# American Institute of Mathematical Sciences

August  2017, 22(6): 2089-2120. doi: 10.3934/dcdsb.2017086

## An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization

 Department of Mathematical Sciences, National Chengchi Uniserstiy, Taipei, 11605, Taiwan

Received  January 2016 Revised  January 2017 Published  March 2017

Let HIV infection be modeled by a dynamical system with a Michaelis-Mente-type immune response. A functional cure refers to driving the system from a stable high-viral-load state to a stable low-viral-load state. This may occur only when at least two stable equilibrium states coexist in the system. This paper analyzes how the number of biologically meaningful equilibrium states varies with system parameters. Meanwhile, it investigates how patients' profiles of immune responses determine their clinical outcomes, with focus on functional curability. The analysis provides a criterion that a functional cure is possible only if the capability of immune stimulation starts to attenuate when the density of infected cells is below a threshold. From treatment viewpoints, such a criterion is crucial because it identifies which patients cannot use a low-viral-load state as a treatment endpoint. The deriving process also provides a method to study functional curability problems with a wider class of immune response functions and functional curability problems of similar virus infections such as chronic hepatitis B virus infection.

Citation: Jeng-Huei Chen. An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086
##### References:
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Perelson, Modeling complex decay profiles of hepatitis B virus during antiviral therapy, Hepatology, 49 (2009), 32-38. [13] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth, Mathematical Biosciences and Engineering, 6 (2009), 283-299. [14] European AIDS Clinical Society (EACS), EACS Guidelines, 2013. Available from: http://www.eacsociety.org/guidelines/eacs-guidelines/eacs-guidelines.html. [15] Global Fact Sheet, UNAIDS. org, 2013. Available from: http://files.unaids.org/en/media/unaids/contentassets/documents/epidemiology/2013/gr2013/20130923_FactSheet_Global_en.pdf. [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, SpringerVerlag, New York, 1975. [17] M. Gopal, Control Systems: Principles and Design, Tata McGraw-Hill Education, New Delhi, 2002. [18] A. V. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. U.S.A, 93 (1996), 7247-7251. [19] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, J. Math. Biol., 60 (2010), 573-590. [20] D. D. Ho, A. U. Neuman, 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. [21] L. Hocqueloux, T. Prazuck, V. Avettand-Fenoel, A. Lafeuillade, B. Cardon, J. P. Viard and C. Rouzioux, Long-term immunovirologic control following antiretroviral therapy interruption in patients treated at the time of primary HIV-1 infection, AIDS, 24 (2010), 1598-1601. [22] H. Korthals Altes, R. M. Ribeiro and R. J. de Boer, The race between initial T-helper expansion and virus growth upon HIV infection influences polyclonality of the response and viral set-point, Proc. Biol. Sci., 270 (2003), 1349-1358. [23] S. R. Lewin, R. M. Ribeiro, T. Walters, G. K. Lau, S. Bowden, S. Locarnini and A. S. Perelson, Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed, Hepatology, 34 (2001), 1012-1020. [24] Y. F. Liaw and C. M. Chu, Hepatitis B virus infection, The Lancet, 373 (2009), 582-592. [25] Y. F. Liaw, J. H. Kao and T. Piratvisuth, et al., Asian-Pacific consensus statement on the management of chronic hepatitis B: A 2012 update, Heptaol. Int., 6 (2012), 531-561. [26] J. D. Lifson, J. L. Rossio, R. Arnaout, L. Li, T. L. Parks, S. K. Schneider, R. F. Kiser, V. J. Coalter, G. Walsh, R. J. Imming, B. Fisher, B. M. Flynn, N. Bischofberger, M. Jr. Piatak, V. M. Hirsch, M. A. Nowak and D. Wodarz, Containment of simian immunodeficiency virus infection: Cellular immune responses and protection from rechallenge following transient postinoculation antiretroviral treatment, J. Virol., 74 (2000), 2584-2593. [27] J. D. Lifson, J. L. Rossio, M. Piatak, T. Parks, L. Li, R. Kiser, V. Coalter, B. Fisher, B. M. Flynn, S. Czajak, V. M. Hirsch, K. A. Reimann, J. E. Schmitz, J. Ghrayeb, N. Bischofberger, M. A. Nowak, R. C. Desrosiers and D. Wodarz, Role of CD8(+) lymphocytes in control of simian immunodeficiency virus infection and resistance to rechallenge after transient early antiretroviral treatment, J. Virol., 75 (2001), 10187-10199. [28] J. Lisziewicz and F. Lori, Structured treatment interruptions in HIV/AIDS therapy, Microbes and Infection, 4 (2002), 207-214. [29] J. Lisziewicz, E. Rosenberg, J. Lieberman et al., Control of HIV despite the discontinuation of antiretroviral therapy, N. Engl. J. Med., 340 (1999), 1683-1684. [30] S. J. Little, A. R. McLean, C. A. Spina, D. D. Richman and D. V. Havlir, Viral dynamics of acute HIV-1 infection, J. Exp. Med., 190 (1999), 841-850. [31] S. Lodi, L. Meyer, A. D. Kelleher, M. Rosinska, J. Ghosn, M. Sannes and K. Porter, Immunovirologic control 24 months after interruption of antiretroviral therapy initiated close to hiv seroconversion, Arch. Intern. Med., 172 (2012), 1252-1255. [32] M. Markowitz, M. Louie, A. Hurley, E. Sun, M. Di. Mascio, A. S. Perelson and D. D. Ho, A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T-cell decay in vivo, J. Virol., 77 (2003), 5037-5038. [33] L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. [34] H. Mohri, S. Bonhoeffer, S. Monard, A. S. Perelson and D. D. Ho, Rapid turnover of T lymphocytes in SIV-infected rhesus macaques, Science, 279 (1998), 1223-1227. [35] M. A. Nowak, S. Bonhoeffer, G. M. Shaw and R. M. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol, 184 (1997), 203-217. [36] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000. [37] G. M. Ortiz, D. F. Nixon, A. Trkola, et al., HIV-1-specific immune responses in subjects who temporarily contain antiretroviral therapy, J. Clin. Invest., 104 (1999), R13-R18. [38] A. S. Perelson, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96-105. [39] 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. [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. [41] A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response, Science, 271 (1996), 497-499. [42] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York, 1964. [43] R. M. Ribeiro, S. Bonhoeffer and M. A. Nowak, The frequency of resistant mutant virus before antiviral therapy, AIDS, 12 (1998), 461-465. [44] R. M. Ribeiro and S. Bonhoeffer, Production of resistant HIV mutants during antiretroviral therapy, Proc. Natl. Acad. Sci. U.S.A., 97 (2000), 7681-7686. [45] R. M. Ribeiro, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microbes Infect., 4 (2002), 829-835. [46] R. M. Ribeiro, L. Qin, L. L. Chavez, D. Li, S. G. Self and A. S. Perelson, Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection, J. Virol., 84 (2010), 6096-6102. [47] L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331. [48] 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), e1000533, 18pp. [49] D. I. Rosenbloom, A. L. Hill, S. A. Rabi, R. F. Siliciano and M. A. Nowak, Antiretroviral dynamics determines HIV evolution and predicts therapy outcome, Nat. Med., 18 (2012), 1378-1385. [50] S. K. Sarin, B. S. Sandhu, B. C. Sharma, M. Jain, J. Singh and V. Malhotra, Beneficial effects of 'lamivudine pulse' therapy in HBeAg-positive patients with normal ALT, J. Viral. hpat., 11 (2004), 552-558. [51] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. [52] N. I. Stilianakis, C. A. Boucher, M. D. De Jong, R. Van Leeuwen, R. Schuurman and R. J. De Boer, Clinical data sets of human immunodeficiency virus type 1 reverse transcriptaseresistant mutants explained by a mathematical model, J. Virol., 71 (1997), 161-168. [53] The Strategies for Management of Antiretroviral Therapy (SMART) Study Group, CD4+ Count-Guided Interruption of Antiretroviral Treatment, N. Engl. J. Med., 355 (2006), 2283-2296. [54] M. A. Thompson, J. A. Aberg, J. F. Hoy, A. Telenti, C. Benson, P. Cahn, J. J. Eron, H. F. Gunthard, S. M. Hammer, P. Reiss, D. D. Richman, G. Rizzardini, D. L. Thomas, D. M. Jacobsen and P. A. Volberding, Antiretroviral treatment of adult HIV infection: 2012 recommendations of the International Antiviral Society-USA panel, JAMA, 308 (2012), 387-402. [55] G. D. Tomaras, N. L. Yates, P. Liu, L. Qin, G. G. Fouda, L. L. Chavez, A. C. Decamp, R. J. Parks, V. C. Ashley, J. T. Lucas, M. Cohen, J. Eron, C. B. Hicks, H. X. Liao, S. G. Self, G. Landucci, D. N. Forthal, K. J. Weinhold, B. F. Keele, B. H. Hahn, M. L. Greenberg, L. Morris, S. 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##### References:
 [1] Available from: http://en.wikipedia.org/wiki/Michaelis-Menten. [2] [3] B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control apporahces, Math. Biosci. Eng., 1 (2004), 223-241. [4] B. Autran, B. Descours, V. Avettand-Fenoel and C. Rouzioux, Elite controllers as a model of functional cure, Curr. Opin. HIV AIDS, 6 (2011), 181-187. [5] S. Bajariz, G. Webb and D. E. Kirschner, Predicting differential responses to structured treatment tnterruptions during HAART, Bull. Math. Biol., 66 (2004), 1093-1118. [6] K. J. Bar, C. Y. Tsao, S. S. Iyer, J. M. Decker, Y. Yang, M. Bonsignori, X. Chen, K. K. Hwang, D. C. Montefiori, H. X. Liao, P. Hraber, W. Fischer, H. Li, S. Wang, S. Sterrett, B. F. Keele, V. V. Ganusov, A. S. Perelson, B. T. Korber, I. Georgiev, J. S. McLellan, J. W. Pavlicek, F. Gao, B. F. Haynes, B. H. Hahn, P. D. Kwong and G. M. Shaw, Early low-titer neutralizing antibodies impede HIV-1 replication and select for virus escape, PLoS Pathog., 8 (2012), e1002721. [7] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. U.S.A., 94 (1997), 6971-6976. [8] S. Bonhoeffer and M. A. Nowak, Pre-existence and emergence of drug resistance in HIV-1 infection, Proc. Biol. Sci., 264 (1997), 631-637. [9] S. Bonhoeffer, M. Rembiszewski, G. M. Ortiz and D. F. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322. [10] G. Carcelain and B. Autran, Immune interventions in HIV infection, Immunological Reviews, 254 (2013), 355-371. [11] J. M. Coffin, HIV population dynamics in vivo: Implications for genetic variation, pathogenesis, and therapy, Science, 267 (1995), 483-489. [12] H. Dahari, E. Shudo, R. M. Ribeiro and A. S. Perelson, Modeling complex decay profiles of hepatitis B virus during antiviral therapy, Hepatology, 49 (2009), 32-38. [13] S. Eikenberry, S. Hews, J. D. Nagy and Y. Kuang, The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth, Mathematical Biosciences and Engineering, 6 (2009), 283-299. [14] European AIDS Clinical Society (EACS), EACS Guidelines, 2013. Available from: http://www.eacsociety.org/guidelines/eacs-guidelines/eacs-guidelines.html. [15] Global Fact Sheet, UNAIDS. org, 2013. Available from: http://files.unaids.org/en/media/unaids/contentassets/documents/epidemiology/2013/gr2013/20130923_FactSheet_Global_en.pdf. [16] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, SpringerVerlag, New York, 1975. [17] M. Gopal, Control Systems: Principles and Design, Tata McGraw-Hill Education, New Delhi, 2002. [18] A. V. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. U.S.A, 93 (1996), 7247-7251. [19] S. Hews, S. Eikenberry, J. D. Nagy and Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, J. Math. Biol., 60 (2010), 573-590. [20] D. D. Ho, A. U. Neuman, 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. [21] L. Hocqueloux, T. Prazuck, V. Avettand-Fenoel, A. Lafeuillade, B. Cardon, J. P. Viard and C. Rouzioux, Long-term immunovirologic control following antiretroviral therapy interruption in patients treated at the time of primary HIV-1 infection, AIDS, 24 (2010), 1598-1601. [22] H. Korthals Altes, R. M. Ribeiro and R. J. de Boer, The race between initial T-helper expansion and virus growth upon HIV infection influences polyclonality of the response and viral set-point, Proc. Biol. Sci., 270 (2003), 1349-1358. [23] S. R. Lewin, R. M. Ribeiro, T. Walters, G. K. Lau, S. Bowden, S. Locarnini and A. S. Perelson, Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed, Hepatology, 34 (2001), 1012-1020. [24] Y. F. Liaw and C. M. Chu, Hepatitis B virus infection, The Lancet, 373 (2009), 582-592. [25] Y. F. Liaw, J. H. Kao and T. Piratvisuth, et al., Asian-Pacific consensus statement on the management of chronic hepatitis B: A 2012 update, Heptaol. Int., 6 (2012), 531-561. [26] J. D. Lifson, J. L. Rossio, R. Arnaout, L. Li, T. L. Parks, S. K. Schneider, R. F. Kiser, V. J. Coalter, G. Walsh, R. J. Imming, B. Fisher, B. M. Flynn, N. Bischofberger, M. Jr. Piatak, V. M. Hirsch, M. A. Nowak and D. Wodarz, Containment of simian immunodeficiency virus infection: Cellular immune responses and protection from rechallenge following transient postinoculation antiretroviral treatment, J. Virol., 74 (2000), 2584-2593. [27] J. D. Lifson, J. L. Rossio, M. Piatak, T. Parks, L. Li, R. Kiser, V. Coalter, B. Fisher, B. M. Flynn, S. Czajak, V. M. Hirsch, K. A. Reimann, J. E. Schmitz, J. Ghrayeb, N. Bischofberger, M. A. Nowak, R. C. Desrosiers and D. Wodarz, Role of CD8(+) lymphocytes in control of simian immunodeficiency virus infection and resistance to rechallenge after transient early antiretroviral treatment, J. Virol., 75 (2001), 10187-10199. [28] J. Lisziewicz and F. Lori, Structured treatment interruptions in HIV/AIDS therapy, Microbes and Infection, 4 (2002), 207-214. [29] J. Lisziewicz, E. Rosenberg, J. Lieberman et al., Control of HIV despite the discontinuation of antiretroviral therapy, N. Engl. J. Med., 340 (1999), 1683-1684. [30] S. J. Little, A. R. McLean, C. A. Spina, D. D. Richman and D. V. Havlir, Viral dynamics of acute HIV-1 infection, J. Exp. Med., 190 (1999), 841-850. [31] S. Lodi, L. Meyer, A. D. Kelleher, M. Rosinska, J. Ghosn, M. Sannes and K. Porter, Immunovirologic control 24 months after interruption of antiretroviral therapy initiated close to hiv seroconversion, Arch. Intern. Med., 172 (2012), 1252-1255. [32] M. Markowitz, M. Louie, A. Hurley, E. Sun, M. Di. Mascio, A. S. Perelson and D. D. Ho, A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T-cell decay in vivo, J. Virol., 77 (2003), 5037-5038. [33] L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. [34] H. Mohri, S. Bonhoeffer, S. Monard, A. S. Perelson and D. D. Ho, Rapid turnover of T lymphocytes in SIV-infected rhesus macaques, Science, 279 (1998), 1223-1227. [35] M. A. Nowak, S. Bonhoeffer, G. M. Shaw and R. M. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol, 184 (1997), 203-217. [36] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000. [37] G. M. Ortiz, D. F. Nixon, A. Trkola, et al., HIV-1-specific immune responses in subjects who temporarily contain antiretroviral therapy, J. Clin. Invest., 104 (1999), R13-R18. [38] A. S. Perelson, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96-105. [39] 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. [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. [41] A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response, Science, 271 (1996), 497-499. [42] L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York, 1964. [43] R. M. Ribeiro, S. Bonhoeffer and M. A. Nowak, The frequency of resistant mutant virus before antiviral therapy, AIDS, 12 (1998), 461-465. [44] R. M. Ribeiro and S. Bonhoeffer, Production of resistant HIV mutants during antiretroviral therapy, Proc. Natl. Acad. Sci. U.S.A., 97 (2000), 7681-7686. [45] R. M. Ribeiro, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microbes Infect., 4 (2002), 829-835. [46] R. M. Ribeiro, L. Qin, L. L. Chavez, D. Li, S. G. Self and A. S. Perelson, Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection, J. Virol., 84 (2010), 6096-6102. [47] L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331. [48] 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), e1000533, 18pp. [49] D. I. Rosenbloom, A. L. Hill, S. A. Rabi, R. F. Siliciano and M. A. Nowak, Antiretroviral dynamics determines HIV evolution and predicts therapy outcome, Nat. Med., 18 (2012), 1378-1385. [50] S. K. Sarin, B. S. Sandhu, B. C. Sharma, M. Jain, J. Singh and V. Malhotra, Beneficial effects of 'lamivudine pulse' therapy in HBeAg-positive patients with normal ALT, J. Viral. hpat., 11 (2004), 552-558. [51] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. [52] N. I. Stilianakis, C. A. Boucher, M. D. De Jong, R. Van Leeuwen, R. Schuurman and R. J. De Boer, Clinical data sets of human immunodeficiency virus type 1 reverse transcriptaseresistant mutants explained by a mathematical model, J. Virol., 71 (1997), 161-168. [53] The Strategies for Management of Antiretroviral Therapy (SMART) Study Group, CD4+ Count-Guided Interruption of Antiretroviral Treatment, N. Engl. J. Med., 355 (2006), 2283-2296. [54] M. A. Thompson, J. A. Aberg, J. F. Hoy, A. Telenti, C. Benson, P. Cahn, J. J. Eron, H. F. Gunthard, S. M. Hammer, P. Reiss, D. D. Richman, G. Rizzardini, D. L. Thomas, D. M. Jacobsen and P. A. Volberding, Antiretroviral treatment of adult HIV infection: 2012 recommendations of the International Antiviral Society-USA panel, JAMA, 308 (2012), 387-402. [55] G. D. Tomaras, N. L. Yates, P. Liu, L. Qin, G. G. Fouda, L. L. Chavez, A. C. Decamp, R. J. Parks, V. C. Ashley, J. T. Lucas, M. Cohen, J. Eron, C. B. Hicks, H. X. Liao, S. G. Self, G. Landucci, D. N. Forthal, K. J. Weinhold, B. F. Keele, B. H. Hahn, M. L. Greenberg, L. Morris, S. 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(Case 1) The graph of function $g(I)$ with $b_1 = 4$, $b_2 = 2$, $M_1 = 5$ and $M_2 = 5$
(Case 2) The graph on the left hand side is the function $g(I)$ with $b_1 = 90$, $b_2 = 80$, $M_1 = 4$ and $M_2 = 6$. The graph of $g(I)$ with $I>-M_1$ is enlarged on the right hand side. Point A is a local maximum with coordinates $(I_2^{*}, g(I_2^{*}))$ = (8.9282, 14.3078). Point B is an infection point with coordinates $(I_1^{**}, g(I_1^{**}))$ = (15.8723, 13.8299). Point C is the intersection point of $y = g(I)$ and $y=b_1-b_2$. Its coordinates is ($I_{{\mathop{\rm int}}}, g(I_{{\mathop{\rm int}}}))$ = (2.000, 10)
Type 1 immune induction function (referred as a function valid in Bonhoeffer sense.) The parameters are $b_1 = 4$, $M_1 =100$, $b_2 = 2.5$, $M_2 = 120$ and $d_E = 0$
Type 2 immune induction function. The parameters are $b_1 = 2.5$, $M_1 =120$, $b_2 = 4$, $M_2 = 100$ and $d_E = 0$
Type 3 immune induction function (referred as a function valid in Adams sense.) The parameters are $b_1 = 6$, $M_1 =1.25$, $b_2 = 5$, $M_2 = 6.25$ and $d_E = 0$. The positive local maximum is located at $I^{*} = 3.55$ (point A) and the inflection point is located at $I^{**} = 6.96.$ (point B)
Type 4 immune induction function. The parameters are $b_1 = 5$, $M_1 =6.25$, $b_2 = 6$, $M_2 = 1.25$ and $d_E = 0$. The positive local minimum is located at $I^{*} = 3.55$ (point A) and the inflection point is located at $I^{**} = 6.96$ (point B)
The graph of the function $z(I)$ in lemma 4.6 with $a=5$ and $b=10$
A demonstration of theorem 4.8. With the immune induction function $g(I)$ valid in Bonhoeffer sense, the functions $H(I)$ and $g(I)$ intersect exactly once at $I = I_e > 0$ with $H(I_e) = g(I_e) < 0$. (They intersect at point A in the graph.) This may occur if and only if the condition $H(0) > g(0)$ holds. Equivalently, this is the condition ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ stated in theorem 4.8. The parameters of this graph are $s_T =5$, $d_T = 0.01$, $\beta =8$, $d_I = 0.7$, $p = 0.7$, $k =5$, $d_V =13$, $c_E =1$, $b_1 =4$, $M_1 =100$, $b_2 = 2.5$, $M_2 =120$ and $d_E =1$
Part (c) of theorem 4.9. It is possible that the functions $H(I)$ and $g(I)$ intersect three times. (They intersect at points A, B and C. The upper half of the function $H(I)$ is not shown in the graph.) Each intersection point leads to one biologically meaningful equilibrium states other than $Q_1^{1}$. This may occur if $H(0) > g(0)$. Equivalently, this is the condition ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ stated in part (c) of theorem 4.9. The parameters of this graph are $s_T =50$, $d_T = 0.01$, $\beta =8$, $d_I = 0.7$, $p = 0.7$, $k =5$, $d_V =13$, $c_E =1$, $b_1 =6$, $M_1 =1.25$, $b_2 = 5$, $M_2 = 6.25$ and $d_E =2.5$
Part (c) of theorem 4.9. It is possible that the functions $H(I)$ and $g(I)$ intersect three times. (They intersect at points A, B and C. The upper half of the function $H(I)$ is not shown in the graph.) Each intersection point leads to one biologically meaningful equilibrium states other than $Q_1^{1}$. This may occur if $H(0) > g(0)$. Equivalently, this is the condition ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ stated in part (c) of theorem 4.9. The parameters of this graph are $s_T =50$, $d_T = 0.01$, $\beta =8$, $d_I = 0.7$, $p = 0.7$, $k =5$, $d_V =13$, $c_E =1$, $b_1 =6$, $M_1 =1.25$, $b_2 = 5$, $M_2 = 6.25$ and $d_E =2.5$
The behavior of $g(I)$ with respective to system parameters
 Type Conditions in parameters 1 (case 1) $M_1 = M_2$ and $b_1 > b_2$ (case 3) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 < b_1M_2$ (case 4) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 = b_1M_2$ (case 5) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 < b_1M_1$ (case 6) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 = b_1M_1$ 2 (case 7) $M_1 = M_2$ and $b_1 < b_2$ (case 8) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 < b_2M_2$ (case 9) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 = b_2M_2$ (case 10) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 < b_2M_1$ (case 11) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 = b_2M_1$ 3 (case 12) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 > b_2M_1$ (case 13) $M_1 < M_2$ and $b_1 = b_2$ (case 2) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 > b_1M_1$ 4 (case 14) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 > b_1M_2$ (case 15) $M_1 > M_2$ and $b_1 = b_2$ (case 16) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 > b_2M_2$
 Type Conditions in parameters 1 (case 1) $M_1 = M_2$ and $b_1 > b_2$ (case 3) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 < b_1M_2$ (case 4) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 = b_1M_2$ (case 5) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 < b_1M_1$ (case 6) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 = b_1M_1$ 2 (case 7) $M_1 = M_2$ and $b_1 < b_2$ (case 8) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 < b_2M_2$ (case 9) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 = b_2M_2$ (case 10) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 < b_2M_1$ (case 11) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 = b_2M_1$ 3 (case 12) $M_1 < M_2, b_1 < b_2$ and $b_1M_2 > b_2M_1$ (case 13) $M_1 < M_2$ and $b_1 = b_2$ (case 2) $M_1 < M_2, b_1 > b_2$ and $b_2M_2 > b_1M_1$ 4 (case 14) $M_1 > M_2, b_1 > b_2$ and $b_2M_1 > b_1M_2$ (case 15) $M_1 > M_2$ and $b_1 = b_2$ (case 16) $M_1 > M_2, b_1 < b_2$ and $b_1M_1 > b_2M_2$
The chosen values of system parameters
 Parameters Value Parameters Value $s_T$ $10^{4} \frac{{cells}}{{ml \cdot day}}$ $c_E$ $1 \frac {cells} {ml \cdot day}$ $d_T$ $0.01 \frac{1}{day}$ $b_1$ $0.3 \frac {1}{day}$ $\beta$ $8 \times 10^{-7} \frac{ml}{{virions \cdot day}}$ $M_1$ $100 \frac {cells}{ml}$ $d_I$ $0.7 \frac {1}{day}$ $b_2$ $0.25 \frac {1}{day}$ $p$ $10^{-5} \frac{ml}{{cells \cdot day}}$ $M_2$ $500 \frac {cells}{ml}$ $k$ $100 \frac {virions}{day}$ $d_E$ $0.1 \frac {1} {day}$ $d_V$ $13 \frac {1}{day}$
 Parameters Value Parameters Value $s_T$ $10^{4} \frac{{cells}}{{ml \cdot day}}$ $c_E$ $1 \frac {cells} {ml \cdot day}$ $d_T$ $0.01 \frac{1}{day}$ $b_1$ $0.3 \frac {1}{day}$ $\beta$ $8 \times 10^{-7} \frac{ml}{{virions \cdot day}}$ $M_1$ $100 \frac {cells}{ml}$ $d_I$ $0.7 \frac {1}{day}$ $b_2$ $0.25 \frac {1}{day}$ $p$ $10^{-5} \frac{ml}{{cells \cdot day}}$ $M_2$ $500 \frac {cells}{ml}$ $k$ $100 \frac {virions}{day}$ $d_E$ $0.1 \frac {1} {day}$ $d_V$ $13 \frac {1}{day}$
Summarized numerical results for $c_E=0$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$, $k =20$, $c_E = 0$ 0.35 Adams Virus eradication $Q_1^{0}$ (b) $c_E = 0$ 8.7912 Adams A functional cure ($Q_2^{0}$, $Q_3^{0}$) (c) $b_1 = 0.4$, $M_2 = 120$ $c_E =0$ 8.7912 Bonhoeffer Elite controller $Q_3^{0}$ (d) $M_2 = 110$, $c_E = 0$ 8.7912 Bonhoeffer High viral load $Q_2^{0}$ (e) $d_E = 0.2$, $c_E = 0$ 8.7912 Adams High viral load $Q_2^{0}$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$, $k =20$, $c_E = 0$ 0.35 Adams Virus eradication $Q_1^{0}$ (b) $c_E = 0$ 8.7912 Adams A functional cure ($Q_2^{0}$, $Q_3^{0}$) (c) $b_1 = 0.4$, $M_2 = 120$ $c_E =0$ 8.7912 Bonhoeffer Elite controller $Q_3^{0}$ (d) $M_2 = 110$, $c_E = 0$ 8.7912 Bonhoeffer High viral load $Q_2^{0}$ (e) $d_E = 0.2$, $c_E = 0$ 8.7912 Adams High viral load $Q_2^{0}$
Summarized numerical results for case $c_E \neq 0$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$ $k = 20$ 0.35 Adams Virus eradication $Q_1^{1}$ (b) No changes 8.7912 Adams A functional cure ($Q_2^{1}$, $Q_4^{1}$) (c) $b_1 = 0.4$, $M_2 = 120$ 8.7912 Bonhoeffer Elite controller $Q_4^{1}$ (d) $\beta$ = $1.6 \times 10^{-7}$ $b_1 = 0.4$, $k = 20$ $M_2 = 120$ 0.35 Bonhoeffer Virus eradication $Q_1^{1}$ (e) $M_1 =1 \times 10^{-4}$ $M_2 = 5 \times 10^{4}$ 8.7912 Adams High viral load $Q_4^{1}$
 Case Varied parameters $R_0$ Immunity Outcomes (a) $\beta$ = $1.6 \times 10^{-7}$ $k = 20$ 0.35 Adams Virus eradication $Q_1^{1}$ (b) No changes 8.7912 Adams A functional cure ($Q_2^{1}$, $Q_4^{1}$) (c) $b_1 = 0.4$, $M_2 = 120$ 8.7912 Bonhoeffer Elite controller $Q_4^{1}$ (d) $\beta$ = $1.6 \times 10^{-7}$ $b_1 = 0.4$, $k = 20$ $M_2 = 120$ 0.35 Bonhoeffer Virus eradication $Q_1^{1}$ (e) $M_1 =1 \times 10^{-4}$ $M_2 = 5 \times 10^{4}$ 8.7912 Adams High viral load $Q_4^{1}$
Functional curability under different conditions
 $g(I)$ valid in Bonhoeffer sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ inconclusive $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $g(I)$ valid in Adams sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ possible $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ inconclusive (with immunity counter-induced) $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ possible with the criterion satisfied
 $g(I)$ valid in Bonhoeffer sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ inconclusive $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ not possible $g(I)$ valid in Adams sense $c_E =0$ and $R_0 < 1$ not possible $c_E = 0$ and $R_0 > 1$ possible $c_E > 0$ and ${R_0} < (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ inconclusive (with immunity counter-induced) $c_E > 0$ and ${R_0} > (1 + p\frac{{{c_E}}}{{{d_I}{d_E}}})$ possible with the criterion satisfied
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