where T is the mean probability of transmission between individuals and k and k 2 are the mean degree and mean square degree of the network. Here, R 0 depends explicitly on the structure of the network (i.e., on k and k 2 ). A single pathogen may, therefore, have very different transmission dynamics depending on the population through which it spreads. If two networks have the same mean degree, k, then the one with the larger variance in degree, k 2 − k 2 , will be more vulnerable to the spread of disease. In compartment models, infected hosts are assumed to have potentially disease-causing contacts with random individuals from the population according to a Poisson process that yields an average contact rate of β per unit time. The mass-action assumption of compartmental models is tantamount to assuming that the underlying contact patterns form a random graph with a Poisson degree distribution. Estimates of R 0 that assume a mass-action model may, therefore, be invalid for populations with non-Poisson contact patterns and, in particular, will underestimate the actual growth rate of the disease in highly heterogeneous networks. [24] noted that if R 0 > 1 in an agent-based model then, due to the stochastic nature of interactions, it is possible that no epidemic occurs or that it ends early if, by chance, the few initially contagious individuals recover before generating new cases.