Dynamics of a two-patch epidemic model with deterministic/stochastic migration and distributed delays

We study the global dynamics of a two-patch epidemic model that integrates spatial migration, Erlang-distributed delays, and environmental stochasticity. The two patches are coupled by migration whose intensities are modulated by an Ornstein-Uhlenbeck (OU) process, capturing stochasticity but mean-reverting fluctuations in population movement, while the infection progression is described by a distributed delay structure represented through a linear-chain formulation. For the deterministic system, we establish threshold-type results that characterize disease extinction and persistence: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number $R_0<1$ , whereas infection persists when $R_0>1$ . For the corresponding stochastic model, we derive explicit sufficient conditions for almost sure exponential extinction by constructing Lyapunov functions and exploiting the Metzler structure of the infected subsystem. Moreover, we prove that the stochastic system admits a stationary distribution when stochastic threshold $R_0^s>1$ , which serves as a stochastic analogue of the endemic equilibrium and describes persistent random fluctuations of infection levels. Numerical simulations are provided to validate the theoretical findings and to quantify how distributed delays and OU-driven migration randomness shape long-term infection burden and stationary prevalence across patches. Notably, adjusting the noise intensity and the mean-reversion rate can redistribute the long-term infection burden between patches, which highlights how migration-driven randomness fundamentally reshapes spatial epidemic patterns.