In this paper, we propose and analyze a stochastic SIVS model with saturated incidence and Lévy jumps. We first prove the existence of a global positive solution of the model. Then, with the help of semimartingale convergence theorem, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. At last, we further study the threshold dynamics of a stochastic SIRS model with saturated or bilinear incidence by a similar method and carry out some numerical simulations to demonstrate our theoretical results. Comparing with the method given by Zhou and Zhang (Physica A 446:204–216, 2016), we find that the method used in this paper is simple and effective.

在本文中，我们提出并分析了具有饱和发生率和Lévy跳变的随机SIVS模型。我们首先证明该模型存在全局正解。然后，借助半mart收敛定理，我们获得了该模型的随机阈值，该阈值完全确定了该流行病的灭绝和持续性。最后，我们通过相似的方法进一步研究了具有饱和或双线性入射的随机SIRS模型的阈值动力学，并进行了一些数值模拟，以证明我们的理论结果。与Zhou和Zhang（Physica A 446：204–216，2016）给出的方法相比，我们发现本文使用的方法简单有效。