For nonnetwork mixing, the threshold for invasion is given by the basic reproductive ratio R 0 , defined as the expected number of secondary infectious cases created by an average primary infectious case in an otherwise wholly susceptible population. In structured populations, this verbal definition is typically altered to be the secondary cases caused by a typical primary case once the dynamical system has settled into its early asymptotic behaviour. As such, the threshold for invasion is R 0 = 1: for values above this, an infection can grow in the population and the disease can successfully invade; for values below it, each chain of infection is doomed to eventual extinction. Values of R 0 can be measured directly during the course of an epidemic by detailed contact tracing; however, there are considerable statistical issues concerning censoring and data quality. Provided there are no short closed loops in the network, R 0 can be defined through a next-generation matrix