In this paper, a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination, nonlinear incidence and white noises under regime switching and Lévy jumps is investigated. A new threshold value is determined. Some basic assumptions with regard to nonlinear incidence, white noises, Markov switching and Lévy jumps are introduced. The threshold conditions to guarantee the extinction and permanence in the mean of the disease with probability one and the existence of unique ergodic stationary distribution for the model are established. Some new techniques to deal with the Markov switching, Lévy jumps, nonlinear incidence and vaccination for the stochastic epidemic models are proposed. Lastly, the numerical simulations not only illustrate the main results given in this paper, but also suggest some interesting open problems.

中文翻译：

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具有疫苗接种和非线性发病率的随机易感感染恢复 (SIRS) 流行病模型在政权转换和 Lévy 跳跃下的动力学

在本文中，研究了在政权转换和 Lévy 跳跃下具有疫苗接种、非线性发生率和白噪声的随机易感感染恢复 (SIRS) 流行病模型。确定新的阈值。介绍了一些关于非线性入射、白噪声、马尔可夫开关和 Lévy 跳跃的基本假设。建立了阈值条件以保证该疾病在概率为 1 的均值中的灭绝和持久性以及模型的唯一遍历平稳分布的存在性。提出了一些新技术来处理随机流行病模型的马尔可夫转换、Lévy 跳跃、非线性发生率和疫苗接种。最后，数值模拟不仅说明了本文给出的主要结果，而且还提出了一些有趣的开放性问题。