Using the technique of edge-based compartmental modelling (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks, in a recent paper (PloS One, 8(2013), e69162), Miller and Volz established an SIR disease network model with heterogeneous infectiousness and susceptibility. The authors provided a numerical example to demonstrate its validity but they did not perform any mathematical analysis of the model. In this paper, we resolve this problem. Using the nature of irreducible cooperative system in the theory of monotonic dynamical system, we prove that the dynamics of the model are completely determined by a critical value ρ₀: When ρ₀ > 0, the disease persists in a globally stable outbreak equilibrium; while when ρ₀ < 0, the disease dies out in the population and the disease free equilibrium is globally stable.

中文翻译：

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具有异构传染性和易感性的SIR网络模型中的临界值

在最近的一篇论文中（PloS One，8（2013），e69162），Miller和Volz使用基于边缘的区室建模技术（EBCM）在网络中传播易感感染恢复（SIR）疾病，Miller和Volz建立了SIR具有异质传染性和易感性的疾病网络模型。作者提供了一个数值示例来证明其有效性，但他们并未对该模型进行任何数学分析。在本文中，我们解决了这个问题。使用在单调动力系统的不可约理论合作系统的性质，我们证明了该模型的动力学是由临界值确定的完全ρ ₀：当ρ ₀ > 0，在一个全局稳定爆发平衡该疾病仍然存在; 而当ρ₀ <0，该疾病在人群中消亡，并且无病平衡在全球范围内是稳定的。