Global Continuation of Epidemic Periodic Waves in Delay-Algebraic Systems

Abstract

We consider a hybrid system of delay-algebraic equations motivated by an epidemic model involving both disease transmission and behavioral adaptation, and we show how the 1 -degree theoretic approach towards global bifurcations of nonlinear systems with two parameters can be adopted to develop a global Hopf bifurcation theory for such a hybrid system.We use the epidemic models, previously developed to explore mechanisms of multiple waves during the acute phase and postpandemic phase of an emerging disease, to illustrate the effectiveness and challenges of applying the global Hopf bifurcation theory to examine the global continuation of local Hopf bifurcations when the critical parameter (implementation delay) moves away from the critical value for onset of the oscillation.We also show how a 3-dimensional ordinary differential competitive system with built-in cyclic symmetry and embedded negative loops arises naturally from our global Hopf bifurcation study.