Fast-variable elimination in stochastic SIRS epidemic model with Itô processes

Abstract

We investigate the dynamics of the SIRS epidemic model through both deterministic and stochastic frameworks, focusing on the effects of random fluctuations in key epidemiological parameters. After presenting the deterministic SIRS model using differential equations, the model is extended to include stochasticity by incorporating Itô stochastic differential equations (SDEs), where infection, recovery, and loss-of-immunity rates are treated as random variables. The corresponding Fokker–Planck equation is derived, capturing the evolution of the probability distribution of the infected population over time. We perform a linear stability analysis of the deterministic model and apply the fast-variable elimination method to reduce the two-variable SDE system to a single slow variable, leading to a simplified Fokker–Planck equation that describes the stationary distribution of the infected population. This reduction reveals that the stochastic epidemic threshold depends on the variances in the infection and recovery rates, but not on the variability in the loss-of-immunity rate. Simulation results, conducted using a Gillespie-type algorithm, validate the theoretical findings.