Threshold dynamics of a two-patch SIS epidemic model with stochastic population movement

This paper investigates the influence of inter-regional population movement on the transmission of infectious diseases through the development of a two-patch SIS epidemic model incorporating stochastic mobility. The theoretical analysis reveals that the long-term behavior of the disease, whether it dies out or persists, is fully characterized by the Lyapunov exponent, $ \lambda $, which quantifies the asymptotic growth rate of the infected population. Specifically, the disease will eventually vanish if $ \lambda < 0 $, whereas it will persist with a unique stationary distribution when $ \lambda > 0 $. A significant methodological challenge arises in establishing the weak convergence of the invariant measure from the stochastic system to its associated two-dimensional boundary system. Owing to the complexity inherent in the boundary system, obtaining the density function of its invariant measure via the Fokker–Planck equation proves to be highly nontrivial. This issue is overcome by employing limiting measures in conjunction with Prokhorov's theorem, thereby offering fresh insights and robust methodologies. Complementary numerical simulations further elucidate the effects of both population movement and environmental fluctuations on disease transmission dynamics.