Stochastic SIR model

The stochastic SIR model is a continuous-time Markov chain with infection jump $(-1,1,0)$ at rate $\beta SI$ and recovery jump $(0,-1,1)$ at rate $\gamma I$. The state space consists of nonnegative integer compartment counts with conserved total population.

An epidemic reaches an absorbing state when no infected individuals remain and there is no external source of infection. Extinction time differs from a simulation horizon at which infected individuals are still present.

With one initial infectious individual, two events require an infection as the first event. If it occurs at time $s$, its density is $\beta N{e}^{-(\beta N+\gamma )s}$ and the next total event rate is $2\beta (N-1)+2\gamma$. Integrating this density times the chance of another event before a fixed horizon gives the two-event probability.

A chain-binomial epidemic model uses discrete time, with conditionally independent binomial distributions for infection and recovery counts given the current compartments. It requires valid probability parameters; replacing $1-e^{-\beta I\delta t}$ by $\beta I\delta t$ is a small-step approximation.