Effects of seasonal variation patterns on recurrent outbreaks in epidemic models

Transmission of infectious diseases often depends on seasonal variability. Mathematical epidemic models driven by seasonal forcing have been widely explored to understand recurrent outbreaks of infectious diseases. Here we present an effective method to examine the impact of seasonal variation patterns on epidemic dynamics. The idea is to represent the seasonal variability as a piecewise constant function and analyze the seasonally forced epidemic model by means of a numerical shooting method for switched dynamical systems. Several illustrative examples demonstrate that our method is useful to elucidate the effects of various types of seasonality in outbreak behavior. First, we clarify an effect of the shape of seasonal forcing by comparing sinusoidal and square wave forcing functions. Second, we demonstrate that not only the intensity of seasonality but also its temporal variation pattern significantly influences the outbreak pattern. Finally, we reveal the mechanisms of transitions between different outbreak patterns in an epidemic model driven by realistic term-time seasonal forcing and one driven by seasonal forcing estimated from real data. Our results suggest that accurately estimated seasonal variability is necessary for better understanding the dynamics of infectious diseases.