We study a SIRD epidemic process among a heterogeneous population that interacts through a network. We give general upper bounds for the size of the epidemic starting from a (small) set of initially infected individuals. Moreover, we characterize the epidemic reproduction numbers in terms of the spectral properties of a relevant matrix based on the network adjacency matrix and the infection rates. We suggest that this can be used to identify sub-networks that have high reproduction numbers before the epidemic reaches and picks up in them. When we base social contact on a random graph with given vertex degrees, we give limit theorems on the fraction of infected individuals. For a given social distancing individual strategies, we establish the epidemic reproduction number $\mathfrak{R}_0$ which can be used to identify network vulnerability and inform vaccination policies. In the second part of the paper we study the equilibrium of the social distancing game and we show that voluntary social distancing will always be socially sub-optimal. Our numerical study using Covid-19 data serves to quantify the absolute and relative utility gaps across age cohorts.

中文翻译：

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异构网络中的大流行风险和平衡社会距离

我们研究了通过网络相互作用的异质人群中的 SIRD 流行过程。我们从一组（小）最初感染的个体开始，给出了流行病规模的一般上限。此外，我们根据基于网络邻接矩阵和感染率的相关矩阵的谱属性来表征流行病再现数。我们建议这可用于在流行病到达并在其中出现之前识别具有高繁殖数量的子网络。当我们基于给定顶点度的随机图建立社会联系时，我们给出了受感染个体比例的极限定理。对于给定的社会疏远个人策略，我们建立了流行病繁殖数 $\mathfrak{R}_0$，可用于识别网络漏洞并告知疫苗接种政策。在论文的第二部分，我们研究了社会疏远博弈的均衡，我们表明自愿的社会疏远在社会上总是次优的。我们使用 Covid-19 数据进行的数值研究有助于量化不同年龄段的绝对和相对效用差距。