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 & Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086
References:
[1]

Available from: http://en.wikipedia.org/wiki/Michaelis-Menten.Google Scholar

[2]

Avaliable from: http://www.iasociety.org/What-we-do/Towards-an-HIV-Cure/Activities/The-Rome-Statement.Google Scholar

[3]

B. M. AdamsH. T. BanksH. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control apporahces, Math. Biosci. Eng., 1 (2004), 223-241. Google Scholar

[4]

B. AutranB. DescoursV. Avettand-Fenoel and C. Rouzioux, Elite controllers as a model of functional cure, Curr. Opin. HIV AIDS, 6 (2011), 181-187. Google Scholar

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S. BajarizG. Webb and D. E. Kirschner, Predicting differential responses to structured treatment tnterruptions during HAART, Bull. Math. Biol., 66 (2004), 1093-1118. Google Scholar

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K. J. BarC. Y. TsaoS. S. IyerJ. M. DeckerY. YangM. BonsignoriX. ChenK. K. HwangD. C. MontefioriH. X. LiaoP. HraberW. FischerH. LiS. WangS. SterrettB. F. KeeleV. V. GanusovA. S. PerelsonB. T. KorberI. GeorgievJ. S. McLellanJ. W. PavlicekF. GaoB. F. HaynesB. H. HahnP. 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. Google Scholar

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S. BonhoefferR. M. MayG. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. U.S.A., 94 (1997), 6971-6976. Google Scholar

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S. Bonhoeffer and M. A. Nowak, Pre-existence and emergence of drug resistance in HIV-1 infection, Proc. Biol. Sci., 264 (1997), 631-637. Google Scholar

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S. BonhoefferM. RembiszewskiG. M. Ortiz and D. F. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322. Google Scholar

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G. Carcelain and B. Autran, Immune interventions in HIV infection, Immunological Reviews, 254 (2013), 355-371. Google Scholar

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J. M. Coffin, HIV population dynamics in vivo: Implications for genetic variation, pathogenesis, and therapy, Science, 267 (1995), 483-489. Google Scholar

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H. DahariE. ShudoR. M. Ribeiro and A. S. Perelson, Modeling complex decay profiles of hepatitis B virus during antiviral therapy, Hepatology, 49 (2009), 32-38. Google Scholar

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S. EikenberryS. HewsJ. 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. Google Scholar

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S. HewsS. EikenberryJ. 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. Google Scholar

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H. Korthals AltesR. M. Ribeiro and de Boer R. J., 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. Google Scholar

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Y. F. LiawJ. H. Kao and T. Piratvisuth, Asian-Pacific consensus statement on the management of chronic hepatitis B: A 2012 update, Heptaol. Int., 6 (2012), 531-561. Google Scholar

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J. D. LifsonJ. L. RossioR. ArnaoutL. LiT. L. ParksS. K. SchneiderR. F. KiserV. J. CoalterG. WalshR. J. ImmingB. FisherB. M. FlynnN. BischofbergerM. Jr. PiatakV. M. HirschM. 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. Google Scholar

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J. D. LifsonJ. L. RossioM. PiatakT. ParksL. LiR. KiserV. CoalterB. FisherB. M. FlynnS. CzajakV. M. HirschK. A. ReimannJ. E. SchmitzJ. GhrayebN. BischofbergerM. A. NowakR. 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. Google Scholar

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H. MohriS. BonhoefferS. MonardA. S. Perelson and D. D. Ho, Rapid turnover of T lymphocytes in SIV-infected rhesus macaques, Science, 279 (1998), 1223-1227. Google Scholar

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A. S. Perelson, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96-105. Google Scholar

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show all references

References:
[1]

Available from: http://en.wikipedia.org/wiki/Michaelis-Menten.Google Scholar

[2]

Avaliable from: http://www.iasociety.org/What-we-do/Towards-an-HIV-Cure/Activities/The-Rome-Statement.Google Scholar

[3]

B. M. AdamsH. T. BanksH. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control apporahces, Math. Biosci. Eng., 1 (2004), 223-241. Google Scholar

[4]

B. AutranB. DescoursV. Avettand-Fenoel and C. Rouzioux, Elite controllers as a model of functional cure, Curr. Opin. HIV AIDS, 6 (2011), 181-187. Google Scholar

[5]

S. BajarizG. Webb and D. E. Kirschner, Predicting differential responses to structured treatment tnterruptions during HAART, Bull. Math. Biol., 66 (2004), 1093-1118. Google Scholar

[6]

K. J. BarC. Y. TsaoS. S. IyerJ. M. DeckerY. YangM. BonsignoriX. ChenK. K. HwangD. C. MontefioriH. X. LiaoP. HraberW. FischerH. LiS. WangS. SterrettB. F. KeeleV. V. GanusovA. S. PerelsonB. T. KorberI. GeorgievJ. S. McLellanJ. W. PavlicekF. GaoB. F. HaynesB. H. HahnP. 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. Google Scholar

[7]

S. BonhoefferR. M. MayG. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. U.S.A., 94 (1997), 6971-6976. Google Scholar

[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. Google Scholar

[9]

S. BonhoefferM. RembiszewskiG. M. Ortiz and D. F. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322. Google Scholar

[10]

G. Carcelain and B. Autran, Immune interventions in HIV infection, Immunological Reviews, 254 (2013), 355-371. Google Scholar

[11]

J. M. Coffin, HIV population dynamics in vivo: Implications for genetic variation, pathogenesis, and therapy, Science, 267 (1995), 483-489. Google Scholar

[12]

H. DahariE. ShudoR. M. Ribeiro and A. S. Perelson, Modeling complex decay profiles of hepatitis B virus during antiviral therapy, Hepatology, 49 (2009), 32-38. Google Scholar

[13]

S. EikenberryS. HewsJ. 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. Google Scholar

[14]

European AIDS Clinical Society (EACS), EACS Guidelines, 2013. Available from: http://www.eacsociety.org/guidelines/eacs-guidelines/eacs-guidelines.html.Google Scholar

[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.Google Scholar

[16]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, SpringerVerlag, New York, 1975.Google Scholar

[17]

M. Gopal, Control Systems: Principles and Design, Tata McGraw-Hill Education, New Delhi, 2002.Google Scholar

[18]

A. V. HerzS. BonhoefferR. M. AndersonR. 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. Google Scholar

[19]

S. HewsS. EikenberryJ. 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. Google Scholar

[20]

D. D. HoA. U. NeumanA. S. PerelsonW. ChenJ. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126. Google Scholar

[21]

L. HocquelouxT. PrazuckV. Avettand-FenoelA. LafeuilladeB. CardonJ. 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. Google Scholar

[22]

H. Korthals AltesR. M. Ribeiro and de Boer R. J., 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. Google Scholar

[23]

S. R. LewinR. M. RibeiroT. WaltersG. K. LauS. BowdenS. Locarnini and A. S. Perelson, Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed, Hepatology, 34 (2001), 1012-1020. Google Scholar

[24]

Y. F. Liaw and C. M. Chu, Hepatitis B virus infection, The Lancet, 373 (2009), 582-592. Google Scholar

[25]

Y. F. LiawJ. H. Kao and T. Piratvisuth, Asian-Pacific consensus statement on the management of chronic hepatitis B: A 2012 update, Heptaol. Int., 6 (2012), 531-561. Google Scholar

[26]

J. D. LifsonJ. L. RossioR. ArnaoutL. LiT. L. ParksS. K. SchneiderR. F. KiserV. J. CoalterG. WalshR. J. ImmingB. FisherB. M. FlynnN. BischofbergerM. Jr. PiatakV. M. HirschM. 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. Google Scholar

[27]

J. D. LifsonJ. L. RossioM. PiatakT. ParksL. LiR. KiserV. CoalterB. FisherB. M. FlynnS. CzajakV. M. HirschK. A. ReimannJ. E. SchmitzJ. GhrayebN. BischofbergerM. A. NowakR. 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. Google Scholar

[28]

J. Lisziewicz and F. Lori, Structured treatment interruptions in HIV/AIDS therapy, Microbes and Infection, 4 (2002), 207-214. Google Scholar

[29]

J. LisziewiczE. Rosenberg and J. Lieberman, Control of HIV despite the discontinuation of antiretroviral therapy, N. Engl. J. Med., 340 (1999), 1683-1684. Google Scholar

[30]

S. J. LittleA. R. McLeanC. A. SpinaD. D. Richman and D. V. Havlir, Viral dynamics of acute HIV-1 infection, J. Exp. Med., 190 (1999), 841-850. Google Scholar

[31]

S. LodiL. MeyerA. D. KelleherM. RosinskaJ. GhosnM. 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. Google Scholar

[32]

M. MarkowitzM. LouieA. HurleyE. SunM. Di. MascioA. 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. Google Scholar

[33]

L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. Google Scholar

[34]

H. MohriS. BonhoefferS. MonardA. S. Perelson and D. D. Ho, Rapid turnover of T lymphocytes in SIV-infected rhesus macaques, Science, 279 (1998), 1223-1227. Google Scholar

[35]

M. A. NowakS. BonhoefferG. 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. Google Scholar

[36] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, New York, 2000.
[37]

G. M. OrtizD. F. Nixon and A. Trkola, HIV-1-specific immune responses in subjects who temporarily contain antiretroviral therapy, J. Clin. Invest., 104 (1999), R13-R18. Google Scholar

[38]

A. S. Perelson, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96-105. Google Scholar

[39]

A. S. PerelsonP. EssungerY. CaoM. VesanenA. HurleyK. SakselaM. Markowitz and D. D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188-191. Google Scholar

[40]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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. Google Scholar

[41]

A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response, Science, 271 (1996), 497-499. Google Scholar

[42]

L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze and E. F. Mischenko, Mathematical Theory of Optimal Processes, Wiley, New York, 1964.Google Scholar

[43]

R. M. RibeiroS. Bonhoeffer and M. A. Nowak, The frequency of resistant mutant virus before antiviral therapy, AIDS, 12 (1998), 461-465. Google Scholar

[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. Google Scholar

[45]

R. M. RibeiroA. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microbes Infect., 4 (2002), 829-835. Google Scholar

[46]

R. M. RibeiroL. QinL. L. ChavezD. LiS. 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. Google Scholar

[47]

L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331. Google Scholar

[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. Google Scholar

[49]

D. I. RosenbloomA. L. HillS. A. RabiR. F. Siliciano and M. A. Nowak, Antiretroviral dynamics determines HIV evolution and predicts therapy outcome, Nat. Med., 18 (2012), 1378-1385. Google Scholar

[50]

S. K. SarinB. S. SandhuB. C. SharmaM. JainJ. Singh and V. Malhotra, Beneficial effects of 'lamivudine pulse' therapy in HBeAg-positive patients with normal ALT, J. Viral. hpat., 11 (2004), 552-558. Google Scholar

[51]

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Figure 1.  (Case 1) The graph of function $g(I)$ with $b_1 = 4$, $b_2 = 2$, $M_1 = 5$ and $M_2 = 5$
Figure 2.  (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)
Figure 3.  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 $d_E = 0$
Figure 4.  Type 2 immune induction function. The parameters are $b_1 = 2.5$, $M_1 =120$, $b_2 = 4$, $M_2 = 100$ and $d_E = 0$
Figure 5.  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)
Figure 6.  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)
Figure 7.  The graph of the function $z(I)$ in lemma 4.6 with $a=5$ and $b=10$
Figure 8.  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 $d_E =1$
Figure 9.  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 $d_E =2.5$
Figure 10.  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 $d_E =2.5$
Table 1.  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 $b_1M_1 > b_2M_2$
Table 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 $ $13 \frac {1}{day}$
Table 3.  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 $Q_2^{0}$
Table 4.  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 $Q_4^{1}$
Table 5.  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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