SIR Compartmental models

The SIR model is the classic compartmental model of infectious-disease spread, dividing a population into Susceptible, Infected and Recovered groups. Susceptible individuals become infected at a rate set by the transmission coefficient β through contact with the infected, and infected individuals recover at rate γ.

Despite its simplicity, the model captures the characteristic epidemic curve and the threshold behaviour governed by the basic reproduction number.

\begin{align*}
  \frac{d s}{dt} &= - beta \cdot i \cdot s \\ 
  \frac{d i}{dt} &= beta \cdot i \cdot s - gamma \cdot i \\ 
  \frac{d r}{dt} &= gamma \cdot i
\end{align*}