In this paper, we consider an SVIR model with general incidence rates which covers many common incidence rate models and is perturbed by environmental white noises. We derive a sufficient and almost necessary condition for extinction and ergodicity for this model. One of the distinctive features is that we introduce a new technique which combines the classic Lyapunov function with the methods proposed by Du et al. (2016) to study the ergodic stationary distribution and obtain threshold number.

中文翻译：

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具有一般发生率的随机SVIR模型中的阈值行为

在本文中，我们考虑了具有一般发生率的SVIR模型，该模型涵盖了许多常见的发生率模型，并且受到环境白噪声的干扰。我们为该模型得出了灭绝和遍历的充分且几乎必要的条件。特色之一是我们引入了一种新技术，该技术将经典的Lyapunov函数与Du等人提出的方法相结合。（2016年）研究遍历平稳分布并获得阈值数。