Corona with the SIR model

The pandemic explained with the help of System Dynamics

By Nils Harder (BSc Interntional Business), Fenja Indorf (BSc International Business) and Prof. Dr. Florian Kapmeier

Closed schools and universities, closed playgrounds and ghost towns bear no resemblance to what are normally lively city centres... why is social distancing a necessary political instrument in these extraordinary times? What parameters influence the speed and severity of the spread of an epidemic? When do we even speak of an epidemic? What lessons have been learnt after the spread of SARS in 2003, the swine and bird flu and MERS? What tools are available to politicians to slow down or even prevent an epidemic?

These and other questions were asked by BSc International Business students of the fifth semester at ESB Business School, under the guidance of Prof. Dr. Florian Kapmeier in the lecture System Dynamics, in which they investigated complex dynamical phenomena. In an assignment they analyzed the spread of SARS in Taiwan in 2003 for which we developed a System Dynamics simulation model. They then transferred insights from the assignment to the current COVID-19 situation in Germany during the lecture.

The starting point of the analysis is the SEIR model, which is well established among epidemiologists. The model divides people into four groups: S stands for the number of healthy people who are susceptible to the disease (Susceptible contacts), E for infected people who are not yet infectious (Exposed), I for infected people who can pass on the disease (Infected) and R for those who have recovered (Recovered). While the SEIR model explicitly takes into account the incubation period, the intuition of the spread of an epidemic can also be explained using the simplified SIR model.

Characteristic of the SIR model are the three stocks – S, I and R (see figure). How an infectious disease spreads depends on the interaction of the resulting nonlinear feedback loops. The three stocks are linked by the two flows infection rate IR and recovery rate RR and through the three generated feedback loops. 

If there is one infected person at the beginning of the infection, the probability for a susceptible person to come into contact with an infected person increases. Consequently, the number of contacts between susceptible and infected persons increases, which increases the infection rate IR and thus the number of infected persons. We have closed a reinforcing feedback loop (R1 Contagion), which causes the number of infected people to increase exponentially. The infection rate IR is determined by the parameter infectivity i and the number of susceptible contacts, which is determined by the parameter contact frequency c and the contacts between infected and susceptible persons. Note that a higher contact frequency c depletes the stock of Susceptible faster through the balancing feedback loop B1. Thus, the greater the infectivity i and the contact frequency c, the stronger the feedback loop R1 and more people get infected. However, people eventually recover, as they become immune or they die. How quickly people recover (recovery rate RR) depends on the average duration of infectivity d: the lower d, the stronger the recovery rate - and the fewer people can be infected. Here, the balancing loop B2 - Recovery is closed, which, if strong enough, can prevent an epidemic.

In general, an infectious disease develops into an epidemic if the infection rate IR is higher than the recovery rate RR (IR>RR). Multiplying the three parameters contact rate c, infectivity i and average duration of infectivity d determines the basic reproduction number R0 (at the beginning of an epidemic), which is currently frequently reported in the media. If R0 is greater than 1, people become infected more quickly than they can recover (one person infects more than one other person). The epidemic therefore takes its course; it first grows exponentially and then the two balancing feedback loops become stronger that weaken the reinforcing loop until a certain percentage of the population is immune. The higher R0, the more people in a population will be infected. Measles, for example, has an R0 of between 12 and 18.

According to current analysis from the Robert Koch Institute, the baseline reproduction number R0 of COVID-19 was between 2.4 and 3.3 in the beginning of the epidemic. In the meantime, the measures taken have made it possible to reduce R0 in Germany significantly; a value of less than 1 is necessary to stop the epidemic. From our analysis we inferred the following for the situation on COVID-19: 1) the probability of an infection on contact is high due to the relatively easy transmission of the virus' due to its nesting in the throat (there is no vaccine yet that can reduce i). 2) the average duration of infectivity of 14 days is also relatively high (there are no drugs yet to reduce d). 3) the only one of the three parameters that we can currently influence is the contact rate c. Therefore, politicians all over the world are trying to reduce this parameter in the best possible way.

Different political approaches are taken all over the world, depending on the political system and the means of control, in containing the spread. China has decided on strict isolation of the infected people and high-risk areas, analyses movement profiles and carries out fever measurements by drones. In view of the technically very affine population, South Korea relies on monitoring the quarantine by digital means. In Germany, the first weeks of contact quarantine are over and, in the meantime, (due to time delays) we observe a slight flattening of the infection curve. However, government representatives agree that our individual behaviour is crucial for the outcome of the COVID-19 epidemic. The broader and more complete our actions are, the greater is our impact on containing the spread of the virus. Through physical distancing and adherence to safe practices, we can drastically affect the outcome of the epidemic, gain valuable time, not overloading the health-care system and save many lives. Considering the high incubation period and number of asymptomatic infected persons, every day, especially at the beginning of an emerging epidemic, is of great importance to reduce the dominance of the reinforcing feedback loop and the infection rate, so that an exponential increase can be prevented.

But what can we expect once the measures are relaxed? Is a second wave likely? How can we return to everyday life? The students discussed these and other questions during an online guest lecture with Prof. Jeroen Struben, PhD, of emlyon business school in France. He developed a more detailed Covid-19 simulation model for decision makers. Those interested may want to explore the effects of different policies for themselves.