In this paper, we formulate a stochastic predator-prey model with Holling III type functional response and infectious predator. By constructing Lyapunov functions, we prove the global existence and uniqueness of the positive solution of the model, and establish the ergodic stationary distribution of the positive solution, which indicates that both the prey and predator will coexist for a long time. We also obtain sufficient conditions for the extinction of the predator and prey population. We finally provide numerical simulations to demonstrate our main results.

In this paper, we represented the optimal control and dynamics of a stochastic SEIR epidemic model with nonlinear incidence and treatment rate. By using the Lyapunov function method, the existence and uniqueness of the global positive solution of the model were proved. The dynamic analysis of the stochastic model was studied and we found that the model has an ergodic stationary distribution when $R_0^s>1$. The disease was extinct when $R_0^e<1$. The optimal solution of the disease was obtained by using the stochastic control theory. The numerical simulation of our conclusion was carried out. The results showed that the disease decreased with the increase of the control variables.