Stability Switches, Hopf Bifurcation and Chaotic Dynamics in Simple Epidemic Model with State-Dependent Delay

Dynamic behavior investigations of infectious disease models are central to improve our understanding of emerging characteristics of model states interaction. Here, we consider a Susceptible-Infected (SI) model with a general state-dependent delay, which covers an immuno-epidemiological model of pathogen transmission, developed in our early study, using a threshold delay to examine the effects of multiple exposures to a pathogen. The analysis in the previous work showed the appearance of forward as well as backward bifurcations of endemic equilibria when the basic reproductive ratio R0 is less than unity. The analysis, in the present work, of the endemically infected equilibrium behavior, through the study of a second order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation on the upper branch of the backward bifurcation diagram and gives the criteria for stability switches. Furthermore, the inclusion of state-dependent delays is shown to entirely change the dynamics of the SI model and give rise to rich behaviors including periodic, torus and chaotic dynamics.