October  2012, 17(7): 2359-2385. doi: 10.3934/dcdsb.2012.17.2359

Mathematical analysis of a HIV model with quadratic logistic growth term

1. 

School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China

2. 

School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China, and Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence cedex

3. 

(MD) INSERM U897, Centre de recherche Epidémiologie et Biostatistique, ISPED, Université de Bordeaux, 33076 Bordeaux Cedex, France, and Centre Hospitalier, Universitaire de Bordeaux, 33076 Bordeaux Cedex, France

Received  December 2011 Revised  May 2012 Published  July 2012

We consider a model of disease dynamics in the modeling of Human Immunodeficiency Virus (HIV). The system consists of three ODEs for the concentrations of the target T cells, the infected cells and the virus particles. There are two main parameters, $N$, the total number of virions produced by one infected cell, and $r$, the logistic parameter which controls the growth rate. The equilibria corresponding to the infected state are asymptotically stable in a region $(\mathcal I)$, but unstable in a region $(\mathcal P)$. In the unstable region, the levels of the various cell types and virus particles oscillate, rather than converging to steady values. Hopf bifurcations occurring at the interfaces are fully investigated via several techniques including asymptotic analysis. The Hopf points are connected through a "snake" of periodic orbits [24]. Numerical results are presented.
Citation: Xinyue Fan, Claude-Michel Brauner, Linda Wittkop. Mathematical analysis of a HIV model with quadratic logistic growth term. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2359-2385. doi: 10.3934/dcdsb.2012.17.2359
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show all references

References:
[1]

Nat. Rev. Immunol., 5 (2005), 101-111. Google Scholar

[2]

Lancet, 355 (2000), 1131-1137. doi: 10.1016/S0140-6736(00)02061-4.  Google Scholar

[3]

Annu. Rev. Med., 53 (2002), 557-593. doi: 10.1146/annurev.med.53.082901.104024.  Google Scholar

[4]

Discrete Cont. Dyn. Syst. Ser. B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.  Google Scholar

[5]

Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[6]

SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.  Google Scholar

[7]

SIAM J. Appl. Math., 42 (1982), 956-963.  Google Scholar

[8]

Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.  Google Scholar

[9]

Nature, 417 (2002), 95-98. doi: 10.1038/417095a.  Google Scholar

[10]

X. Y. Fan, C.-M. Brauner and L. Lorenzi, Two dimensional stability in a HIV model with logistic growth term,, to appear., ().   Google Scholar

[11]

Lecture Notes in Mathematics, 1309, Springer-Verlag, Berlin, 1988.  Google Scholar

[12]

Nat. Med., 5 (1999), 512-517. Google Scholar

[13]

Chelsea Pub. Co., 2000. Google Scholar

[14]

Reprint of the second edition, Birkhaüser, Boston, Mass., 1982.  Google Scholar

[15]

London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.  Google Scholar

[16]

Nature, 373 (1995), 123-126. doi: 10.1038/373123a0.  Google Scholar

[17]

Nat. Rev. Immunol., 2 (2002), 547-556. Google Scholar

[18]

Semin. Immunol., 17 (2005), 231-237. doi: 10.1016/j.smim.2005.02.003.  Google Scholar

[19]

J. Acquir. Immune. Defic. Syndr., 45 (2007), 483-493. doi: 10.1097/QAI.0b013e3180654836.  Google Scholar

[20]

2$^nd$ edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[21]

Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.  Google Scholar

[22]

AIDS, 19 (2005), 279-286. Google Scholar

[23]

J. Clin. Invest., 119 (2009), 997. Google Scholar

[24]

J. Differ. Equations, 43 (1982), 419-450. doi: 10.1016/0022-0396(82)90085-7.  Google Scholar

[25]

Nat. Rev. Immunol., 10 (2009), 11-23. doi: 10.1038/nri2674.  Google Scholar

[26]

Oxford University Press, Oxford, 2000.  Google Scholar

[27]

N. Engl. J. Med., 328 (1993), 327-335. doi: 10.1056/NEJM199302043280508.  Google Scholar

[28]

Nat. Rev. Immunol., 2 (2002), 28-36. Google Scholar

[29]

Nature, 387 (1997), 188-191. doi: 10.1038/387188a0.  Google Scholar

[30]

Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[31]

SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.  Google Scholar

[32]

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Nat. Med., 9 (2003), 727-728. Google Scholar

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Math. Biosci., 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026.  Google Scholar

[35]

J. Differ. Equations, 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.  Google Scholar

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