The global transmission and control of influenza: Supporting Text 2

The epidemic percolation network (EPN) is a directed random network that represents the final outcomes of a stochastic epidemic model [1]. To generate an EPN, we imagine that every individual in the model is infected and draw arrows from each individual i to all persons with whom i would make infectious contact while he or she was infected. We define infectious contact to be a contact sufficient to infect a susceptible person, so an arrow from i to j means that infection of i implies infection of j, either by i or by someone who makes infectious contact with j before i does. An epidemic begins with one or more persons infected from outside the population, who we call imported infections. For a given set of imported infections and a given realization of the EPN, the set of people who will eventually be infected is precisely the union of the out-components of all imported infections. Since infectious contact is a stochastic process, the EPN is a random network. It can be shown that the size distribution of outbreaks starting with the infection of a node i is identical to the distribution of out-component sizes of i in the probability space of EPNs. In the limit of a large population, the epidemic threshold corresponds to the emergence of a giant strongly-connected component in the EPN, the probability of a large epidemic given a single randomly-chosen initial infection is equal to the proportion of the nodes contained in the giant in-component of the EPN, and the attack rate of a large epidemic