In this paper, we consider a stochastic general epidemic model with relapse, graded cure and nonlinear incidence rate. We show the existence and uniqueness of a global positive solution for the model under some mild conditions. We also give some sufficient conditions for ensuring the existence of an ergodic stationary distribution for the model. Moreover, we analyse the extinction of the disease described by the model. Finally, we provide some numerical simulations to illustrate our main results.

中文翻译：

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具有复发和治愈的随机非线性流行病模型的遍历平稳分布

在本文中，我们考虑了一个具有复发、分级治愈和非线性发病率的随机一般流行病模型。我们展示了在一些温和条件下模型的全局正解的存在性和唯一性。我们还给出了确保模型存在遍历平稳分布的一些充分条件。此外，我们分析了模型描述的疾病的灭绝。最后，我们提供了一些数值模拟来说明我们的主要结果。