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Seasonal variation in environmental variables, and in rates of contact amongindividuals, are fundamental drivers of infectious disease dynamics. Unlikemost periodically-forced physical systems, for which the precise pattern offorcing is typically known, underlying patterns of seasonal variation intransmission rates can be estimated approximately at best, and only the periodof forcing is accurately known. Yet solutions of epidemic models dependstrongly on the forcing function, so dynamical predictions---such as changes inepidemic patterns that can be induced by demographic transitions or massvaccination---are always subject to the objection that the underlying patternsof seasonality are poorly specified. Here, we demonstrate that the keybifurcations of the standard epidemic model are invariant to the shape ofseasonal forcing if the amplitude of forcing is appropriately adjusted.Consequently, analyses applicable to real disease dynamics can be conductedwith a smooth, idealized sinusoidal forcing function, and qualitative changesin epidemic patterns can be predicted without precise knowledge of theunderlying forcing pattern. We find similar invariance in a seasonally forcedpredator-prey model, and conjecture that this phenomenon---and the associatedrobustness of predictions---might be a feature of many other periodicallyforced dynamical systems.

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