by Janina Schöneberger

This explorable illustrates the dynamics of the SIRS epidemic model, a generic model that captures disease dynamics in a populations or related contagion phenomena.

The Model:

Susceptible individuals (S) can be infected by coming in contact with other infected (I) individuals. Once infected they can transmit the disease until they recover (R) and become immune. After some time immunity wanes and individuals become susceptible again.

The system can therefore be described by three reactions:

[\begin{align} S + I & \rightarrow 2 I\newline I & \rightarrow R\newline R & \rightarrow S \end{align}]

corresponding to the transmission of the disease, recovery to the immune state and losing immunity, respectively. Each of these reactions occurs at a corresponding rate: the disease transmission rate, the recovery rate, and the waning immunity rate, respectively.

The Simulation:

The simulation starts with a fully suscpetible population of 400 individuals and a seed of a few infected individuals. The infectious state is encoded by color. With the sliders you can change the magnitude of the transmission, recovery and waning immunity rate. An infected individual can transmit the disease to susceptibles within a small radius. Recovery and loss of immunity occur spontaneously at the specified rate.

When you press play, an epidemic will spread through the population. The system will go into a quasi-equilibrium state in which new infections and the supply of new susceptibles by waning immunity balance. The equilibrium state depends on the rate parameters: When the transmission rate or the waning immunity rate are too low, the disease will die out. Likewise if the recovery rate is too high the epidemic cannot be sustained.


When we consider more than one population we can model the effect of movements between sub-populations. In the simulation you can chose 1,2 or 4 sub-populations and vary the migration rate. For some parameter values, the epidemic may die out in isolated sub-populations. These subpopulations are particulary susceptible to the import of infections.