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 $Ω(W)$, where $W$ is the CutWidth of the graph, and also of order $Ω(\log n)$, 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 $r$.