Behavior-induced oscillations in epidemic outbreaks with distributed memory: beyond the linear chain trick using numerical methods

We consider a model for an infectious disease outbreak, when the depletion of susceptible is negligible, and assume that individuals adapt their behavior according to the information they receive about the new cases. In line with the information index approach, we assume that individuals react to the past information according to a memory kernel that is continuously distributed in the past. We analyze equilibria and their stability, with analytical results for selected cases. Thanks to the recently developed pseudospectral approximation of delay equations, we study numerically the long-term dynamics of the model for memory kernels defined by Gamma distributions with general non-integer shape parameter, extending the analysis beyond what is allowed by the linear chain trick. In agreement with previous studies, we show that behavior adaptation alone can cause sustained waves of infections even in an outbreak scenario, and notably in the absence of other processes like demographic turnover, seasonality or waning of immunity. Our analysis gives a more general insight into how the period and peak of the epidemic waves depend on the shape of the memory kernel and how the level of minimal contact (essential service) impacts on the stability of the behavior-induced positive equilibrium.