Dynamic analysis of a stochastic SVEIR epidemic model with saturated incidence and partial vaccination

In this paper, a stochastic SVEIR epidemic model with saturated incidence and partial immunity is proposed. First, we define the basic reproduction number R-0 and study the global asymptotic stability of disease-free equilibrium and endemic equilibrium of the deterministic epidemic model. Then we establish the existence and uniqueness of global positive solution, along with delineating sufficient conditions for disease extinction. By constructing appropriate Lyapunov functions, we analyze the asymptotic behavior of solutions to the stochastic epidemic model around the disease-free equilibrium and endemic equilibrium points of the deterministic epidemic model. Under certain conditions, the solution of the stochastic model fluctuates around the disease-free equilibrium point and the endemic equilibrium point, with the intensity of fluctuation proportional to the intensity of white noise. The existence of ergodic stationary distribution is proved by the Khasminskii method. Finally, numerical simulations are presented to illustrate our analysis results.