Stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion

The objective of this paper is to study stationary distribution of a stochastic SIQR epidemic model with saturated incidence and degenerate diffusion. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory will be used to obtain the existence of a unique stable stationary distribution. A threshold dynamic determined by the basic reproduction number R0s is established: the disease can be eradicated almost surely if R0s<1, whereas if R0s>1, it has an endemic stationary distribution which leads to the stochastic persistence of the disease.