TY - JOUR
T1 - Complex Dynamics in a Unified SIR and HIV Disease Model
T2 - A Bifurcation Theory Approach
AU - Yu, Pei
AU - Zhang, Wenjing
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2019/10/1
Y1 - 2019/10/1
N2 - 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.
AB - 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.
KW - A unified SIR and HIV disease model
KW - Bogdanov–Takens bifurcation
KW - Homoclnic orbit
KW - Hopf bifurcation
KW - Limit cycle
KW - Recurrent infection
KW - Stability
KW - The simplest normal form
UR - http://www.scopus.com/inward/record.url?scp=85065172349&partnerID=8YFLogxK
U2 - 10.1007/s00332-019-09550-7
DO - 10.1007/s00332-019-09550-7
M3 - Article
AN - SCOPUS:85065172349
SN - 0938-8974
VL - 29
SP - 2447
EP - 2500
JO - Journal of Nonlinear Science
JF - Journal of Nonlinear Science
IS - 5
ER -