2020

The number of infected people in the USA is explo­ding almost unche­cked: with 123k regis­tered cases today, Sunday morning, they are 30k higher than in Italy and 42k higher than in China. My mathe­ma­tical model, which I built for the obser­va­tion in Germany, does not work for the US numbers. So I have reba­lanced it, like a bathroom scale that is suddenly to be used for weig­hing an apple harvest weig­hing tons.

The general condi­tions in the coun­tries that are curr­ently very strongly affected by corona – USA, Italy, China, Germany, Spain, Iran and France – differ extre­mely. There are a handful of varia­bles, some of which are unknown or cannot even be put into figures:

• Start of the epidemic (first case detected)
• Age struc­ture (demo­graphy)
• The tests (when, how much, who is tested)
• Quality of health­care (see also Fontblog: Some Interactive Coronavirus Maps
• Politics

We know from the media that the rela­tive number of tests in the USA is lower than in Germany … which leads to a compu­ta­tio­nally high death rate (too few regis­tered cases vs. very precisely regis­tered deaths). Experts also believe that the reac­tion of US poli­tics to the crisis came too late and that the president’s zigzag­ging course exacer­bates the situa­tion. In some regions, for example New York, the health care system has already reached its limits.

To adjust my table to US condi­tions, I have read­justed the first column. It shows the unche­cked deve­lop­ment of corona infec­tions, as it can be calcu­lated from inter­na­tional obser­va­tions in the last months: doub­ling of cases every 3 days. Here it is important to know that the number of regis­tered cases of course does NOT reflect the number of infected persons, because not the whole popu­la­tion is tested every 3 days, but rather a sample value, which nevertheless results in a rela­tive growth trend, and is there­fore suitable as a basis for poli­tical decisions.

Yesterday’s coro­na­virus spreadsheet: regis­tered Cases and Deaths in the USA

Assuming that the amount of testing in the US has been ramped up day by day since the begin­ning of March and that it is quite resi­lient, I have norma­lized my first column to the first 3 days of last week, i.e. March 23-25. You can see from the yellow high­lighted fields that they show the same values in column 1 and 2: theory and prac­tice are put on the same level. Starting from these three days, I extra­po­lated the projec­tions of the first column into the past and into the future … i.e. I stret­ched the doub­lings per 3 days in the table up and down.

In retro­s­pect, on some days the unreal moment arises that the number of regis­tered cases is higher than the actual sprea­ding speed of the coro­na­virus … which is simply because more tests were carried out and for this reason alone, the number of regis­tra­tions increased. However, it is much more important to look into the future, i.e. the ques­tion: is there a gap between projected cases and actual cases? Or, to put it another way: are the measures ordered by poli­ti­cians taking effect?

The good news is that the growth rate of regis­tered cases is also decli­ning in the US, from 2.0 in the last 3 days, to 1.9 and yesterday to 1.8. Considering that the amount of testing conti­nues to increase, the dece­le­ra­tion of infec­tions could even be below 1.8.

Yesterday’s coro­na­virus curves for US, predicted deaths vs. confirmed deaths, March 28, 2020

The trend in deaths in the US is similar to that in Germany, but with a growth rate of 1.7 it is two tenths higher than in Germany (1.5).

Author’s note: The above values are purely specu­la­tive esti­ma­tions using simple mathe­ma­tical model­ling (based on regis­tered cases/deaths) and are not confirmed by health autho­ri­ties nor any other national public autho­rity.


Daily Covid-19 Stats Nº 19

Today’s coro­na­virus spreadsheet: regis­tered Cases and Deaths in Germany and Berlin

[Preliminary remark: I am curr­ently working on a statistic for the Covid-19 situa­tion in the USA, which has worsened drama­ti­cally in the last four days. Why this is so, I hope to be able to answer on Sunday morning.]

End of Day Summary (19). Angela Merkel announced today in an audio podcast for the first time when the reduc­tion of precau­tio­nary measures could begin: If it is fore­see­able that the number of regis­tered cases will double in 10 days; curr­ently the number doubles in 5 days (e.g.: yesterday, 27 March 49k, 22 March 25k).

Today the number of regis­tered cases in Germany has increased by 7,163, unfor­tu­n­a­tely with an unch­anged high growth rate for 5 days: ×1.5 per 3 days. The number of deaths is increa­sing faster than the number of cases, but with a lower growth rate than yesterday, ×1.5 instead of ×1.6 per 2 days. In Berlin we recorded the 9th death and 200 new cases. The values have flat­tened out, but this is rather due to the fact that the autho­ri­ties have delayed repor­ting over the weekend.

Today’s coro­na­virus curves for Germany, predicted cases vs. confirmed cases, March 28, 2020

The curve of confirmed cases (blue) is flatter than the statis­tical predic­tion (gray), but the growth rate did not change. Today’s news announced that about 1200 pati­ents are in inten­sive care in Germany. There are curr­ently 6,000 ICUs avail­able, which means that our health system reaches its limits above 100,000 regis­tered cases … but this barrier must still be treated with caution. Some hospi­tals are already over­loaded.

Today’s coro­na­virus curves for Germany, predicted deaths vs. confirmed deaths, March 28, 2020

The number of deaths is still rising dyna­mi­cally, the growth lowered a bit, from ×1.6 to ×1.5

Author’s note: The above values are purely specu­la­tive esti­ma­tions using simple mathe­ma­tical model­ling (based on regis­tered cases/deaths) and are not confirmed by health autho­ri­ties nor any other national public autho­rity.


Daily Covid-19 Stats Nº 18

Today’s coro­na­virus spreadsheet: regis­tered Cases and Deaths in Germany and Berlin today

[Preliminary remark: I am curr­ently working on a statistic for the Covid-19 situa­tion in the USA, which has worsened drama­ti­cally in the last three days. Why this is so, I hope to be able to answer on Sunday morning.]

End of Day Summary (18). The number of regis­tered cases in Germany has increased today by 5,809, unfor­tu­n­a­tely with an unch­anged high growth rate for 5 days: ×1.5 per 3 days. The number of deaths is increa­sing faster than the number of cases, but with a lower growth rate than yesterday, ×1.5 instead of ×1.6 per 2 days. In Berlin we recorded the 8th death, which means that the reality corre­sponds exactly with the projec­tion.

Today’s coro­na­virus curves for Germany, predicted cases vs. confirmed cases, March 27, 2020

The curve of confirmed cases (blue) is flatter than the statis­tical predic­tion (gray), but the growth rate did not change. Today’s news announced that about 1200 pati­ents are in inten­sive care in Germany. There are curr­ently 6,000 ICUs avail­able, which means that our health system reaches its limits above 100,000 regis­tered cases … but this barrier must still be treated with caution. Some hospi­tals are already over­loaded.

Today’s coro­na­virus curves for Germany, predicted deaths vs. confirmed deaths, March 27, 2020

The number of deaths is still rising dyna­mi­cally, the growth is unch­anged: ×1.6.

Author’s note: The above values are purely specu­la­tive esti­ma­tions using simple mathe­ma­tical model­ling (based on regis­tered cases/deaths) and are not confirmed by health autho­ri­ties nor any other national public autho­rity.


Daily Covid-19 Stats Nº 17

Today’s coro­na­virus spreadsheet: regis­tered Cases and Deaths in Germany and Berlin today

End of Day Summary (17). The number of regis­tered cases in Germany has increased today by 5,888, with a growth rate iden­tical to that of the last 3 days (×1.5 per 3 days). The number of deaths is increa­sing faster than the number of cases, but with a growth rate than yesterday, ×1.6 instead of ×1.7 per 2 days. In Berlin we recorded the 5th death, which means that the reality corre­sponds exactly with the projec­tion.

Today’s coro­na­virus curves for Germany, predicted cases vs. confirmed cases, March 26, 2020

The curve of confirmed cases (blue) is flatter than the statis­tical predic­tion (gray), but the growth rate did not change. Today’s news announced that about 1200 pati­ents are in inten­sive care in Germany. There are curr­ently 5,000 ICUs avail­able, which means that our health system reaches its limits above 100,000 regis­tered cases … but this barrier must still be treated with caution.

The number of deaths is still rising dyna­mi­cally, but today the growth rate reduced by 0.1 point from ×1.7 to ×1.6.

Author’s note: The above values are purely specu­la­tive esti­ma­tions using simple mathe­ma­tical model­ling (based on regis­tered cases/deaths) and are not confirmed by health autho­ri­ties nor any other national public autho­rity.


Some Interactive Coronavirus Maps

(Abbildung: © Kurzgesagt, via Our World in Data)

The media focuses on events. The news of print and tele­vi­sion are all about moments and not able to show how our world changes, how trends change. However, trends are essen­tial, espe­ci­ally with the sprea­ding coro­na­virus pandemic. We all want to know when we can meet in person again, when work and leisure no longer need to be restricted.

The excel­lent stats-website Our World in Data from the Oxford Martin School focuses on slow but long-lasting deve­lop­ments. Their mission: to make data and rese­arch on the world’s largest problems under­stand­able and acces­sible. And so it is no surprise that Our World in Data provides very helpful info graphics on the corona crisis. And they are well desi­gned. And free. Some of them are inter­ac­tive, are updated on a daily basis and can even be embedded, which I have done below.

1. Total confirmed Covid-19 cases

The follo­wing visua­li­sa­tion shows the number of total confirmed cases (in abso­lute numbers) and daily new confirmed cases for all coun­tries that report their figures. The chart is inter­ac­tive. The data is shown as the world­wide figures by default but can be explored by country: just click on ⊕ Add Country within the chart. The date here reflects to the date of repor­ting, not necessa­rily the confirmed case figures on that given day.

2. Total confirmed Covid-19 deaths

Here’s a similar visua­li­sa­tion like above, showing the number of total confirmed deaths abso­lute numbers). Also this chart is inter­ac­tive and can be explored by country: just click on ⊕ Add Country within the chart.

3. Trajectories since the 100th confirmed case

The next chart answers ques­tions like: Did the number of confirmed cases rise faster in China, Italy, South Korea, or the US? The star­ting point for each country is the day that parti­cular country had reached 100 confirmed cases. The grey lines show trajec­to­ries for a doub­ling time of 2 days and a doub­ling time of 3 days. Countries that follow a steeper rise have seen a doub­ling time faster than that.

This visua­li­sa­tion is also inter­ac­tive, so make sure to play with the region selector in the upper right corner.

4. Healthcare capa­city: Medical doctors, beds

To respond to the pandemic, the capa­city of the health­care system if of great impor­t­ance. The follo­wing two maps show the number of medical doctors and hospital beds rela­tive to the size of each country’s popu­la­tion.

5. The current case fata­lity rate of Covid-19

The case fata­lity rate (CFR) can help us under­stand more about the seve­rity of the disease, and how best to respond. There is no single figure of CFR for any parti­cular disease, because CFR varies by loca­tion, and is chan­ging over time.

The follo­wing chart shows the CFR for coun­tries which have more than 100 confirmed cases, because CFR is a parti­cu­larly poor metric to under­stand morta­lity risk with a small sample size. We see this if we look at the trajec­tory of cases and deaths in Iran: on February 24th it had 2 confirmed cases and 2 deaths, which would have a CFR of 100%. With time its CFR begins to fall as the number of confirmed cases increases, but it’s not until it reaches hund­reds of cases that the CFR falls below 20%.

Also in this inter­ac­tive graphic the data can be explored by country: just click on ⊕ Add Country.