# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021238
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## Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics

 1 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 2 Center for Basic Teaching and Experiment, Nanjing University of Science and Technology, Jiangyin 214443, China 3 Aliyun School of Big Data, Changzhou University, Changzhou, 213164, China

* Corresponding author: Zhipeng Qiu

Received  April 2021 Revised  August 2021 Early access October 2021

Fund Project: Zhipeng Qiu is supported by the National Natural Science Foundation of China (12071217). Ting Guo is supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX20_0243)

HIV infects active uninfected CD4$^+$ T cells, and the active CD4$^+$ T cells are transformed from quiescent state in response to antigenic activation. Activation effect of the CD4$^+$ T cells may play an important role in HIV infection. In this paper, we formulate a mathematical model to investigate the activation effect of CD4$^+$ T cells on HIV dynamics. In the model, the uninfected CD4$^+$ T cells are divided into two pools: quiescent and active, and the stimuli rate of quiescent cells by HIV is described by saturated form function. We derive the basic reproduction number $R_0$ and analyze the existence and the stability of equilibria. Numerical simulations confirm that the system may have backward bifurcation and Hopf bifurcation. The results imply that $R_0$ cannot completely determine the dynamics of the system and the system may have complex dynamics, which are quite different from the models without the activation effect of CD4$^+$ T cells. Some numerical results are further presented to assess the activation parameters on HIV dynamics. The simulation results show that the changes of the activation parameters can cause the system periodic oscillation, and activation rate by HIV may induce the supercritical Hopf bifurcation and subcritical Hopf bifurcation. Finally, we proceed to investigate the effect of activation on steady-state viral loads during antiretroviral therapy. The results indicate that, viral load may exist and remain high level even if antiretroviral therapy is effective to reduce the basic reproduction number below 1.

Citation: Linghui Yu, Zhipeng Qiu, Ting Guo. Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021238
##### References:
 [1] P. Aavania and L. J. S. Allen, The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection, Appl. Math. Model., 75 (2019), 210-222.  doi: 10.1016/j.apm.2019.05.028. [2] K. B. Blyuss and L. B. Nicholson, The role of tunable activation thresholds in the dynamics of autoimmunity, J. Theor. Biol., 308 (2012), 45-55.  doi: 10.1016/j.jtbi.2012.05.019. [3] K. B. Blyuss and L. B. Nicholson, Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease, J. Theor. Biol., 375 (2015), 13-20. [4] C. Bunce and E. B. Bell, CD45RC isoforms define two types of CD4 memory T cells, one of which depends on persisting antigen, J. Exp. Med., 185 (1997), 767-776.  doi: 10.1084/jem.185.4.767. [5] E. B. Bell, S. M. Sparshott and C. Bunce, CD4+ T-cell memory, CD45R subsets and the persistence of antigen-a unifying concept, Trends in Immunology, 19 (1998), 60-64.  doi: 10.1016/S0167-5699(97)01211-5. [6] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Aced. Sci., 94 (1997), 6971-6976.  doi: 10.1073/pnas.94.13.6971. [7] M. Bukrinsky, T. Stanwick, M. Dempsey and M. Stevenson, Quiescent T lymphocytes as an inducible virus reservoir in HIV-1 infection, Science, 254 (1991), 423-427.  doi: 10.1126/science.1925601. [8] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.  doi: 10.1006/bulm.2001.0266. [9] A. H. Courtney, W. L. Lo and A. Weiss, Tcr signaling: Mechanisms of initiation and propagation, Trends Biochem. Sci., 43 (2018), 108-123.  doi: 10.1016/j.tibs.2017.11.008. [10] S. Cemerski and A. Shaw, Immune synapses in T-cell activation, Curr. Opin. Immunol., 18 (2006), 298-304.  doi: 10.1016/j.coi.2006.03.011. [11] D. A. Cantrell and K. A. Smith, Transient expression of interleukin 2 receptors, Consequences for T cell growth, J. Exp. Med., 158 (1983), 1895-1911.  doi: 10.1084/jem.158.6.1895. [12] D. Cantrell, T cell antigen receptor signal transduction pathways, Annu. Rev. Immunol., 14 (1996), 259-274.  doi: 10.1146/annurev.immunol.14.1.259. [13] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [14] C. Ding, Z. Qiu and H. Zhu, Multi-host transmission dynamics of schistosomiasis and its optimal control, Math. Biosci. Eng., 12 (2015), 983-1006.  doi: 10.3934/mbe.2015.12.983. [15] R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theor. Biol., 190 (1998), 201-214. [16] C. Doyle and J. L. Strominger, Interaction between CD4 and class Ⅱ MHC molecules mediates cell adhesion, Nature, 330 (1987), 256-259.  doi: 10.1038/330256a0. [17] Q. Deng, Z. Qiu, T. Guo and L. Rong, Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance, Discrete Contin. Dyn. Syst. B, 26 (2021), 3543-3562.  doi: 10.3934/dcdsb.2020245. [18] R. J. De Boer, P. Hogeweg, H. F. Dullens, R. A. De Weger and W. Den Otter, Macrophage T lymphocyte interactions in the anti-tumor immune response: A mathematical model, J. Immunol., 134 (1985), 2748-2758. [19] R. J. De Boer, D. Homann and A. S. Perelson, Different dynamics of CD4++ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection, J. Immunol., 171 (2003), 3928-3935. [20] A. Fauci, Multifactorial nature of human immunodeficiency virus disease: Implications for therapy, Science, 262 (1993), 1011-1018.  doi: 10.1126/science.8235617. [21] F. Fatehi Chenar, Y. N. Kyrychko and K. B. Blyuss, Mathematical model of immune response to hepatitis B, J. Theor. Biol., 447 (2018), 98-110.  doi: 10.1016/j.jtbi.2018.03.025. [22] T. Guo, Z. Qiu and L. Rong, Modeling the role of macrophages in HIV persistence during antiretroviral therapy, J. Math. Bio., 81 (2020), 369-402.  doi: 10.1007/s00285-020-01513-x. [23] S. D. Gowda, B. S. Stein, N. Mohagheghpour, C. J. Benike and E. G. Engleman, Evidence that T cell activation is required for HIV-1 entry in CD4+ lymphocytes, J. Immunol., 142 (1989), 773-780. [24] A. Hashimoto-Tane and T. Saito, Dynamic regulation of TCR-microclusters and the microsynapse for T cell activation, Front. Immunol., 7 (2016). doi: 10.3389/fimmu.2016.00255. [25] D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.  doi: 10.1038/373123a0. [26] A. L. Hill, D. I. S. Rosenbloom, M. A. Nowak and R. F. Siliciano, Insight into treatment of HIV infection from viral dynamics models, Immunol. Rev., 285 (2018), 9-25.  doi: 10.1111/imr.12698. [27] C. Janeway, P. Travers, M. Walport and M. Schlomchik, Immunobiology: The Immune System in Health and Disease, Garland Science Publishing, New York, 2005. [28] P. S. Kim and P. P. Lee, T cell state transition produces an emergent change detector, J. Theor. Biol., 275 (2011), 59-69.  doi: 10.1016/j.jtbi.2011.01.031. [29] A. Lanzavecchia, G. Iezzi and A. Viola, From TCR engagement to T cell activation: A kinetic view of T cell behavior, Cell, 96 (1999), 1-4. [30] L. Lu, K. Ikizawa, D. Hu, M. B. F. Werneck, K. W. Wucherpfennig and H. Cantor, Regulation of activated CD4+ T cells by NK Cells via the Qa-1-NKG2A inhibitory pathway,, Immunity, 26 (2007), 593-604.  doi: 10.1016/j.immuni.2007.03.017. [31] D. A. Mitchell, X. Cui, R. J. Schmittling, L. Sanchez-Perez, D. J. Snyder, K. L. Congdon, G. E. Archer, A. Desjardins, A. H. Friedman, H. S. Friedman, J. E. Herndon, R. E. McLendon, D. A. Reardon, J. J. Vredenburgh, D. D. Bigner and J. H. Sampson, Monoclonal antibody blockade of IL-2 receptor Á during lymphopenia selectively depletes regulatory T cells in mice and humans, Blood, 118 (2011), 3003-3012.  doi: 10.1182/blood-2011-02-334565. [32] S. J. Merrill, A model of the role of natural killer cells in immune surveillance, J. Math. Biol., 12 (1981), 363-373.  doi: 10.1007/BF00276923. [33] H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia CML and T cell interaction, J. Theor. Biol., 227 (2004), 513-523.  doi: 10.1016/j.jtbi.2003.11.024. [34] J. S. McDougal, A. Mawle, S. P. Cort, J. K. Nicholson, G. D. Cross, J. A. Scheppler-Campbell, D. Hicks and J. Sligh, Cellular tropism of the human retrovirus HTLV-Ⅲ/LAV. I. Role of T cell activation and expression of the T4 antigen, J. Immunol., 135 (1985), 3151-3162. [35] J. M. Murray, G. Kaufmann, A. D. Kelleher and D. A. Cooper, A model of primary HIV-1 infection, Math. Biosic., 154 (1998), 57-85.  doi: 10.1016/S0025-5564(98)10046-9. [36] A. R. McLean and M. A. Nowak, Models of interactions between HIV and other pathogens, J. theor. Biol., 155 (1992), 69-86.  doi: 10.1016/S0022-5193(05)80549-1. [37] J. S. McDougal, A. Mawle, S. P. Cort, J. K. Nicholson, G. D. Cross, J. A. Scheppler-Campbell, D. Hicks and J. Sligh, Cellular tropism of the human retrovirus HTLV-Ⅲ/LAV, J. Immunol., 135 (1985), 3151-3162. [38] A. R. McLean and T. B. L. Kirkwood, A model of human immunodeficiency virus infection in T helper cell clones, J. Theor. Biol., 147 (1990), 177-203.  doi: 10.1016/S0022-5193(05)80051-7. [39] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [40] 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. [41] L. Rong and A. S. Perelson, Modeling latently infected cell activation: Viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy, PLOS Comput. Biol., 5 (2009), 18pp. doi: 10.1371/journal.pcbi.1000533. [42] H. A. Van Den Berg and D. A. Rand, Dynamicsof T cell activation threshold tuning, J. Theor. Bio., 228 (2004), 397-416. [43] N. Vrisekoop, I. Den Braber, A. B. De Boer, A. F. C. Ruiter and M. T. Ackerman, Sparse production but preferential incorporation of recently produced naive T cells in the human peripheral pool, Proc Natl Aced Sci., 105 (2008), 6115-6120.  doi: 10.1073/pnas.0709713105. [44] L. Santarpia, A. K. El-Naggar, G. J. Cote, J. N. Myers and S. I. Sherman, Phosphatidylinositol 3-kinase/akt and ras/raf-mitogen-activated protein kinase pathway mutations in anaplastic thyroid cancer, J. Clin. Endocrinol. Metab., 93 (2008), 278-284.  doi: 10.1210/jc.2007-1076. [45] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bio., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [46] R. A. Weiss, How does HIV cause aids?, Science, 260 (1993), 1273-1279.  doi: 10.1126/science.8493571. [47] S. Wang, P. Hottz, M. Schechter and L. Rong, Modeling the slow CD4+ T cell decline in HIV-Infected individuals, PLoS Comput. Biol., 11 (2015). doi: 10.1371/journal.pcbi.1004665. [48] D. Zagury, J. Bernard, R. Leonard, R. Cheynier, M. Feldman, P. S. Sarin and R. C. Gallo, Long-term cultures of HTLV-Ⅲ-infected t cells: A model of cytopathology of T-cell depletion in AIDS, Science, 231 (1986), 850-853.  doi: 10.1126/science.2418502. [49] UNAIDS, Fact sheet-Latest statistics on the status of the AIDS epidemic, http://www.unaids.org/en/resources/fact-sheet, 2016.

show all references

##### References:
 [1] P. Aavania and L. J. S. Allen, The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection, Appl. Math. Model., 75 (2019), 210-222.  doi: 10.1016/j.apm.2019.05.028. [2] K. B. Blyuss and L. B. Nicholson, The role of tunable activation thresholds in the dynamics of autoimmunity, J. Theor. Biol., 308 (2012), 45-55.  doi: 10.1016/j.jtbi.2012.05.019. [3] K. B. Blyuss and L. B. Nicholson, Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease, J. Theor. Biol., 375 (2015), 13-20. [4] C. Bunce and E. B. Bell, CD45RC isoforms define two types of CD4 memory T cells, one of which depends on persisting antigen, J. Exp. Med., 185 (1997), 767-776.  doi: 10.1084/jem.185.4.767. [5] E. B. Bell, S. M. Sparshott and C. Bunce, CD4+ T-cell memory, CD45R subsets and the persistence of antigen-a unifying concept, Trends in Immunology, 19 (1998), 60-64.  doi: 10.1016/S0167-5699(97)01211-5. [6] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Aced. Sci., 94 (1997), 6971-6976.  doi: 10.1073/pnas.94.13.6971. [7] M. Bukrinsky, T. Stanwick, M. Dempsey and M. Stevenson, Quiescent T lymphocytes as an inducible virus reservoir in HIV-1 infection, Science, 254 (1991), 423-427.  doi: 10.1126/science.1925601. [8] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.  doi: 10.1006/bulm.2001.0266. [9] A. H. Courtney, W. L. Lo and A. Weiss, Tcr signaling: Mechanisms of initiation and propagation, Trends Biochem. Sci., 43 (2018), 108-123.  doi: 10.1016/j.tibs.2017.11.008. [10] S. Cemerski and A. Shaw, Immune synapses in T-cell activation, Curr. Opin. Immunol., 18 (2006), 298-304.  doi: 10.1016/j.coi.2006.03.011. [11] D. A. Cantrell and K. A. Smith, Transient expression of interleukin 2 receptors, Consequences for T cell growth, J. Exp. Med., 158 (1983), 1895-1911.  doi: 10.1084/jem.158.6.1895. [12] D. Cantrell, T cell antigen receptor signal transduction pathways, Annu. Rev. Immunol., 14 (1996), 259-274.  doi: 10.1146/annurev.immunol.14.1.259. [13] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [14] C. Ding, Z. Qiu and H. Zhu, Multi-host transmission dynamics of schistosomiasis and its optimal control, Math. Biosci. Eng., 12 (2015), 983-1006.  doi: 10.3934/mbe.2015.12.983. [15] R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theor. Biol., 190 (1998), 201-214. [16] C. Doyle and J. L. Strominger, Interaction between CD4 and class Ⅱ MHC molecules mediates cell adhesion, Nature, 330 (1987), 256-259.  doi: 10.1038/330256a0. [17] Q. Deng, Z. Qiu, T. Guo and L. Rong, Modeling within-host viral dynamics: The role of CTL immune responses in the evolution of drug resistance, Discrete Contin. Dyn. Syst. B, 26 (2021), 3543-3562.  doi: 10.3934/dcdsb.2020245. [18] R. J. De Boer, P. Hogeweg, H. F. Dullens, R. A. De Weger and W. Den Otter, Macrophage T lymphocyte interactions in the anti-tumor immune response: A mathematical model, J. Immunol., 134 (1985), 2748-2758. [19] R. J. De Boer, D. Homann and A. S. Perelson, Different dynamics of CD4++ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection, J. Immunol., 171 (2003), 3928-3935. [20] A. Fauci, Multifactorial nature of human immunodeficiency virus disease: Implications for therapy, Science, 262 (1993), 1011-1018.  doi: 10.1126/science.8235617. [21] F. Fatehi Chenar, Y. N. Kyrychko and K. B. Blyuss, Mathematical model of immune response to hepatitis B, J. Theor. Biol., 447 (2018), 98-110.  doi: 10.1016/j.jtbi.2018.03.025. [22] T. Guo, Z. Qiu and L. Rong, Modeling the role of macrophages in HIV persistence during antiretroviral therapy, J. Math. Bio., 81 (2020), 369-402.  doi: 10.1007/s00285-020-01513-x. [23] S. D. Gowda, B. S. Stein, N. Mohagheghpour, C. J. Benike and E. G. Engleman, Evidence that T cell activation is required for HIV-1 entry in CD4+ lymphocytes, J. Immunol., 142 (1989), 773-780. [24] A. Hashimoto-Tane and T. Saito, Dynamic regulation of TCR-microclusters and the microsynapse for T cell activation, Front. Immunol., 7 (2016). doi: 10.3389/fimmu.2016.00255. [25] D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.  doi: 10.1038/373123a0. [26] A. L. Hill, D. I. S. Rosenbloom, M. A. Nowak and R. F. Siliciano, Insight into treatment of HIV infection from viral dynamics models, Immunol. Rev., 285 (2018), 9-25.  doi: 10.1111/imr.12698. [27] C. Janeway, P. Travers, M. Walport and M. Schlomchik, Immunobiology: The Immune System in Health and Disease, Garland Science Publishing, New York, 2005. [28] P. S. Kim and P. P. Lee, T cell state transition produces an emergent change detector, J. Theor. Biol., 275 (2011), 59-69.  doi: 10.1016/j.jtbi.2011.01.031. [29] A. Lanzavecchia, G. Iezzi and A. Viola, From TCR engagement to T cell activation: A kinetic view of T cell behavior, Cell, 96 (1999), 1-4. [30] L. Lu, K. Ikizawa, D. Hu, M. B. F. Werneck, K. W. Wucherpfennig and H. Cantor, Regulation of activated CD4+ T cells by NK Cells via the Qa-1-NKG2A inhibitory pathway,, Immunity, 26 (2007), 593-604.  doi: 10.1016/j.immuni.2007.03.017. [31] D. A. Mitchell, X. Cui, R. J. Schmittling, L. Sanchez-Perez, D. J. Snyder, K. L. Congdon, G. E. Archer, A. Desjardins, A. H. Friedman, H. S. Friedman, J. E. Herndon, R. E. McLendon, D. A. Reardon, J. J. Vredenburgh, D. D. Bigner and J. H. Sampson, Monoclonal antibody blockade of IL-2 receptor Á during lymphopenia selectively depletes regulatory T cells in mice and humans, Blood, 118 (2011), 3003-3012.  doi: 10.1182/blood-2011-02-334565. [32] S. J. Merrill, A model of the role of natural killer cells in immune surveillance, J. Math. Biol., 12 (1981), 363-373.  doi: 10.1007/BF00276923. [33] H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia CML and T cell interaction, J. Theor. Biol., 227 (2004), 513-523.  doi: 10.1016/j.jtbi.2003.11.024. [34] J. S. McDougal, A. Mawle, S. P. Cort, J. K. Nicholson, G. D. Cross, J. A. Scheppler-Campbell, D. Hicks and J. Sligh, Cellular tropism of the human retrovirus HTLV-Ⅲ/LAV. I. Role of T cell activation and expression of the T4 antigen, J. Immunol., 135 (1985), 3151-3162. [35] J. M. Murray, G. Kaufmann, A. D. Kelleher and D. A. Cooper, A model of primary HIV-1 infection, Math. Biosic., 154 (1998), 57-85.  doi: 10.1016/S0025-5564(98)10046-9. [36] A. R. McLean and M. A. Nowak, Models of interactions between HIV and other pathogens, J. theor. Biol., 155 (1992), 69-86.  doi: 10.1016/S0022-5193(05)80549-1. [37] J. S. McDougal, A. Mawle, S. P. Cort, J. K. Nicholson, G. D. Cross, J. A. Scheppler-Campbell, D. Hicks and J. Sligh, Cellular tropism of the human retrovirus HTLV-Ⅲ/LAV, J. Immunol., 135 (1985), 3151-3162. [38] A. R. McLean and T. B. L. Kirkwood, A model of human immunodeficiency virus infection in T helper cell clones, J. Theor. Biol., 147 (1990), 177-203.  doi: 10.1016/S0022-5193(05)80051-7. [39] M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. [40] 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. [41] L. Rong and A. S. Perelson, Modeling latently infected cell activation: Viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy, PLOS Comput. Biol., 5 (2009), 18pp. doi: 10.1371/journal.pcbi.1000533. [42] H. A. Van Den Berg and D. A. Rand, Dynamicsof T cell activation threshold tuning, J. Theor. Bio., 228 (2004), 397-416. [43] N. Vrisekoop, I. Den Braber, A. B. De Boer, A. F. C. Ruiter and M. T. Ackerman, Sparse production but preferential incorporation of recently produced naive T cells in the human peripheral pool, Proc Natl Aced Sci., 105 (2008), 6115-6120.  doi: 10.1073/pnas.0709713105. [44] L. Santarpia, A. K. El-Naggar, G. J. Cote, J. N. Myers and S. I. Sherman, Phosphatidylinositol 3-kinase/akt and ras/raf-mitogen-activated protein kinase pathway mutations in anaplastic thyroid cancer, J. Clin. Endocrinol. Metab., 93 (2008), 278-284.  doi: 10.1210/jc.2007-1076. [45] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bio., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [46] R. A. Weiss, How does HIV cause aids?, Science, 260 (1993), 1273-1279.  doi: 10.1126/science.8493571. [47] S. Wang, P. Hottz, M. Schechter and L. Rong, Modeling the slow CD4+ T cell decline in HIV-Infected individuals, PLoS Comput. Biol., 11 (2015). doi: 10.1371/journal.pcbi.1004665. [48] D. Zagury, J. Bernard, R. Leonard, R. Cheynier, M. Feldman, P. S. Sarin and R. C. Gallo, Long-term cultures of HTLV-Ⅲ-infected t cells: A model of cytopathology of T-cell depletion in AIDS, Science, 231 (1986), 850-853.  doi: 10.1126/science.2418502. [49] UNAIDS, Fact sheet-Latest statistics on the status of the AIDS epidemic, http://www.unaids.org/en/resources/fact-sheet, 2016.
Schematic diagram of model (2)
Forward and backward bifurcations of system (2). (a) Forward bifurcation; (b) Backward bifurcation. Blue solid line indicates that infection-free equilibrium $E_0$ is L.A.S., red dotted line indicates that the equilibrium $E_0$ is a saddle, black solid line represents the positive equilibrium $E_*^1$ is L.A.S., and red dashed line represents positive equilibrium $E_*^2$ is unstable
Stable positive equilibrium $E_*^1$ when $R_0>1$. (a) Time series of quiescent uninfected CD4$^+$ T cells $Q$; (b) Phase diagram of system (2). When $k_{1} = 0.5$, solution trajectory of the system converges to the positive equilibrium $E_*^1$.
Periodic orbit at $E_*^1$ when $R_0>1$. (a) Time series of quiescent uninfected CD4$^+$ T cells $Q$; (b) Phase diagram of system (2). When $k_1 = 0.6$, solution trajectory of the system presents periodic oscillation at the positive equilibrium $E_*^1$.
Numerical solutions of the model (2). Parameter values are from Table 2, and $k_1 = 1$, $k_s = 10^4$. Red dashed line represents $\alpha_Q = 0.5$, blue solid line represents $\alpha_Q = 0.3$, yellow dashed line represents $\alpha_Q = 0.15$ and green solid line represents $\alpha_Q = 0.01$.
Numerical solutions of the model (2). Parameter values are from Table 2, and $\alpha_Q = 0.15$, $k_s = 10^4$. Yellow solid line represents $k_1 = 10^3$, red dashed line represents $k_1 = 10^2$, black solid line represents $k_1 = 10$, blue dashed line represents $k_1 = 1$ and green solid line represents $k_1 = 10^{-1}$.
Numerical solutions of the model (2). Parameter values are from Table 2, and $\alpha_Q = 0.15$, $k_1 = 1$. Yellow solid line represents $k_s = 10^8$, red dashed line represents $k_s = 10^6$, black solid line represents $k_s = 10^5$, blue dashed line represents $k_s = 10^4$ and green solid line represents $k_s = 10^2$.
HIV dynamics in the case of antiretroviral therapy $\epsilon$. (a)$R_0$ vs drug efficacy. (b)Virus load vs drug efficacy for the quiescent cell model. Parameters are fixed in the Table 2, and $k_1 = 10$, $k_s = 10^4$ and $\alpha_Q = 0.1$. Red line represents $V_*^1$ and blue line represents $V_*^2$.
Four typical patterns of dynamical behaviors of system.
 Pattern Range of R0(Range of p) Steady states of system (2) 1 0 < R0 < R00 < 1(0 < p < p00) E0 is G.A.S. 2 0 < R0 < Rc < 1 < R00(p00 < p < pc < p0) E0 is L.A.S. 3 0 < Rc < R0 < 1 < R00(p00 < p < p0 < pc) E0 is L.A.S.; E*2 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.). 4 R0 > 1(p > p0) E0 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.).
 Pattern Range of R0(Range of p) Steady states of system (2) 1 0 < R0 < R00 < 1(0 < p < p00) E0 is G.A.S. 2 0 < R0 < Rc < 1 < R00(p00 < p < pc < p0) E0 is L.A.S. 3 0 < Rc < R0 < 1 < R00(p00 < p < p0 < pc) E0 is L.A.S.; E*2 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.). 4 R0 > 1(p > p0) E0 is unstable; E*1 is either L.A.S. or unstable(when Re(ηi(E*1)) < 0, i = 1; 2; 3; 4, E*1 is L.A.S.).
Parameters notations and values used.
 Symbol Description Value References $\lambda$ Generation rate of quiescent uninfected CD4$^+$ T cells $4.8\times10^3$ $cells\cdot ml^{-1}\cdot day^{-1}$ see text $\alpha_{Q}$ Stimulation rate by other antigens varied see text $k_{1}$ Stimulation rate by HIV varied see text $k_{s}$ Half-maximal stimulation threshold of HIV varied see text $\xi$ Revert rate of active uninfected CD4$^+$ T cells $0.15$ $day^{-1}$ [11] $d_{Q}$ Death rate of quiescent uninfected CD4$^+$ T cells $0.001$ $day^{-1}$ [19,43] $d_{T}$ Death rate of active uninfected CD4$^+$ T cells $0.01$ $day^{-1}$ [47] $d_{I}$ Death rate of infected CD4$^+$ T cells $1$ $day^{-1}$ [47] p Virus production rate of CD4$^+$ T cells $2000$ $virions\cdot cells^{-1}\cdot day^{-1}$ [47] c Clearance rate of free viruses $23$ $day^{-1}$ [47] $\beta$ Infection rate of active uninfected CD4$^+$ T cells by HIV $7.4\times10^{-8}$ $ml\cdot virion^{-1}\cdot day^{-1}$ [22,47]
 Symbol Description Value References $\lambda$ Generation rate of quiescent uninfected CD4$^+$ T cells $4.8\times10^3$ $cells\cdot ml^{-1}\cdot day^{-1}$ see text $\alpha_{Q}$ Stimulation rate by other antigens varied see text $k_{1}$ Stimulation rate by HIV varied see text $k_{s}$ Half-maximal stimulation threshold of HIV varied see text $\xi$ Revert rate of active uninfected CD4$^+$ T cells $0.15$ $day^{-1}$ [11] $d_{Q}$ Death rate of quiescent uninfected CD4$^+$ T cells $0.001$ $day^{-1}$ [19,43] $d_{T}$ Death rate of active uninfected CD4$^+$ T cells $0.01$ $day^{-1}$ [47] $d_{I}$ Death rate of infected CD4$^+$ T cells $1$ $day^{-1}$ [47] p Virus production rate of CD4$^+$ T cells $2000$ $virions\cdot cells^{-1}\cdot day^{-1}$ [47] c Clearance rate of free viruses $23$ $day^{-1}$ [47] $\beta$ Infection rate of active uninfected CD4$^+$ T cells by HIV $7.4\times10^{-8}$ $ml\cdot virion^{-1}\cdot day^{-1}$ [22,47]
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