Childhood diseases, such as measles and whooping cough, exhibit cyclical behaviour, and the frequencies of their recurrent epidemics change over decades or centuries. A key mechanism underlying these epidemic cycles is seasonal variation in contact rates stemming mainly from school terms (known as "term-time forcing"). Early mechanistic models represented seasonality with a simple sine wave for mathematical convenience, which raises the question: to what extent do inferences about transitions in epidemic cycles stemming from mechanistic models depend on accurate representation of the seasonal forcing function's shape?

Papst and Earn addressed this question by introducing a family of functions smoothly connecting sinusoidal and term-time forcing. They showed that the asymptotic epidemic dynamics predicted by these models can be mapped onto one another, enabling comparison of historical periods with different seasonal patterns. However, their method relies on adjusting the amplitude of forcing by following a period-doubling bifurcation in the system. This key bifurcation does not occur for all diseases, notably whooping cough, limiting the method's applicability.

Papst and Earn focused on asymptotic epidemic behaviour; here, we turn our attention to the transient dynamics in order to extend their method to diseases like whooping cough. We show that the transient periods of damped oscillations onto cyclical attractors can also be made invariant across different seasonal forcing schemes, and so invariance is a more general principle in these systems. This finding supports the ongoing search for a more general framework for understanding shifts in epidemic frequencies observed in childhood infectious diseases.

Keywords: Childhood infectious diseases, mechanistic modelling