Survival and extinction of epidemics on random graphs with general degree

In this paper we establish the necessary and sufficient criterion for the contact process on Galton–Watson trees (resp., random graphs) to exhibit the phase of extinction (resp., short survival). We prove that the survival threshold λ₁ for a Galton–Watson tree is strictly positive if and only if its offspring distribution ξ has an exponential tail, that is, Eecξ<∞ for some $c>0$, settling a conjecture by Huang and Durrett (2018). On the random graph with degree distribution μ, we show that if μ has an exponential tail, then for small enough λ the contact process with the all-infected initial condition survives for n¹⁺ᵒ⁽¹⁾-time whp (short survival), while for large enough λ it runs over eΘ(n)-time whp (long survival). When μ is subexponential, we prove that the contact process whp displays long survival for any fixed λ>0.