where l j is the force of infection of strain j and S n is the part of the population that is immune to strain n. Infected individuals recovering from strain n enter a class entirely immune to n (e.g. S 0 ! S 1 , S 1 ! S 12 ). Immunity to a given strain is lost at a rate s. Figure 2 illustrates the flows of individuals between the various immunity classes. The model is further complicated by stratification by age (children and adults) with heterogeneous mixing, differential infectiousness and susceptibility by age and a realistic infectiousness profile. The full details of the model are described in the electronic supplementary material. The effect of seasonal forcing is included through a time-varying contact parameter,