We consider the contact process (SIS epidemic) on finite undirected graphs and study the relationship between the expected epidemic lifetime, the infection and cure rates, and properties of the graph. In particular, we show the following: 1) if the ratio of cure rate to infection rate exceeds the spectral radius of the graph, then the epidemic dies our quickly. 2) If the ratio of cure rate to infection rate is smaller than a generalisation of the isoperimetric constant, then the epidemic is long-lived. These results suffice to establish thresholds on certain classes of graphs with homogeneous node degrees. In addition, we obtain thresholds for epidemics on power-law graphs. Finally, we use these techniques to study the efficacy of different schemes for distributing curing resources among the nodes.