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

Jie Lou; Tommaso Ruggeri; Claudio Tebaldi

doi:10.3934/mbe.2006.3.313

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2006, Volume 3, Issue 2: 313-324. Doi: 10.3934/mbe.2006.3.313

This issue Previous Article The Effects of Vertical Transmission on the Spread of HIV/AIDS in the Presence of Treatment Next Article Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model

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

1.
Department of Mathematics, Shanghai University, 99 Shangda Road Shanghai 200444, P. R.
2.
Department of Mathematics and, Research Center of Applied Mathematics (CIRAM), University of Bologna, Via Saragozza 8, 40123 Bologna
3.
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received: August 31, 2005

Revised: December 31, 2005

Published: January 2006

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Abstract

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.

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Mathematics Subject Classification: 92C60.

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