Heterogeneity can markedly increase final outbreak size in the SIR model of epidemics

We study the SIR model of epidemics on positively correlated heterogeneous networks with population variability, and explore the dependence of the final outbreak size on the network heterogeneity strength and basic reproduction number $R_0$ -- the ratio between the infection and recovery rates per individual. We reveal a critical value $R_0^c$, above which the maximal outbreak size is obtained at zero heterogeneity, but below which, the maximum is obtained at finite heterogeneity strength. This second-order phase transition, universal for all network distributions with finite standardized moments indicates that, network heterogeneity can greatly increase the final outbreak size. We also show that this effect can be enhanced by adding population heterogeneity, in the form of varying inter-individual susceptibility and infectiousness. Notably, our results provide key insight as to the predictability of the well-mixed SIR model for the final outbreak size, in realistic scenarios.