Human to human transmissible infectious diseases spread in a population using human interactions as its transmission vector. The early stages of such an outbreak can be modeled by a graph whose edges encode these interactions between individuals, the vertices. This article attempts to account for the case when each individual entails in different kinds of interactions which have therefore different probabilities of transmitting the disease. The majority of these results can be also stated in the language of percolation theory. The main contributions of the article are: (1) Extend to this setting some results which were previously known in the case when each individual has only one kind of interactions. (2) Find an explicit formula for the basic reproduction number \(R_0\) which depends only on the probabilities of transmitting the disease along the different edges and the first two moments of the degree distributions of the associated graphs. (3) Motivated by the recent Covid-19 pandemic, we use the framework developed to compute the \(R_0\) of a model disease spreading in populations whose trees and degree distributions are adjusted to several different countries. In this setting, we shall also compute the probability that the outbreak will not lead to an epidemic. In all cases we find such probability to be very low if no interventions are put in place.

人与人之间的传染性疾病以人际互动为传播媒介在人群中传播。这种爆发的早期阶段可以通过一个图来建模，该图的边编码个体之间的这些交互，即顶点。本文试图解释每个人都需要进行不同类型的相互作用的情况，这些相互作用因此具有不同的传播疾病的可能性。这些结果中的大部分也可以用渗透理论的语言来表述。这篇文章的主要贡献是：（1）将一些先前已知的结果扩展到这个设置，这些结果在每个个体只有一种交互的情况下是已知的。(2) 求基本再生数\(R_0\)的显式公式这仅取决于沿不同边传播疾病的概率以及相关图的度分布的前两个矩。(3) 受最近 Covid-19 大流行的启发，我们使用开发的框架来计算模型疾病在人群中传播的\(R_0\)，其树和度数分布已调整到几个不同的国家。在这种情况下，我们还将计算爆发不会导致流行病的概率。在所有情况下，如果不采取干预措施，我们发现这种可能性非常低。