where N is the total population size. The precise way in which vaccination is implemented within the model ensures that a fixed number of individuals (v) are vaccinated per day, of which a fraction p are completely protected, and assumes that each person only receives one course of vaccine. If multiple doses of vaccine are required, or if protection only develops some time after vaccination, these can be included by delaying the time, T, at which immunization begins to take effect. Even this simplest of models confirms two simple rules-of-thumb regarding successful vaccination campaigns that seek to minimize the number of cases ( figure 1) . Firstly, that vaccination should begin as early as possible, so that susceptibles are depleted by vaccination before many cases arise; and secondly, that vaccination should be performed as rapidly as possible-both of which have been discussed before for a range of control measures [20, 21] . From figure 1 it is also clear that an early start to a vaccination campaign is far more beneficial than faster vaccination of the population. It should be stressed that when considering an ongoing epidemic, the critical vaccination threshold for the elimination of an endemic infection or prevention of epidemic invasion (¼1 2 1/R 0 ) [17] no longer plays such a clear role. Instead, the primary aim should be to immunize many people in as short a time as possible, subject to trade-offs from economic costs or adverse effects of vaccination (such as that observed for smallpox [12, 20] ). Here, and in all the figures that follow, we have considered a wide range of vaccination speeds ( y-axis in figure 1, line colours in figure 2, x-axes in figure 4 ); while some of these are extremely rapid and may be practically unachievable, the associated results are shown to provide a clearer picture of the most optimistic control scenario.