Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion

In this paper, we study a stochastic SIR epidemic model with distributed delay and degenerate diffusion. Firstly, we transform the stochastic model into an equivalent system which contains three equations. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We verify the densities of the distributions of the solutions can converge in L¹ to an invariant density. Then we establish sufficient conditions for extinction of the disease. Some examples and numerical simulations are introduced to illustrate our analytical results.