This paper proposes a framework for modelling financial contagion that is based on SIR (Susceptible-Infected-Recovered) transmission models from epidemic theory. This class of models addresses two important features of contagion modelling, which are a common shortcoming of most existing empirical approaches, namely the direct modelling of the inherent dependencies involved in the transmission mechanism, and an associated canonical measure of crisis severity. The proposed methodology naturally implies a control mechanism, which is required when evaluating prospective immunisation policies that intend to mitigate the impact of a crisis. It can be implemented not only as a way of learning from past experiences, but also at the onset of a contagious financial crisis. The approach is illustrated on a number of currency crisis episodes, using both historical final outcome and temporal data. The latter require the introduction of a novel hierarchical model that we call the Hidden Epidemic Model (HEM), and which embeds the stochastic financial epidemic as a latent process. The empirical results suggest, among others, an increasing trend for global transmission of currency crises over time.