In this paper, a stochastic SIRS epidemic model with saturating contact rate is constructed. First, for the deterministic system, the stability of the equilibria is discussed by using eigenvalue theory. Second, for the stochastic system, the threshold conditions of disease extinction and persistence are established. Our results indicate that a large environmental noise intensity can suppress the spread of disease. Conversely, if the intensity of environmental noise is small, the system has a stationary solution which indicates the disease is persistent. Eventually, we introduce some computer simulations to validate the theoretical results.

中文翻译：

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具有饱和接触率的随机SIRS流行病模型的动力学分析

本文建立了具有饱和接触率的随机SIRS流行病模型。首先，对于确定性系统，使用特征值理论讨论了均衡的稳定性。其次，对于随机系统，确定了疾病灭绝和持久性的阈值条件。我们的结果表明，较大的环境噪声强度可以抑制疾病的传播。相反，如果环境噪声的强度较小，则系统具有固定的解决方案，表明该疾病持续存在。最终，我们引入了一些计算机模拟来验证理论结果。