In this paper, we study the dynamical behaviors of a SVI epidemic model with half saturated incidence rate. Firstly, the local asymptotic stability of the endemic and disease-free equilibria of the deterministic model are studied. Then for stochastic model, we show that there is a critical value $R_0^s$ which can determine the extinction and the persistence in the mean of the disease. Furthermore, by constructing a series of suitable Lyapunov functions, we prove that if $R_0^s>1,$ then there exists an ergodic stationary distribution to the stochastic SVI model. It is worth mentioning that we obtain an exact expression of the probability density function of the stochastic model around the unique endemic equilibrium of the deterministic system by solving the corresponding Fokker-Planck equation, which is guaranteed by a new critical ${\widehat{R}}_0^s$. Finally, some numerical simulations illustrate the analytical results.

在本文中，我们研究了具有半饱和发生率的SVI流行病模型的动力学行为。首先，研究了确定性模型的地方病和无病平衡点的局部渐近稳定性。然后对于随机模型，我们表明存在临界值${\mathrm{[R}}_0^s$可以确定疾病的灭绝和持续性。此外，通过构造一系列合适的Lyapunov函数，我们证明了${\mathrm{[R}}_0^s>\mathrm{1个}，$则随机SVI模型存在遍历平稳分布。值得一提的是，我们通过求解相应的Fokker-Planck方程，获得了确定性系统唯一地方性均衡周围随机模型的概率密度函数的精确表达式，这一点得到了新的临界条件的保证。${\widehat{\mathrm{[R}}}_0^s$。最后，一些数值模拟说明了分析结果。