Stochastic SIS Epidemic: Master Equation & Extinction

Finite-size fluctuations drive stochastic extinction even above the deterministic threshold — discrete Gillespie SIS vs. ODE

Parameters

100
0.30
0.20
10
5
R₀ = β/γ = 1.50  |  Threshold: R₀ > 1 (above)  |  Endemic level I* = —  |  Time: 0  |  Extinctions: 0
Stochastic SIS Model

States: S (susceptible) + I (infected) = N (fixed). Transitions:
• Infection: I → I+1 at rate β·I·S/N (mass-action)
• Recovery: I → I-1 at rate γ·I

Deterministic ODE: dI/dt = βI(N−I)/N − γI → endemic equilibrium I* = N(1−γ/β) for R₀=β/γ > 1.

Stochastic fluctuations cause absorption at I=0 (extinction) even for R₀ > 1. The quasi-stationary distribution (QSD) has a peak near I* with fluctuations ∝ 1/√N. Mean time to extinction: T_ext ~ exp(N·I(ρ)) where I(ρ) is a large-deviation rate function.

Left histogram: stationary distribution of I (aggregated from all trajectories after burnin). The absorbing state (I=0) accumulates probability over time — extinction is inevitable but can take exponentially long.