Example 3: Stochastic SIS model

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 \((\nu dt\)\) per infected individual.

The ODE representation of the model is:

\[\frac{dS}{dt}=-\beta \frac{SI}{N}+\nu I\] \[\frac{dI}{dt}=\beta \frac{SI}{N}-\nu I\]

The analytical solution to this ODE can be shown to be:

\[I(t)=I^* \frac{1}{1+(I^*/I_0 -1)\exp(-(\beta -\nu )t)}\]

where \(I_0\) is the initial number of infected individuals and \(I^*=N(1-\nu / \beta )\). 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 \(\beta ⁄\nu \). At this point, we say that the disease is endemic in the population.

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