 Compartmental models (epidemiology)

Compartmental models divide a biological population into states and specify transfers between them. A compartment might contain susceptible individuals, infective individuals or recovered individuals. The model must state what a compartment represents and whether its variable is a count or a fraction.

For a closed, homogeneously mixed SIR model, let $N=S+I+R$ and use force of infection $\beta I/N$ and recovery rate $\gamma$. ThenEvery transfer leaves one compartment and enters another, so summing the equations gives $\dot{N}=0$. This conservation of population follows from the specified closed model; demographic gains, deaths or external arrivals can change the total. An SEIR model adds an exposed state to distinguish infection from the ability to transmit.

A deterministic ordinary differential equation describes continuous compartment sizes. A stochastic epidemic model instead treats infection and recovery as random events between integer states. The latter can exhibit epidemic extinction even when the corresponding deterministic trajectory initially grows.