  Recent progress in the study of the contact process [2] has verified that the
extinction-survival threshold $\lambda_1$ on a Galton-Watson tree is strictly
positive if and only if the offspring distribution $\xi$ has an exponential
tail. In this paper, we derive the first-order asymptotics of $\lambda_1$ for
the contact process on Galton-Watson trees and its corresponding analog for
random graphs. In particular, if $\xi$ is appropriately concentrated around its
mean, we demonstrate that $\lambda_1(\xi) \sim 1/\mathbb{E} \xi$ as
$\mathbb{E}\xi\rightarrow \infty$, which matches with the known asymptotics on
the $d$-regular trees. The same result for the short-long survival threshold on
the Erd\H{o}s-R\'enyi and other random graphs are shown as well.
