Stochastic Epidemic — Quasi-Stationary Distribution

The stochastic SIS model on a finite population of N individuals has a crucial difference from its deterministic counterpart: the disease-free state I=0 is an absorbing state — once reached, the epidemic ends forever. For R₀ > 1, the deterministic model has a stable endemic equilibrium at I* = N(1 − 1/R₀). But stochastically, the epidemic will eventually go extinct (in expected time ~ exp(αN) for some α > 0). Before extinction, the system is metastable, fluctuating around the quasi-stationary distribution (QSD) — the conditional distribution given non-extinction. The QSD peaks near I* with width ~ √N. When N is small, extinction is fast; large N makes QSD long-lived. This demonstrates how demographic stochasticity fundamentally changes epidemic dynamics.