  We study the contact process on random graphs with low infection rate
$\lambda$. For random $d$-regular graphs, it is known that the survival time is
$O(\log n)$ below the critical $\lambda_c$. By contrast, on the
Erd\H{o}s-R\'enyi random graphs $\mathcal G(n,d/n)$, rare high-degree vertices
result in much longer survival times. We show that the survival time is
governed by high-density local configurations. In particular, we show that
there is a long string of high-degree vertices on which the infection lasts for
time $n^{\lambda^{2+o(1)}}$. To establish a matching upper bound, we introduce
a modified version of the contact process which ignores infections that do not
lead to further infections and allows for a shaper recursive analysis on
branching process trees, the local-weak limit of the graph. Our methods,
moreover, generalize to random graphs with given degree distributions that have
exponential moments.
