Peak R 0 , the amplitude of variation of R 0 and the duration of immunity are all strongly constrained. Figure 3 illustrates that increasing R 0 is offset by a longer duration of immunity reducing the susceptible population. The narrowness of the well-fitting parameter region is a result of the sudden changes of behaviour generated by changes in these parameters. Figure 5 also illustrates this feature. The closest fitting region is found for d between 0.15 and 0.3, but patches of well-fitting solutions are scattered a range of values of R p and d owing to the sensitivity to the system to temporal forcing. We note that a change in the mode of forcing from sinusoidal to school-term leads to generally broader ranges of acceptable parameter values (see the electronic supplementary material), perhaps indicating that the presence of this mechanism is a strong contributor to the variable annual behaviour observed in the ILI dataset. Because of the difficulties in knowing the 'true' incidence rate, the mean AAR is not precisely known and a range of 10-20% is often quoted. To allow for this uncertainty, we investigated allowing the information distance calculation to be based on the best-fit mean AAR from the range 10-20%, rather than precisely 15 per cent per year. Resulting best-fit parameter regions for R 0 against D and u against f were not significantly changed, owing almost certainly to the dominance of qualitative model behaviour as described above.