where K km defines the number of cases in individuals with k contacts from an individual with m contacts during the early stages of the epidemic. Here and elsewhere in this section we use square brackets to represent the numbers of different types on the network; hence, [m] is the number of individuals with m edges in the network and [km] is the number of edges between individuals with k and m contacts, respectively. In addition, p is the probability of infection eventually passing across the edge between a susceptible-infectious pair (for Markovian recovery rate γ and transmission rate τ this is given by p = τ/(τ + γ)). The basic reproductive ratio is given by the dominant eigenvalue of the next-generation matrix