The values of other parameters are listed in table 1 and discussed in the electronic supplementary material. In discussing the behaviour of the model, periodicity refers to the periodicity of the overall case rate with time, rather than for an individual strain. Simulations were run using a population of 60 million, approximating the population of the UK. Owing to the large population, a deterministic model was used. Tests using the corresponding stochastic models showed no qualitative variation from the deterministic dynamics and the presence of a continuous low level external force of infection precluded the possibility of extinction. Figure 3 illustrates the behaviour of the model and aspects of the information distance between model and data as a function of R p and D. The region of best fits is located in a narrow diagonal band spanning 1.6 , R p , 2.5 and 3 , D , 8 (figure 3d ). Along this band, increasing reproduction number is compensated for by a longer period of effective immunity that decreases the susceptible proportion of the population, giving a constant mean attack rate. The acceptable region is bounded in part by the duration of the model epidemics. The lowest and highest values of R p generate epidemics that are too broad and too narrow, compared with the target distribution. The fit of the model is strongly constrained by the dynamics of the model, which exhibits a wide range of periodic behaviour across quite small changes in parameter values (figure 3a). While AAR changes smoothly with the parameters, abrupt changes in periodicity lead to qualitative changes in the distributions of AAR and epidemic duration and hence the KL distance. Best-fit behaviour is associated with long-period behaviour of the model (4þ years). Here, the model generates a range of AARs clustered around the mean and matching the target distribution. The qualitatively different forms of behaviour are well characterized by the total KL distance measure (figure 4). KL distances less than 200 correspond to realistic behaviour with appropriate mean attack rate distributions (figure 4a). KL distance between 200 and 300 match either with realistic behaviour interspersed with large-scale epidemic episodes or with realistic behaviour but with a mean attack rate displaced from the target value ( figure 4b) . Larger values represent dynamics and mean attack rates greatly different from the observed time series (figure 4c). Figure 5 illustrates a strong sensitivity to the amplitude of variation of the contact parameter, d, with the best-fit lying in the range 0.15-0.3. Although the mean AAR is not strongly dependent on d (figure 5a), the bifurcation behaviour of the model means realistic solutions (resembling figure 4a,b) can be found for higher amplitudes of seasonal variation but not in a contiguous region (figure 5b). Solutions with smaller seasonal variations are rejected on the epidemic duration component of the information distance. Low amplitudes generate broad epidemics which do not match the target distribution.