American Institute of Mathematical Sciences

Advanced
2006, 3(2): 313-324. doi: 10.3934/mbe.2006.3.313

Modeling Cancer in HIV-1 Infected Individuals: Equilibria, Cycles and Chaotic Behavior

Jie Lou 1, , Tommaso Ruggeri 2, and  Claudio Tebaldi 3,

1. 

Department of Mathematics, Shanghai University, 99 Shangda Road Shanghai 200444, P. R., China
2. 

Department of Mathematics and, Research Center of Applied Mathematics (CIRAM), University of Bologna, Via Saragozza 8, 40123 Bologna, Italy
3. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received  September 2005 Revised  January 2006 Published  February 2006

For HIV-infected individuals, cancer remains a significant burden. Gaining insight into the epidemiology and mechanisms that underlie AIDS-related cancers can provide us with a better understanding of cancer immunity and viral oncogenesis. In this paper, an HIV-1 dynamical model incorporating the AIDS-related cancer cells was studied. The model consists of three components, cancer cells, healthy CD4+ T lymphocytes and infected CD4+ T lymphocytes, and can have six steady states. We discuss the existence, the stability properties and the biological meanings of these steady states, in particular for the positive one: cancer-HIV-healthy cells steady state. We find conditions for Hopf bifurcation of the positive steady state, leading to periodic solutions, sequences of period doubling bifurcations and appearance of chaos. Further, chaos and periodic behavior alternate. Our results are consistent with some clinical and experimental observations.
Keywords: chaos., HIV-1 infection, Hopf bifurcation, stability, steady state, cancer.
Mathematics Subject Classification: 92C6.
Citation: Jie Lou, Tommaso Ruggeri, Claudio Tebaldi. Modeling Cancer in HIV-1 Infected Individuals: Equilibria, Cycles and Chaotic Behavior. Mathematical Biosciences & Engineering, 2006, 3 (2) : 313-324. doi: 10.3934/mbe.2006.3.313
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