Invariant predictions of epidemic patterns from radically different forms of seasonal forcing

Seasonal variation in environmental variables, and in rates of contact among individuals, are fundamental drivers of infectious disease dynamics. Unlike most periodically-forced physical systems, for which the precise pattern of forcing is typically known, underlying patterns of seasonal variation in transmission rates can be estimated approximately at best, and only the period of forcing is accurately known. Yet solutions of epidemic models depend strongly on the forcing function, so dynamical predictions---such as changes in epidemic patterns that can be induced by demographic transitions or mass vaccination---are always subject to the objection that the underlying patterns of seasonality are poorly specified. Here, we demonstrate that the key bifurcations of the standard epidemic model are invariant to the shape of seasonal forcing if the amplitude of forcing is appropriately adjusted. Consequently, analyses applicable to real disease dynamics can be conducted with a smooth, idealized sinusoidal forcing function, and qualitative changes in epidemic patterns can be predicted without precise knowledge of the underlying forcing pattern. We find similar invariance in a seasonally forced predator-prey model, and conjecture that this phenomenon---and the associated robustness of predictions---might be a feature of many other periodically forced dynamical systems.