Dynamical behavior and optimal control of a stochastic SAIRS epidemic model with two saturated incidences

Stochastic models are widely used to investigate the spread of epidemics in a complex environment. This paper extends a deterministic SAIRS epidemic model to a stochastic case with limited patient capacity and exposure. We first study the dynamical properties of the model under certain conditions, including persistence, extinction, and ergodic. Then, we introduce vaccination and isolation into the model as control variables. The optimal control strategies are obtained based on the Pontryagin minimum principle. Finally, numerical simulations are given to illustrate our theoretical results.