  We provide a dynamic policy for the rapid containment of a contagion process
modeled as an SIS epidemic on a bounded degree undirected graph with n nodes.
We show that if the budget $r$ of curing resources available at each time is
${\Omega}(W)$, where $W$ is the CutWidth of the graph, and also of order
${\Omega}(\log n)$, then the expected time until the extinction of the epidemic
is of order $O(n/r)$, which is within a constant factor from optimal, as well
as sublinear in the number of nodes. Furthermore, if the CutWidth increases
only sublinearly with n, a sublinear expected time to extinction is possible
with a sublinearly increasing budget $r$.
