The More Drastic the Disease, the Sooner it Declines

Since the 1930s researchers have been able to track and predict the spread of epidemics using mathematical models. Prof. Dr. Jens Timmer’s work group at the University of Freiburg’s Institute of Physics has calculated how SARS-CoV-2, better known as corona virus, will develop over the coming weeks.

For almost 90 years computational models have been providing insights into the spread of epidemics. Photo: Romolo Tavani/stock.adobe.com

When do epidemics subside? How quickly can people with the infection pass it on to others? Researchers are attempting to answer questions like this with mathematics, or more precisely differential equations. For almost 90 years now such computational models have been providing insights into epidemics, now they are also being used on the corona virus. The simplest scientific models for analyzing the course of a disease assumes that there are three populations, explains Jens Timmer: healthy individuals who could become infected; infected individuals who could infect the healthy ones; and finally those who have either died or recovered from the infection and are now immune to it. In models based on this assumption, the researchers also consider incubation periods and the possibility of reinfection after recovery.

Equations describe the trends over time

Timmer’s research focuses on system biology. This represents biological processes using mathematical formulae. For instance, working with doctors in the LiSyM research network, the Freiburg physicist decoded the signal paths of liver cells. On account of the current spread of the new corona virus, Timmer together with his PhD students Lukas Refisch, Marcus Rosenblatt and Christian Tönsing are translating the classification of individuals for diseases into a mathematical model of differential equations — one for each of the three populations. With these equations the researchers describe how the current epidemic will progress over time.

Ebola less dangerous than AIDS

“Analysis of these equations provides initial insights for every disease,” says Timmer: an epidemic ends not because of the lack of healthy people who could become infected, but because of the lack of infected individuals to pass it on. “This brings us to the conclusion: a local infection has a that much smaller chance of turning into a pandemic – in other words a global epidemic – the faster it develops fatally. Or – looked at positively – the faster those who are infected get well and gain immunity.” This is why in global terms the Ebola virus was less dangerous than the HIV which triggers AIDS, explains Timmer, as people can be free of symptoms with the latter for years, but still infectious.

Exponential progress of corona virus

The mathematical model has two parameters. The first one describes how efficiently those who are already infected can pass it on to healthy people – this is where those all-important hygiene and quarantine measures come in. The second describes how quickly those who are infected no longer need to be included in the calculations, because of death or immunization. “The relationship of these two parameters determines the level of infection in the population as a whole.” Graphically, the daily number of new cases and of those who have died and are immunized after recovery takes a bell-shape, reflecting the exponential growth at the start. In this phase it is impossible to conclude how severe the epidemic will be in the end.

“Over the course of the corona virus in China it became clear that the bell-shaped pattern of daily fatalities has passed well beyond its peak,” Timmer sums up, “so the worst is over.” The figures and models indicate that the incubation period is about one week and the death rate is roughly three per cent of those infected and mostly affects people aged 70 and over with a weakened immune system. In Europe, where the epidemic broke out about a month later than in China, the number of people infected is still in the phase of exponential growth. “Therefore it’s still not possible to make reliable statements about future developments,” Timmer says. One recommendation can however already be taken from the mathematical models: “Calculations made by American colleagues about the epidemic in Wuhan suggest that a quarantine of two weeks still isn’t enough. The models recommend a quarantine period of three weeks.”

Annette Kollefrath-Persch

Information from the University of Freiburg on the corona virus