  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.
