The Reaction–Diffusion Stochastic Susceptible‐Infected‐Vaccinated Epidemic Model With Nonlocal Delay

ABSTRACT In this paper, a stochastic reaction–diffusion susceptible‐infected‐vaccinated (SIV) epidemic model with nonlocal delay is proposed. Using the Banach fixed point theorem combined with a truncation method, we establish the existence and uniqueness of positive mild solutions. The long‐term behavior of the system is characterized by a real‐valued parameter . Specifically, if , the disease will become extinct; if , the system has an invariant probability measure. Furthermore, the mean square exponential stability was proved by using the Gronwall inequality, and the critical delay that guarantees the stability of the mild solution was derived. Finally, numerical examples are given to illustrate the theoretical results.