In this paper, we analyze a higher-order stochastically perturbed multigroup staged-progression model for the transmission of HIV with saturated incidence rate. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. In addition, we make up adequate conditions for complete eradication and wiping out the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, examples and numerical simulations are introduced to validate our theoretical results.

中文翻译：

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具有饱和发病率和高阶扰动的随机多组阶段进展 HIV 模型的动态行为

在本文中，我们分析了具有饱和发病率的 HIV 传播的高阶随机扰动多组阶段进展模型。通过建立合适的随机 Lyapunov 函数，我们获得了系统正解的遍历平稳分布存在和唯一性的充分条件。此外，我们还为彻底消灭和消灭传染病创造了充足的条件。在生物学解释中，平稳分布的存在意味着疾病将长期流行并持续存在。最后，通过实例和数值模拟来验证我们的理论结果。