A stochastic waterborne disease model is analyzed to reveal the effects of the white noise and telegraph noise. For this stochastic model, a critical quantity \(R_0^s\) is derived, which depends on not only model parameters but also environmental noises. If \(R_0^s>1\), then this model admits an ergodic stationary distribution. One of the most important contributions is the suitable Lyapunov function depending on the state of Markov chains is constructed. Theoretical results and numerical simulations provide a clear insight of the impact of environmental noises on the dynamical behavior of the model.

中文翻译：

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环境噪声对水传播病原体模型稳定性的影响

分析随机水传播疾病模型以揭示白噪声和电报噪声的影响。对于这个随机模型，推导出一个临界量\(R_0^s\)，它不仅取决于模型参数，还取决于环境噪声。如果\(R_0^s>1\)，则该模型承认遍历平稳分布。最重要的贡献之一是根据马尔可夫链的状态构造合适的李雅普诺夫函数。理论结果和数值模拟提供了对环境噪声对模型动力学行为影响的清晰洞察。