We first introduce the most basic model of vaccination in response to an infection that conforms to the simple SIR-type (susceptible -infectious -recovered) paradigm ( [17] [18] [19] ); this model will form the basic template for all the work that follows. We assume that the uncontrolled epidemic obeys the simple SIR model dynamics, such that susceptible individuals (of which there are S) can become infected and infectious by interaction with infected individuals (of which there are I), infected individuals recover at a constant rate and enter the recovered class (of which there are R) after which they are assumed immune for life. (We stress that all results presented are qualitatively invariant to the precise model formulation, in particular using a model with gamma-distributed exposed and infectious classes more reminiscent of the 2009 H1N1 pandemic [16] .) In all the models that follow, we assume frequency-dependent mixing (such that the number of epidemiologically relevant contacts is independent of population size) and ignore the demographics of birth and death [17] -a reasonable assumption given the rapid time-frame of an epidemic. To this simple model, we add vaccination at a constant rate, v; we assume that individuals are vaccinated independently of their disease status but individuals are only vaccinated once, vaccination begins at time T, and a proportion p of vaccinated individuals are successfully immunized.