# American Institute of Mathematical Sciences

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
##### References:
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Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells,, Math. Biosci., 165 (2000), 27. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar [6] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313. doi: 10.1137/S0036139902406905. Google Scholar [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. Google Scholar [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, (2000). Google Scholar [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. 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] B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry,", Lecture Notes in Mathematics, 1309 (1988). Google Scholar [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. Google Scholar [13] F. R. Gantmakher, "The Theory of Matrices,", Chelsea Pub. Co., (2000). Google Scholar [14] P. Hartman, "Ordinary Differential Equations,", Reprint of the second edition, (1982). Google Scholar [15] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,", London Mathematical Society Lecture Note Series, 41 (1981). Google Scholar [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. doi: 10.1038/373123a0. Google Scholar [17] S. C. Jameson, Maintaining the norm: T-cell homeostasis,, Nat. Rev. Immunol., 2 (2002), 547. Google Scholar [18] S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful,, Semin. Immunol., 17 (2005), 231. doi: 10.1016/j.smim.2005.02.003. Google Scholar [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. doi: 10.1097/QAI.0b013e3180654836. Google Scholar [20] T. Kato, "Perturbation Theory for Linear Operators,", 2$^{nd}$ edition, (1976). Google Scholar [21] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Second edition, 112 (1998). Google Scholar [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. Google Scholar [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). Google Scholar [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. doi: 10.1016/0022-0396(82)90085-7. Google Scholar [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. doi: 10.1038/nri2674. Google Scholar [26] M. A. Nowak and R. M. May, "Virus Dynamics. Mathematical Principles of Immunology and Virology,", Oxford University Press, (2000). Google Scholar [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. doi: 10.1056/NEJM199302043280508. Google Scholar [28] A. S. Perelson, Modelling viral and immune system dynamics,, Nat. Rev. Immunol., 2 (2002), 28. Google Scholar [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. doi: 10.1038/387188a0. Google Scholar [30] A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A. Google Scholar [31] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [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. doi: 10.1126/science.271.5255.1582. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.mbs.2005.12.026. Google Scholar [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. doi: 10.1006/jdeq.1994.1063. Google Scholar

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. Google Scholar [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. doi: 10.1016/S0140-6736(00)02061-4. Google Scholar [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. doi: 10.1146/annurev.med.53.082901.104024. Google Scholar [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. doi: 10.3934/dcdsb.2011.15.545. Google Scholar [5] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells,, Math. Biosci., 165 (2000), 27. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar [6] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313. doi: 10.1137/S0036139902406905. Google Scholar [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. Google Scholar [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, (2000). Google Scholar [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. 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] B. Fiedler, "Global Bifurcation of Periodic Solutions with Symmetry,", Lecture Notes in Mathematics, 1309 (1988). Google Scholar [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. Google Scholar [13] F. R. Gantmakher, "The Theory of Matrices,", Chelsea Pub. Co., (2000). Google Scholar [14] P. Hartman, "Ordinary Differential Equations,", Reprint of the second edition, (1982). Google Scholar [15] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,", London Mathematical Society Lecture Note Series, 41 (1981). Google Scholar [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. doi: 10.1038/373123a0. Google Scholar [17] S. C. Jameson, Maintaining the norm: T-cell homeostasis,, Nat. Rev. Immunol., 2 (2002), 547. Google Scholar [18] S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful,, Semin. Immunol., 17 (2005), 231. doi: 10.1016/j.smim.2005.02.003. Google Scholar [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. doi: 10.1097/QAI.0b013e3180654836. Google Scholar [20] T. Kato, "Perturbation Theory for Linear Operators,", 2$^{nd}$ edition, (1976). Google Scholar [21] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Second edition, 112 (1998). Google Scholar [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. Google Scholar [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). Google Scholar [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. doi: 10.1016/0022-0396(82)90085-7. Google Scholar [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. doi: 10.1038/nri2674. Google Scholar [26] M. A. Nowak and R. M. May, "Virus Dynamics. Mathematical Principles of Immunology and Virology,", Oxford University Press, (2000). Google Scholar [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. doi: 10.1056/NEJM199302043280508. Google Scholar [28] A. S. Perelson, Modelling viral and immune system dynamics,, Nat. Rev. Immunol., 2 (2002), 28. Google Scholar [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. doi: 10.1038/387188a0. Google Scholar [30] A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells,, Math. Biosci., 114 (1993), 81. doi: 10.1016/0025-5564(93)90043-A. Google Scholar [31] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [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. doi: 10.1126/science.271.5255.1582. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.mbs.2005.12.026. Google Scholar [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. doi: 10.1006/jdeq.1994.1063. Google Scholar
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