Endemic Bubbles Generated by Delayed Behavioral Response: Global Stability and Bifurcation Switches in an SIS Model

Abstract

During infectious disease outbreaks, people may reduce their contact numbers or take other precautions to prevent transmission. The change in their behavior can be directly or indirectly triggered by the density of infected individuals in the population. In this paper, we investigate an SIS (susceptible-infected-susceptible) model where the transmission rate is a decreasing function of the prevalence of the disease (determined by a reduction function $h$), with the assumption that such a change in the transmission rate occurs with some time delay. We prove that if the basic reproduction number $R_0$ is less than one, then the disease-free equilibrium is globally asymptotically stable, while for $R_0>1$ a unique endemic equilibrium exists and the disease uniformly persists, regardless of the delay or the specific form of $h$. However, characterized by the shape of the response function $h$, various dynamics are possible if $R_0>1$. Roughly speaking, if $h$ is decreasing slowly (weak response), then the endemic equilibrium is absolutely globally asymptotically stable. When the slope of $h$ is larger (strong response), then the endemic equilibrium loses its stability and periodic orbits appear via Hopf bifurcation as $R_0$ increases. Further increasing $R_0$, the endemic equilibrium regains its stability, forming an interesting structure in the bifurcation diagram that we call an endemic bubble.