Modeling Cancer in HIV-1 Infected Individuals: Equilibria, Cycles and Chaotic Behavior
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Department of Mathematics, Shanghai University, 99 Shangda Road Shanghai 200444, P. R.
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Department of Mathematics and, Research Center of Applied Mathematics (CIRAM), University of Bologna, Via Saragozza 8, 40123 Bologna
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Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino
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Received:
01 September 2005
Accepted:
29 June 2018
Published:
01 February 2006
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MSC :
92C60.
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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.
Citation: Jie Lou, Tommaso Ruggeri, Claudio Tebaldi. Modeling Cancer in HIV-1 Infected Individuals: Equilibria, Cycles and Chaotic Behavior[J]. Mathematical Biosciences and Engineering, 2006, 3(2): 313-324. doi: 10.3934/mbe.2006.3.313
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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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