Research | WIAS | 17-12-2020

Understanding the crisis with mathematics

During the Corona crisis the whole world is suddenly looking not only at the research results of virologists, but also at those of mathematicians: how fast is the virus spreading through the population? What are the routes of transmission of the infection? And when can we go back to the stadium? Mathematicians at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) have also been looking into the pandemic.

Photo: © Barbaroga -

In March, the coronavirus pandemic in Germany suddenly became the center of media attention. Discussions flared up about exponential growth, with terms like “doubling time” and reproduction number” being thrown around. The whole of society was calling to “flatten the curve,” and mathematics was suddenly a part of the daily news.

In order to gain a clearer picture of the current events, Dr. Markus Kantner of WIAS spent his vacation time working intensively on the mathematical modeling of epidemics. The public was increasingly arguing the merits of all kinds of intervention strategies based on simulations against a backdrop of TV images from Northern Italy, the debate about controlled infection to achieve “herd immunity” of the population in Great Britain, and the uncertain prospect of a vaccine being developed any time soon. Kantner reports, “There were numerous model calculations in which the influence of special measures was analyzed: What will happen if we close the schools? Or the shops?” The simulations showed that, while the contact reduction measures could slow the infection process down, they would always be followed by a second wave once they ended. The mathematician therefore wanted a better answer to the question “what is our best course of action?”

Kantner applied methods from optimal control theory to an epidemiological model tailored to specific aspects of Covid-19. The task was to calculate the optimal time curve for mean contact reduction that would, on the one hand, minimize the number of deaths and, on the other hand, rule out a second wave once the intervention period ended. Furthermore, the socioeconomic costs of the intervention should be kept as low as possible (see box).

The mathematician collected his results into a manuscript and discussed them with WIAS colleague Dr. Thomas Koprucki via video conference. Kantner reports, “I hardly had to explain anything. Thomas was already fully involved in the topic.” Koprucki had been following the current publications and modeling activities of other research groups on the topic, and could immediately relate to his colleague’s manuscript. Together, they continued analyzing the results and completed the manuscript from their respective home offices.

Does that mean the mathematicians have now developed a proposal on how best to control the course of the pandemic? Kantner responds: “It’s not that simple. Our results are based on certain model assumptions. In the case of a macroscopic course of infection, our path is optimal but, at the same time, it is either extremely expensive or extremely dangerous. So, ‘optimal’ does not necessarily mean ‘good’.” Instead, the mathematicians suggest continuing to contain the virus until a state has been reached in which the infection event is determined by individual cases and stochastic fluctuations. Koprucki stresses, “We should get to a point where we can track down individual chains of infection and targetedly break them.” Only a manageable portion of the population would be affected by this intervention, while life would be largely normal for the vast majority. Koprucki adds, “In this way, our results support the position of the major research organisations and scientific societies in Germany.”

Text: Gesine Wiemer
Translation: Peter Gregg