  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.
