In the present work, we introduce a stochastic SIRI epidemic model with nonlinear relapse. We give sufficient conditions for extinction and persistence of the disease. We also study the existence of a stationary distribution and the ergodicity of the solutions. As a special case of our results that under some conditions on noise intensities, we obtain the threshold ${\mathcal{R}}_{\beta }$ for the disease. Finally, we provide some computer simulations to illustrate our theoretical findings.

中文翻译：

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非线性复发和再感染对SIRI流行病模型推导随机阈值的影响

在当前的工作中，我们介绍了具有非线性复发的随机SIRI流行病模型。我们为疾病的灭绝和持续提供了充分的条件。我们还研究了平稳分布的存在性和解的遍历性。作为我们结果的特例，在某些情况下根据噪声强度，我们获得了阈值${\mathcal{[R}}_{\beta }$对于这种疾病。最后，我们提供一些计算机模拟来说明我们的理论发现。