This paper deals with the spread of diseases over both a population network and an infrastructure network. We develop a layered networked spread model for a susceptible-infected-susceptible (SIS) pathogen-borne disease spreading over a human contact network and an infrastructure network, and refer to it as a layered networked susceptible-infected-water-susceptible (SIWS) model. The SIWS network is in the healthy state (also referred to as the disease-free equilibrium) if none of the individuals in the population are infected nor is the infrastructure network contaminated; otherwise, we say that the network is in the endemic state (also referred to as the endemic equilibrium). First, we establish sufficient conditions for local exponential stability and global asymptotic stability (GAS) of the healthy state. Second, we provide sufficient conditions for existence, uniqueness, and GAS of the endemic state. Building off of these results, we provide a necessary, and sufficient, condition for the healthy state to be the unique equilibrium of our model. Third, we show that the endemic equilibrium of the SIWS model is worse than that of the networked SIS model without any infrastructure network, in the sense that at least one subpopulation has strictly larger infection proportion at the endemic equilibrium in the former model than that in the latter. Fourth, we study an observability problem, and, assuming that the measurements of the sickness-levels of the human contact network are available, provide a necessary and sufficient condition for estimation of the pathogen levels in the infrastructure network. Furthermore, we provide another sufficient, but not necessary, condition for estimation of pathogen levels in the infrastructure network.

中文翻译：

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具有基础设施网络的多层 SIS 模型

本文涉及疾病在人口网络和基础设施网络上的传播。我们开发了一种分层网络传播模型，用于在人类接触网络和基础设施网络上传播的易感感染易感 (SIS) 病原体传播疾病，并将其称为分层网络易感染水敏感 (SIWS)模型。如果人口中没有个人被感染，基础设施网络也没有受到污染，则 SIWS 网络处于健康状态（也称为无病平衡）；否则，我们说网络处于地方病状态（也称为地方病平衡）。首先，我们为健康状态的局部指数稳定性和全局渐近稳定性（GAS）建立充分条件。第二，我们为流行状态的存在性、唯一性和 GAS 提供了充分条件。基于这些结果，我们为健康状态成为我们模型的独特平衡提供了必要且充分的条件。第三，我们表明 SIWS 模型的流行均衡比没有任何基础设施网络的网络化 SIS 模型更差，从某种意义上说，至少一个亚群在前一种模型中的流行均衡时的感染比例严格大于后者。第四，我们研究了一个可观察性问题，假设人类接触网络的疾病水平的测量是可用的，为基础设施网络中病原体水平的估计提供了充分必要条件。此外，我们提供另一个足够的，