In this work, we have identified a minimal set of mechanisms necessary to match the long-term temporal behaviour seen in ILI time series. We find that an age-structured population and multiple strains with cross-immunity are necessary to recreate the distributions of AAR and epidemic duration seen in time-series data. Nonlinear models of this type with temporal forcing are well known for having complex bifurcation structures affecting their periodic behaviour. On the timescale of a single disease season or single epidemic, these would have little effect on parameter estimation. Over many seasons, however, the periodicity of the underlying model is crucially important. We would argue that capturing the long-term trends in behaviour is at least as important as the detail of individual seasonal outbreaks and our fitting approach focuses on these aspects. The complex bifurcation structure of the model means that the information distance is not necessarily a smooth function of the parameters. This makes finding a unique set of parameters giving an overall minimum distance impossible. Although mean attack rate and duration generally vary smoothly, the period of the model solution changes discontinuously (inevitably, as it only takes values that are multiples of the annual forcing period). Solutions with longer periods generate a wider range of AAR and epidemic durations over an extended period of time and are therefore capable of fitting the target data distributions better. As a result, the best fits are strongly associated with longer periods in model solutions and the quality of fit of the model can change abruptly over small changes in parameter values.