Flattening the curve for the Coronavirus

In order to better understand the development of the Coronavirus pandemic, I decided to do my own statistical analysis. I only considered the number of deaths in various countries, as this number is probably more reliable than the number of infections. I then made the assumption that this number grows exponentially, but that the coefficient of growth changes linearly with time:

Simple model for the number of deaths N(t). The exponent α(t) is assumed to change linearly with time.

It is now possible to make a simple fit for each country and to estimate when the curve will flatten and the number of deaths will stop increasing.

I was surprised how well it worked. Below is the result for Italy

Number of deaths from the Coronavirus in Italy

The analogous plot for Switzerland looks as follows

Number of deaths from the Coronavirus in Switzerland

Unfortunately, the United States is not even close to flattening the curve

Number of deaths from the Coronavirus in the United States

It is now possible to compare different countries both in terms of actual deaths

or in terms of the fitted curves

The curves tell us an important message: If strict measures are implemented to stop the spread of the virus, this actually helps. It is the only thing we can do at the moment.

As public service, I have made the Python script for the analysis available. It automatically downloads the data from the Center for Humanitarian Data, performs the analysis and plots the data.

Here is a scientific study of the efficacy of different measures from Imperial College: Estimating the number of infections and the impact of nonpharmaceutical interventions on COVID-19 in 11 European countries.

Here is a useful tool for a more detailed analysis of the disease from the University of Basel: https://neherlab.org/covid19/

BTW, there is a good reason be be wary of exponential growth, as this delightful video shows:

When to Pull the Emergency Brake?

Important announcement: COVID-19 is a deadly and extremely contagious disease, which will kill hundreds of millions of people unless we manage suppress it. Please cooperate with the authorities, follow the instructions from health professionals, and avoid unnecessary contacts with other people. Social distancing is important!

The actions currently being taken worldwide to stop the spread of the coronavirus can best be compared to pulling the emergency brake on the world economy. Interestingly, the same governments that refused to sacrifice even the tiniest amount of economic growth to save the environment, are suddenly prepared to implement a complete lockdown of public life, costing trillions of dollars and causing a global recession. It is important to understand why.

The argument for implementing drastic measures against the coronavirus is compelling: If we fail to stop the spread of the virus early, we will lose control of the situation and there will be nothing left to decide. Even if stepping on the brake now will not necessarily solve the problem, it might at least buy us some time to think about what to do next. Currently, it is not clear whether the spread of the virus can be contained.

The same argument can be made for stopping global warming and environmental destruction. If we wait until the polar caps have melted and a large amount of species have gone extinct, there will be very little left to decide. At that point, we will run out of sensible options and will simply be struggling to survive by reacting to disasters as best as we can. It would make a lot more sense to act now.

One important difference between the coronavirus and global warming is the time scale. People are dying from the virus today and the full extent of the disaster will become obvious within a couple of months. This makes the problem politically relevant. With climate change, most people have the feeling that the problem is not urgent and that we still have plenty of time to act.

Bengt Ekerot and Max von Sydow in The Seventh Seal by Ingmar Bergman

Unfortunately, I believe that there is a more fundamental reason for our different reactions to climate change and the coronavirus: a primeval fear of death. Most people are concerned about the future of their children, but they are simply terrified by the idea of contracting a deadly virus. It is difficult to think rationally in the face of death.

The global pandemic caused by the coronavirus will be over in 1-2 years. Despite the current drop in greenhouse gas emissions, the struggle to stop climate change and to decarbonize society will have to continue. Whether we like it or not, society’s response to Covid-19 offers some important lessons for the future:

  • People a mainly motivated by fear.
  • Drastic political measures are possible in the case of an emergency.
  • If scientific models give you early warning of an impending disaster, it makes sense to act quickly and decisively.
  • Never listen to economists when they talk about global challenges.

Less than two weeks ago, some economists were still warning against spending too much money on fighting Covid-19. Apart from the fact that no economist has ever understood exponential growth, they always seem to overestimate their ability to predict the economic consequences of disruptive developments. It simply does not make sense to extrapolate if the entire society sinks into chaos and anarchy.

Unfortunately, this phenomenon is all too familiar in climate politics. When William Nordhaus claims to be able to predict the economic consequences of 4°C of global warming, he is either mad or almost criminally ignorant. We all know that the uncertainties in such a prediction are large enough to render it meaningless. Nonetheless, it serves as a convenient excuse for inaction.

The economy emerging from the coronavirus pandemic will have to be much more sustainable and resilient than the one we have today. This will not be possible if we only focus on maximizing GDP.

PS: If you really want to understand exponential growth, here is an excellent cartoon.

Leben in Zeiten des Coronavirus – danke liebe Jugend.

Gestern erreichte mich ein Aufruf zur Nachbarschaftshilfe, der mich sehr berührt hat. Da im Moment keine Klimastreiks oder andere Aktionen durchgeführt werden können, überlegen sich die jungen Klimastreikenden, wie sie den Alten und Schwachen der Gesellschaft helfen können. Das Besondere mit dem Coronavirus ist ja, dass es für junge gesunde Menschen praktisch keine Gefahr darstellt. Somit macht es Sinn, wenn sie z. B. die Einkäufe der Alten erledigen.

Ist dies selbstverständlich? Diejenigen, die von der Klimaerwärmung am stärksten betroffen sein werden, zeigen ihre Solidarität mit der Generation, die das Problem verursacht hat. Der Homo oeconomicus würde anders handeln und das Coronavirus als Chance sehen, sich Vorteile zu verschaffen oder die Politik zu erpressen. Die Jungen sind aber nicht zynisch genug, um dies zu tun.

Es gibt eine interessante Parallele zwischen der Klimaerwärmung und der Verbreitung des Coronavirus: Politisches Zögern ist in beiden Fällen tödlich, wie Elizabeth Sawin in einem Tweet schön gezeigt hat.


Der Grund dafür ist, dass beide Prozesse eine Eigendynamik besitzen. Bei der Verbreitung einer Grippe ist die Anzahl Neuansteckungen proportional zur Anzahl der Kranken (exponentielles Wachstum) aber der Wachstumskoeffizient ist davon abhängig, wie viele Menschen noch angesteckt werden können. Wenn eine kranke Person im Durchschnitt 1.3 andere Personen ansteckt, fangen die Fallzahlen dann an zu sinken, wenn 25% der Bevölkerung Immunität entwickelt hat, da 1.3×(1-0.25)<1. Wenn stattdessen im Durchschnitt 2.3 Personen angesteckt werden, müssen zuerst 60% der Bevölkerung krank werden, bevor die Ansteckungskurve abflacht.

Das Problem mit dem Coronavirus ist, dass es viel ansteckender und viel tödlicher ist als das normale Grippevirus. Dies führt dazu, dass mehr Menschen krank werden und viele davon sterben. Wenn zu viele Menschen gleichzeitig krank werden, nimmt die Mortalität zusätzlich zu, da die Spitäler nicht mehr in der Lage sind die Kranken zu behandeln. Dieser Zustand ist in Italien schon eingetreten und es ist wohl nur eine Frage der Zeit, bis andere Länder so weit sind. Dies wäre dann der Kipppunkt (tipping point) der Pandemie: Endliche Ressourcen und exponentielles Wachstum vertragen sich schlecht.

Eine sehr gute Zusammenfassung der Gefahren des Coronavirus ist hier zu finden: Coronavirus: Why You Must Act Now. Den Artikel gibt es auch in deutscher Übersetzung.

Ohne drastische Massnahmen gegen das Coronavirus, werden viele alte Menschen sterben. Wenn wir nichts gegen die Klimaerwärmung unternehmen, werden noch mehr junge Menschen sterben. Worin liegt der Unterschied? Nur darin, dass die Alten mehr politisches Gewicht haben und die katastrophalen Wirkungen des Coronavirus innerhalb einer Legislaturperiode sichtbar werden.

Die Coronapandemie wird wohl in 1-2 Jahren vorbei sein. Es ist zu hoffen, dass der Begriff Solidarität etwas länger in Erinnerung bleibt.

Hier ein interessanter Artikel zu unserem Umgang mit Klimaerwärmung und Coronavirus.

Doing what is necessary

Here is a powerful and emotional interview with Christiana Figueres. She makes it perfectly clear that future of humanity lies on our hands and that we must choose whether we want our children and grandchildren to be able to breathe or not.

It has been said before:

It is not enough that we do our best; sometimes we must do what is required.

Winston Churchill

Reality strikes back!

In 480 BC, the Persian King Xerxes I led his huge army across a ponton bridge over the Hellespont to launch his invasion of Greece. According to Herodotus, the first bridge was destroyed by a storm, prompting the king to behead the builders and punish the sea by having his soldiers whip the waves with chains. Powerful leaders have always had problems with accepting the laws of physics and the forces of nature.


This is what makes the recent decision by the Federal Reserve to lower the interest rate so interesting. Rather than accepting that the Coronavirus will slow down the world economy, with a corresponding drop in share prices, a desperate and futile attempt was made to keep Wall Street happy. The mad king did not behead anyone this time, but the actions of the Fed were about as effective as whipping the seas with iron chains.

If you still believe that stock markets are related to the real economy, the recent worldwide drop in share prices is not surprising. After all, many companies – such as airlines, travel companies, and manufacturing companies with global supply chains – will be harmed by a possible global pandemic. The reasons are easy to understand and not very much can be done about it, apart from trying to limit the spread of the virus. In particular, lowering interest rates is not going to solve anything, as the companies suffer from a lack of demand and not from a lack of capital. Predictably, Wall Street reacted to the decision by the Federal Reserve with even more losses. To quote Bloomberg News: “Rate-cut magic doesn’t work on markets adjusting to economic reality.

The problem for the US government is that the stock market is directly linked to the apparent value of people’s retirement savings. President Trump has repeatedly boasted about how he has made people richer by adding trillions to their 401(k)s. To see this fictitious wealth being wiped out by a virus from China is apparently more than he can bear and will probably reduce his chances getting reelected. Acting like a mad dictator, he demanded reality to be changed.

Sadly, the actions of the Federal Reserve also illustrate why it is so difficult to solve the climate crisis. For many decades, voters have been promised the fruits of an eternally growing economy. The assumption was that if we start saving for our retirement early enough and invest the money wisely, the markets will generate enough return on investment so secure our pensions. Nobody seems to have considered the possibility that the stock market could become saturated with capital, i.e. that there would be more capital available than investment opportunities. Once it happens, the financial markets turn into gigantic Ponzi schemes, where useless pieces of paper are traded at ever higher prices. It also means that the money for our pensions is not available anymore.

The failure of the financial markets is only a symptom. The underlying problem is that we are living above our means and that the economy is unable to maintain our high level of material consumption, even if we are prepared to sacrifice the future our children by ignoring environmental costs. There is no alternative to system change, as was again pointed out by Joseph Stiglitz on Project Syndicate recently: Solidarity Now. Climate Change has only made the need for change more urgent.


Here is an in-depth look at the Fed’s response: The Fed’s Baffling Response to the Coronavirus Explained.

The following post by George Monbiot has a similar message: Real Life