  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.
