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 and Continuous Dynamical Systems - B, 2012, 17 (7) : 2359-2385. doi: 10.3934/dcdsb.2012.17.2359
References:
[1]

R. Antia, V. V. Ganusov and R. Ahmed, The role of models in understanding CD8+ T-cell memory, Nat. Rev. Immunol., 5 (2005), 101-111.

[2]

A. Babiker, S. Darby, D. De Angelis, D. Kwart, K. Porter, V. Beral, J. Darbyshire, N. Day, N. Gill and R. Coutinho, et. al, Time from HIV-1 seroconversion to AIDS and death before widespread use of highly-active antiretroviral therapy: A collaborative re-analysis, Lancet, 355 (2000), 1131-1137. doi: 10.1016/S0140-6736(00)02061-4.

[3]

J. N. Blankson, D. Persaud and R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection, Annu. Rev. Med., 53 (2002), 557-593. doi: 10.1146/annurev.med.53.082901.104024.

[4]

C.-M. Brauner, D. Jolly, L. Lorenzi and R. Thiébaut, Heterogeneous viral environment in a HIV spatial model, Discrete Cont. Dyn. Syst. Ser. B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.

[5]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.

[6]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[7]

D. Dellwo, H. B. Keller, B. J. Matkowsky and E. L. Reiss, On the birth of isolas, SIAM J. Appl. Math., 42 (1982), 956-963.

[8]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

[9]

D. C. Douek, J. M. Brenchley, M. R. Betts, D. R. Ambrozak, B. J. Hill, Y. Okamoto, J. P. Casazza, J. Kuruppu, K. Kunstman and S. Wolinsky, et. al, HIV preferentially infects HIV-specific CD4+ T cells, Nature, 417 (2002), 95-98. doi: 10.1038/417095a.

[10]

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

[11]

B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry," Lecture Notes in Mathematics, 1309, Springer-Verlag, Berlin, 1988.

[12]

D. Finzi, J. Blankson, J. D. Siliciano, J. B. Margolick, K. Chadwick, T. Pierson, K. Smith, J. Lisziewicz, F. Lori, C. Flexner and others, Latent infection of CD4+ T cells provides a mechanism for lifelong persistence of HIV-1, even in patients on effective combination therapy, Nat. Med., 5 (1999), 512-517.

[13]

F. R. Gantmakher, "The Theory of Matrices," Chelsea Pub. Co., 2000.

[14]

P. Hartman, "Ordinary Differential Equations," Reprint of the second edition, Birkhaüser, Boston, Mass., 1982.

[15]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.

[16]

D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, et. al, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126. doi: 10.1038/373123a0.

[17]

S. C. Jameson, Maintaining the norm: T-cell homeostasis, Nat. Rev. Immunol., 2 (2002), 547-556.

[18]

S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful, Semin. Immunol., 17 (2005), 231-237. doi: 10.1016/j.smim.2005.02.003.

[19]

L. E. Jones and A. S. Perelson, Transient viremia, plasma viral load, and reservoir replenishment in HIV-infected patients on antiretroviral therapy, J. Acquir. Immune. Defic. Syndr., 45 (2007), 483-493. doi: 10.1097/QAI.0b013e3180654836.

[20]

T. Kato, "Perturbation Theory for Linear Operators," 2$^nd$ edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.

[21]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

[22]

Y. Levy, H. Gahery-Segard, C. Durier, A. S. Lascaux, C. Goujard, V. Meiffredy, C. Rouzioux, R. E. Habib, M. Beumont-Mauviel and J. G. Guillet, et. al, Immunological and virological efficacy of a therapeutic immunization combined with interleukin-2 in chronically HIV-1 infected patients, AIDS, 19 (2005), 279-286.

[23]

Y. Levy, C. Lacabaratz, L. Weiss, J. P. Viard, C. Goujard, J. D. Lelievre, F. Boue, J. M. Molina, C. Rouzioux and V. Avettand-Fenoel, et. al, Enhanced T cell recovery in HIV-1-infected adults through IL-7 treatment, J. Clin. Invest., 119 (2009), 997.

[24]

J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differ. Equations, 43 (1982), 419-450. doi: 10.1016/0022-0396(82)90085-7.

[25]

A. J. McMichael, P. Borrow, G. D. Tomaras, N. Goonetilleke and B. F. Haynes, The immune response during acute HIV-1 infection: Clues for vaccine development, Nat. Rev. Immunol., 10 (2009), 11-23. doi: 10.1038/nri2674.

[26]

M. A. Nowak and R. M. May, "Virus Dynamics. Mathematical Principles of Immunology and Virology," Oxford University Press, Oxford, 2000.

[27]

G. Pantaleo, C. Graziosi and A. S. Fauci, New concepts in the immunopathogenesis of human immunodeficiency virus infection, N. Engl. J. Med., 328 (1993), 327-335. doi: 10.1056/NEJM199302043280508.

[28]

A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36.

[29]

A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188-191. doi: 10.1038/387188a0.

[30]

A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.

[31]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[32]

A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.

[33]

J. D. Siliciano, J. Kajdas, D. Finzi, T. C. Quinn, K. Chadwick, J. B. Margolick, C. Kovacs, S. J. Gange and R. F. Siliciano, Long-term follow-up studies confirm the stability of the latent reservoir for HIV-1 in resting CD4+ T cells, Nat. Med., 9 (2003), 727-728.

[34]

L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026.

[35]

H.-R. Zhu and H. L. Smith, Stable periodic orbits for a class of three dimensional competitive systems, J. Differ. Equations, 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.

show all references

References:
[1]

R. Antia, V. V. Ganusov and R. Ahmed, The role of models in understanding CD8+ T-cell memory, Nat. Rev. Immunol., 5 (2005), 101-111.

[2]

A. Babiker, S. Darby, D. De Angelis, D. Kwart, K. Porter, V. Beral, J. Darbyshire, N. Day, N. Gill and R. Coutinho, et. al, Time from HIV-1 seroconversion to AIDS and death before widespread use of highly-active antiretroviral therapy: A collaborative re-analysis, Lancet, 355 (2000), 1131-1137. doi: 10.1016/S0140-6736(00)02061-4.

[3]

J. N. Blankson, D. Persaud and R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection, Annu. Rev. Med., 53 (2002), 557-593. doi: 10.1146/annurev.med.53.082901.104024.

[4]

C.-M. Brauner, D. Jolly, L. Lorenzi and R. Thiébaut, Heterogeneous viral environment in a HIV spatial model, Discrete Cont. Dyn. Syst. Ser. B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.

[5]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.

[6]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.

[7]

D. Dellwo, H. B. Keller, B. J. Matkowsky and E. L. Reiss, On the birth of isolas, SIAM J. Appl. Math., 42 (1982), 956-963.

[8]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000.

[9]

D. C. Douek, J. M. Brenchley, M. R. Betts, D. R. Ambrozak, B. J. Hill, Y. Okamoto, J. P. Casazza, J. Kuruppu, K. Kunstman and S. Wolinsky, et. al, HIV preferentially infects HIV-specific CD4+ T cells, Nature, 417 (2002), 95-98. doi: 10.1038/417095a.

[10]

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

[11]

B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry," Lecture Notes in Mathematics, 1309, Springer-Verlag, Berlin, 1988.

[12]

D. Finzi, J. Blankson, J. D. Siliciano, J. B. Margolick, K. Chadwick, T. Pierson, K. Smith, J. Lisziewicz, F. Lori, C. Flexner and others, Latent infection of CD4+ T cells provides a mechanism for lifelong persistence of HIV-1, even in patients on effective combination therapy, Nat. Med., 5 (1999), 512-517.

[13]

F. R. Gantmakher, "The Theory of Matrices," Chelsea Pub. Co., 2000.

[14]

P. Hartman, "Ordinary Differential Equations," Reprint of the second edition, Birkhaüser, Boston, Mass., 1982.

[15]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge-New York, 1981.

[16]

D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, et. al, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126. doi: 10.1038/373123a0.

[17]

S. C. Jameson, Maintaining the norm: T-cell homeostasis, Nat. Rev. Immunol., 2 (2002), 547-556.

[18]

S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful, Semin. Immunol., 17 (2005), 231-237. doi: 10.1016/j.smim.2005.02.003.

[19]

L. E. Jones and A. S. Perelson, Transient viremia, plasma viral load, and reservoir replenishment in HIV-infected patients on antiretroviral therapy, J. Acquir. Immune. Defic. Syndr., 45 (2007), 483-493. doi: 10.1097/QAI.0b013e3180654836.

[20]

T. Kato, "Perturbation Theory for Linear Operators," 2$^nd$ edition, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976.

[21]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

[22]

Y. Levy, H. Gahery-Segard, C. Durier, A. S. Lascaux, C. Goujard, V. Meiffredy, C. Rouzioux, R. E. Habib, M. Beumont-Mauviel and J. G. Guillet, et. al, Immunological and virological efficacy of a therapeutic immunization combined with interleukin-2 in chronically HIV-1 infected patients, AIDS, 19 (2005), 279-286.

[23]

Y. Levy, C. Lacabaratz, L. Weiss, J. P. Viard, C. Goujard, J. D. Lelievre, F. Boue, J. M. Molina, C. Rouzioux and V. Avettand-Fenoel, et. al, Enhanced T cell recovery in HIV-1-infected adults through IL-7 treatment, J. Clin. Invest., 119 (2009), 997.

[24]

J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differ. Equations, 43 (1982), 419-450. doi: 10.1016/0022-0396(82)90085-7.

[25]

A. J. McMichael, P. Borrow, G. D. Tomaras, N. Goonetilleke and B. F. Haynes, The immune response during acute HIV-1 infection: Clues for vaccine development, Nat. Rev. Immunol., 10 (2009), 11-23. doi: 10.1038/nri2674.

[26]

M. A. Nowak and R. M. May, "Virus Dynamics. Mathematical Principles of Immunology and Virology," Oxford University Press, Oxford, 2000.

[27]

G. Pantaleo, C. Graziosi and A. S. Fauci, New concepts in the immunopathogenesis of human immunodeficiency virus infection, N. Engl. J. Med., 328 (1993), 327-335. doi: 10.1056/NEJM199302043280508.

[28]

A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28-36.

[29]

A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz and D. D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188-191. doi: 10.1038/387188a0.

[30]

A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.

[31]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[32]

A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.

[33]

J. D. Siliciano, J. Kajdas, D. Finzi, T. C. Quinn, K. Chadwick, J. B. Margolick, C. Kovacs, S. J. Gange and R. F. Siliciano, Long-term follow-up studies confirm the stability of the latent reservoir for HIV-1 in resting CD4+ T cells, Nat. Med., 9 (2003), 727-728.

[34]

L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026.

[35]

H.-R. Zhu and H. L. Smith, Stable periodic orbits for a class of three dimensional competitive systems, J. Differ. Equations, 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.

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