Complex Dynamics in a Unified SIR and HIV Disease Model: A Bifurcation Theory Approach

Abstract

This paper is concerned with complex dynamical behaviors of a simple unified SIR and HIV disease model with a convex incidence and four real parameters. Due to the complex nature of the disease dynamics, our goal is to explore bifurcations including multistable states, limit cycles, and homoclinic loops in the whole parameter space. The first contribution is the proof of the existence of multiple limit cycles giving rise from Hopf bifurcation, which further induces bistable or tristable states because of the coexistence of stable equilibria and periodic motion. Next, we propose that the existence of Bogdanov–Takens (BT) bifurcation yields the bifurcation of homoclinic loops, which provides a new mechanism for generating disease recurrence, for example, the relapse–remission, viral blip cycles in HIV infection. Last, we present a novel method for the derivation of the normal forms of codimension two and three BT bifurcations. The method is based on the simplest normal form theory from Yu’s previous works.