A computational-analytic framework for assessing stationarity and ergodicity for a stochastic SIS epidemic model

This paper presents a stochastic $\mathcal{SIS}$ (Susceptible-Infected-Susceptible) epidemic model with generalized incidence function and two separate Brownian noise sources. Using analytical methods from stochastic calculus, we derive explicit threshold conditions governing disease extinction and endemic persistence. These theoretical results are validated through systematic numerical simulations, including phase-space trajectory analysis under varying epidemiological scenarios. A central contribution is the development of a novel computational-analytic framework to rigorously verify the model’s long-term statistical properties: (i) stationarity through distributional convergence tests, and (ii) ergodicity via empirical moment estimation. The combined analytical-numerical approach provides actionable insights for epidemic modeling while advancing methodology for stochastic dynamical systems.