Stochastic SIS Model — Endemic Persistence

Stochastic SIS model: Each individual is Susceptible or Infectious. Transitions: S→I at rate βIS/N, I→S at rate γI. The ODE predicts an endemic state at I* = N(1−1/R₀) for R₀ = βN/γ > 1, but the stochastic system always goes extinct — the disease-free state (I=0) is absorbing.

Quasi-stationary distribution (QSD): Conditioned on non-extinction, the system fluctuates around I* with approximately Gaussian distribution (width ~ √N). The mean extinction time scales exponentially: τ_ext ~ exp(N·f(R₀)), so for large N the endemic state persists quasi-permanently. Below threshold (R₀ < 1), τ_ext ~ O(log N) — extinction is rapid.