The behaviour of our model is quite sensitive to the assumed external force of infection, e. The level of external forcing assumed for most of this work ( §2) is negligible compared with the average force of infection generated by the indigenous population. However, long-period and chaotic solutions for seasonally forced SIR models generate very low infection prevalence during epidemic troughs, even low levels of importation of infectives strongly encourages annual and biennial behaviours and removes highly chaotic solutions from the optimal region. The effect of importation rate can be seen in figure 6 . For external forces of infection above about 10 25 yr -1 , only annual and biennial solutions are found. Optimal behaviour is found for an external force of infection of approximately 6 Â 10 27 yr -1 . Figure 7 explores the sensitivity of the model to the degree of heterogeneity and cross-immunity in the population. The choice of R p and D lies in the wellfitting band in figure 3d. It is clear that a wide range of values for these parameters allow the model to fit the behaviour of the time series quite well. The model's qualitative behaviour (in terms of its periodicity) is more stable with respect to these mechanisms and variation within the closest fit region mainly affects the mean attack rate. There is a broadly inverse relationship between the well-fitting values of the parameters u and f. Increasing the assortativity of mixing concentrates infections more strongly in age groups, decreasing the available susceptibles and hence the attack rate. Increasing cross-immunity has an equivalent effect by increasing the effective loss of susceptibles caused by any single infection event and hence reducing the attack rate. As can be seen from figure 7a, values of the cross-immunity parameter f outside the range 0.3-0.6 drive the model into unfavourable periodicities, giving very poor fits. This suggests that a model with two strains interacting via cross-immunity is necessary to reproduce the dynamics seen in influenza time series and that it is insufficient to have two independent strains (f ¼ 0) or two antigenically identical strains (i.e. f ¼ 12equivalent to a single strain model). Similarly, extreme values of u also lead to poorly fitting model behaviour, suggesting that a uniformly mixing population (u ¼ 0) would also not generate matching behaviour.