Abstract Population mobility in an SIRS epidemic model is considered via graph Laplacian diffusion. We show the existence and uniqueness of solutions to the SIRS model defined on weighed graph. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1. By constructing Lyapunov function, we show that the endemic equilibrium is globally asymptotically stable for the model with the same diffusion rates if the basic reproduction number is greater than 1. With numerical simulations, we apply our generalized weighed graph to Watts-Strogatz network, where the degree of node is illustrated to determine the peak number of infectious population. It also indicates that the network has an impact on small-time behavior of epidemic transmission.

中文翻译：

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加权网络SIRS流行病模型

摘要 SIRS 流行病模型中的人口流动是通过图拉普拉斯扩散来考虑的。我们展示了在加权图上定义的 SIRS 模型的解的存在性和唯一性。通过上下解的方法，我们证明了如果基本繁殖数小于1，无病平衡是渐近稳定的。通过构造Lyapunov函数，我们证明了具有以下特征的模型的地方病平衡是全局渐近稳定的如果基本再生数大于 1，则扩散率相同。通过数值模拟，我们将广义加权图应用于 Watts-Strogatz 网络，其中节点的程度被说明以确定感染人口的峰值数量。这也表明网络对疫情传播的小时间行为有影响。