We introduce a weighted configuration model graph, whereedge weightscorrespond to the probability of infection in an epidemic on the graph. On these graphs, we study the development of a Susceptible–Infectious–Recovered epidemic using both Reed–Frost and Markovian settings. For the special case of having two different edge types, we determine thebasic reproduction numberR0, theprobability of a major outbreak, and therelative final size of a major outbreak. Results are compared with those for a calibrated unweighted graph. The degree distributions are based on both theoretical constructs and empirical network data. In addition, bivariate standard normal copulas are used to model the dependence between the degrees of the two edge types, allowing for modeling the correlation between edge types over a wide range. Among the results are that the weighted graph produces much richer results than the unweighted graph. Also, whileR0always increases with increasing correlation between the two degrees, this is not necessarily true for the probability of a major outbreak nor for the relative final size of a major outbreak. When using copulas we see that these can produce results that are similar to those of the empirical degree distributions, indicating that in some cases a copula is a viable alternative to using the full empirical data.

中文翻译：

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加权网络上的 SIR 流行病

我们引入了一个加权配置模型图，其中边权重对应于图表上流行病中的感染概率。在这些图表上，我们使用 Reed-Frost 和 Markovian 设置研究了易感-感染-恢复流行病的发展。对于具有两种不同边缘类型的特殊情况，我们确定基本再生数R0， 这大爆发的可能性, 和大爆发的相对最终规模. 将结果与校准的未加权图的结果进行比较。度分布基于理论构造和经验网络数据。此外，使用二元标准正态 copula 对两种边缘类型的度数之间的依赖关系进行建模，从而可以对广泛范围内的边缘类型之间的相关性进行建模。结果是加权图比未加权图产生更丰富的结果。另外，虽然R0总是随着两个程度之间相关性的增加而增加，这对于大爆发的概率和大爆发的相对最终规模不一定正确。当使用 copula 时，我们看到这些可以产生类似于经验度分布的结果，这表明在某些情况下，copula 是使用完整经验数据的可行替代方案。