All simulated epidemics were seeded with 10 index cases chosen at random, in order to avoid early stochastic extinction. Averages of 10 000 simulations are presented, consisting of 100 epidemic runs on each of 100 different network realizations. Proportions across all plots are relative to the total network size NZ10 000. The simulation time step used is DtZ0.04. Smaller time steps produced effectively identical results (not shown). The s.e. values of the averages of simulation outputs are at least three to four orders of magnitude smaller than the measurement itself and therefore are not discussed further.