SIAM Journal on Applied Mathematics, Volume 80, Issue 2, Page 814-838, January 2020.
This paper investigates a stochastic SIRS epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature. We classify the extinction and permanence by introducing $\lambda$, a real-valued threshold. We show that if $\lambda<0$, then the disease will eventually disappear (i.e., the disease-free state is globally asymptotically stable); if the threshold value $\lambda>0$, the epidemic becomes strongly stochastically permanent. This result substantially generalizes and improves the related results in the literature. Moreover, the mathematical development in this paper is interesting in its own right. The essential difficulties lie in that the dynamics of the susceptible class depend explicitly on the removed class resulting in a three-dimensional system rather than a two-dimensional system. Consequently, the methodologies developed in the literature are not applicable here. One of the main ingredients in the analyses is this: Though it is not possible to compare solutions in the interior and on the boundary for all $t\in[0,\infty)$, approximation in a long but finite interval $[0,T]$ can be carried out. Then, using the ergodicity of the solution on the boundary and exploiting the mutual interplay between the distance of solutions in the interior and solutions on the boundary and the exponential decay or growth (depending on the sign of the Lyapunov exponent), one can classify the behavior of the system. The convergence to the invariant measure is established under the total variation norm together with the corresponding rate of convergence. To demonstrate, some numerical examples are provided to illustrate our results.

中文翻译：

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具有一般发生率的随机SIRS模型的长期分析

SIAM应用数学杂志，第80卷，第2期，第814-838页，2020年1月。
本文研究了一种随机的SIRS流行病模型，其发病率足够普遍，涵盖了迄今为止文献中考虑的许多发病率模型。我们通过引入$ \ lambda $（一个实值阈值）对灭绝和持久性进行分类。我们证明，如果$ \ lambda <0 $，则疾病最终将消失（即，无病状态在全局上是渐近稳定的）；如果阈值$ \ lambda> 0 $，则该流行病在随机性上变得很持久。该结果实质上概括并改进了文献中的相关结果。此外，本文的数学发展本身就很有趣。根本的困难在于，易感类的动态性明确取决于所移除的类，从而导致产生三维系统而不是二维系统。因此，文献中开发的方法不适用于此处。分析中的主要成分之一是：尽管不可能比较所有$ t \ in [0，\ infty）$的内部和边界解，但要在较长但有限的间隔$ [0中进行近似，T] $可以执行。然后，利用边界上解的遍历性，利用内部解和边界上解的距离与指数衰减或增长（取决于李雅普诺夫指数的符号）之间的相互影响，可以对系统的行为。不变测度的收敛是在总变分范数和相应的收敛速度下建立的。为了演示，提供了一些数值示例来说明我们的结果。