Environmental perturbations are unavoidable in the propagation of infectious diseases. In this paper, we introduce the stochasticity into the susceptible–infected–recovered (SIR) model via the parameter perturbation method. The stochastic disturbances associated with the disease transmission coefficient and the mortality rate are presented with two perturbations: Gaussian white noise and Lévy jumps, respectively. This idea provides an overview of disease dynamics under different random perturbation scenarios. By using new techniques and methods, we study certain interesting asymptotic properties of our perturbed model, namely: persistence in the mean, ergodicity and extinction of the disease. For illustrative purposes, numerical examples are presented for checking the theoretical study.

中文翻译：

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具有双重扰动的随机SIR传染病模型的动态表征

在传染病的传播过程中，环境扰动是不可避免的。在本文中，我们通过参数扰动方法将随机性引入易感-感染-恢复（SIR）模型。与疾病传播系数和死亡率相关的随机扰动有两种扰动：分别是高斯白噪声和 Lévy 跳跃。这个想法提供了在不同随机扰动情景下疾病动态的概述。通过使用新技术和方法，我们研究了我们的扰动模型的某些有趣的渐近特性，即：均值的持久性、遍历性和疾病的消退。为了说明的目的，给出了数值例子来检查理论研究。