In the next example we consider a model of infection transmission within a closed popluation (no births or deaths): the susceptible-infected-susceptible (SIS) model.
In this model we consider two types of event: Infection events, which occur in each time step with probability \(((\beta I dt)/N\)\) per susceptible individual, and Recovery events, which occur in each time step with a probability \((\sigma dt\)\) per infected individual.
The ODE representation of the model is:
\[\frac{dS}{dt}=-\beta \frac{SI}{N}+\sigma I\] \[\frac{dI}{dt}=\beta \frac{SI}{N}-\sigma I\]
The analytical solution to this ODE can be shown to be:
\[I(t)=I^* \frac{1}{1+(I^*/I_0 -1)\exp(-(\beta -\sigma )t)}\]
where \(I_0\) is the initial number of infected individuals and \(I^*=N(1-\sigma / \beta )\). After sufficient time, the infection and recovery processes balance each other out. This occurs when the number of infectious individuals reaches its equilibrium value \(I^*\). We can write the equilibrium value as
\[I^*= N \left(1-\frac{1}{R_0} \right)\]
since \(R_0\) is defined as \(\beta ⁄\sigma \). At this point, we say that the disease is endemic in the population.