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In this paper, we establish the necessary and sufficient criterion for thecontact process on Galton-Watson trees (resp. random graphs) to exhibit thephase of extinction (resp. short survival). We prove that the survivalthreshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and onlyif its offspring distribution $\xi$ has an exponential tail, i.e., $\mathbb{E}e^{c\xi}<\infty$ for some $c>0$, settling a conjecture by Huang and Durrett[12]. On the random graph with degree distribution $\mu$, we show that if $\mu$has an exponential tail, then for small enough $\lambda$ the contact processwith the all-infected initial condition survives for $n^{1+o(1)}$-time w.h.p.(short survival), while for large enough $\lambda$ it runs over$e^{\Theta(n)}$-time w.h.p. (long survival). When $\mu$ is subexponential, weprove that the contact process w.h.p. displays long survival for any fixed$\lambda>0$.

Survival and extinction of epidemics on random graphs with general degrees