We consider an SIS-type epidemic process that evolves on a known graph. We assume that a fixed curing budget can be allocated at each instant to the nodes of the graph, towards the objective of minimizing the expected extinction time of the epidemic. We provide a lower bound on the optimal expected extinction time as a function of the available budget, the epidemic parameters, the maximum degree, and the CutWidth of the graph. For graphs with large CutWidth (close to the largest possible), and under a budget which is sublinear in the number of nodes, our lower bound scales exponentially with the size of the graph.