April  2021, 26(4): 2257-2272. doi: 10.3934/dcdsb.2020212

Viral dynamics of HIV-1 with CTL immune response

1. 

School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

2. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

* Corresponding author: Michael Y. Li

Dedicated to Professor Sze-Bi Hsu on the occasion of his retirement.

Received  June 2019 Revised  November 2019 Published  April 2021 Early access  July 2020

In this paper, we investigate an in-host model for the viral dynamics of HIV-1 infection and its interaction with the CTL immune response. The model is sufficiently general to allow nonlinear forms for both viral infection and CTL response. Threshold parameters are identified that completely determine the global dynamics and outcomes of the virus-target cell-CTL interactions. Impacts of key parameter values for CTL functions and viral budding rate on the HIV-1 viral load and CD4 count are investigated using numerical simulations. Results support clinical evidence for important differences between HIV-1 nonprogressors and progressors.

Citation: Aiping Wang, Michael Y. Li. Viral dynamics of HIV-1 with CTL immune response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2257-2272. doi: 10.3934/dcdsb.2020212
References:
[1]

J. B. AlimontiT. B. Ball and K. R. Fowke, Mechanisms of CD$4^+$ T lymphocyte cell death in human immunodeficiency virus infection and AIDS, J. Gen. Virol., 84 (2003), 1649-1661.  doi: 10.1099/vir.0.19110-0.

[2]

R. A. ArnaoutM. A. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic T lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354.  doi: 10.1098/rspb.2000.1149.

[3]

G. J. ButlerS. B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM J. Appl. Math., 45 (1983), 435-449.  doi: 10.1137/0145025.

[4]

J. M. Conway and R. M. Ribeiro, Modeling the immune response to HIV infection, Curr. Opin. Syst. Biol., 12 (2018), 61-69. 

[5]

J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proc. Natl. Acad. Sci. USA, 112 (2015), 5467-5472.  doi: 10.1073/pnas.1419162112.

[6]

T. DumrongpokaphanY. LenburyR. Ouncharoen and Y. S. Xu, An intracellular delay-differential equation model of the HIV infection and immune control, Math. Model. Nat. Phenom. Epidemiol., 2 (2007), 75-99.  doi: 10.1051/mmnp:2008012.

[7]

H. I. FreedmanM. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600.  doi: 10.1007/BF02218848.

[8]

Y. Gao, P. F. McKay and J. F. S. Mann, Advances in HIV-1 vaccine development, Viruses, 10 (2018), 167. doi: 10.3390/v10040167.

[9]

J. C. Gea-BanaclocheS. A. Migueles and L. Martino, Maintenance of large numbers of virus-specific CD8 + T cells in HIV-infected progressors and long-term nonprogressors, J. Immunol., 165 (2000), 1082-1092.  doi: 10.4049/jimmunol.165.2.1082.

[10]

B. S. Goh, Global stability in many-species systems, Amer. Natur., 111 (1997), 135-143.  doi: 10.1086/283144.

[11]

H. Gomez-AcevedoM. Y. Li and S. Jacobson, Multi-stability In a model for CTL response to HTLV-1 infection and its implications to HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696.  doi: 10.1007/s11538-009-9465-z.

[12]

R. A. GrutersC. A. van Baalen and A. D. M. E. Osterhaus, The advantage of early recognition of HIV-infected cells by cytotoxic T-lymphocytes, Vaccine, 20 (2002), 2011-2015.  doi: 10.1016/S0264-410X(02)00089-0.

[13]

H. GuoM. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. 

[14]

H. GuoM. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.

[15]

J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York, 1969.

[16]

S. M. HammerM. E. Sobieszczyk and H. Janes, Efficacy trial of a DNA/rAd5 HIV-1 preventive vaccine, N. Engl. J. Med., 369 (2013), 2083-2092.  doi: 10.1056/NEJMoa1310566.

[17]

S. B. Hsu, On global stability of a predator-prey systems, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.

[18]

S. ImlachS. McBreen and T. Shirafuji, Activated peripheral CD8 lymphocytes express CD4 in vivo and are targets for infection by human immunodeficiency virus type 1, J. Virol., 75 (2001), 11555-11564.  doi: 10.1128/JVI.75.23.11555-11564.2001.

[19]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[20]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57.

[21]

J. P. LaSalle, The Stability of Dynamical System, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.

[22]

J. Lang and M. Y. Li, Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, J. Math. Biol., 65 (2012), 181-199.  doi: 10.1007/s00285-011-0455-z.

[23]

M. Y. LiJ. R. GraefL. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.

[24]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.

[25]

M. Y. LiZ. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.  doi: 10.1016/j.jmaa.2009.09.017.

[26]

M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonl. Anal. RWA, 17 (2014), 147-160.  doi: 10.1016/j.nonrwa.2013.11.002.

[27]

J. LinR. Xu and X. Tian, Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses, Math. Biosci. Engin., 16 (2018), 292-319.  doi: 10.3934/mbe.2019015.

[28]

S. A. MiguelesA. C. Laborico and W. L Shupert, HIV-specific CD8+ T cell proliferation is coupled to perforin expression and is maintained in nonprogressors, Nature Immunol., 3 (2002), 1061-1068.  doi: 10.1038/ni845.

[29]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8.

[30]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74.

[31] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. 
[32]

G. S. Ogg and X. Jin, Quantitation of HIV-1-specific cytotoxic T lymphocytes and plasma load of viral RNA, Science, 279 (1998), 2103-2106.  doi: 10.1126/science.279.5359.2103.

[33]

A. S. PerelsonD. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD$4^+$ T cells, Math. Biosci., 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.

[34]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.

[35]

A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96. doi: 10.1186/1741-7007-11-96.

[36]

L. B. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011.

[37]

H. ShuL. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302.  doi: 10.1137/120896463.

[38]

V. Simon and D. D. Ho, HIV-1 dynamics in vivo: Implications for therapy, Nat. Rev. Microbiol., 1 (2003), 181-190. 

[39]

G. D. Tomarasa and B. F. Haynes, HIV-1-specific antibody responses during acute and chronic HIV-1 infection, Curr. Opin. HIV AIDS, 4 (2009), 373-379.  doi: 10.1097/COH.0b013e32832f00c0.

[40]

J. Overbaugh and L. Morris, The Antibody Response against HIV-1, Cold Spring Harb. Perspect. Med., 2 (2012), a007039. doi: 10.1101/cshperspect.a007039.

[41]

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

[42]

Y. WangY. C. ZhouF. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. 

[43]

J. Weber, The pathogenesis of HIV-1 infection, Br. Med. Bull., 58 (2001), 61-72.  doi: 10.1093/bmb/58.1.61.

[44]

D. Wodarz and M. A. Nowak, Correlates of cytotoxic t-lymphocyte-mediated virus control: Implications for immunosuppressive infections and their treatment, Phil. Trans. R. Soc. Lond. B, 355 (2000), 1059-1070.  doi: 10.1098/rstb.2000.0643.

[45]

H. Y. Zhu and X. F. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, DCDS B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.

show all references

References:
[1]

J. B. AlimontiT. B. Ball and K. R. Fowke, Mechanisms of CD$4^+$ T lymphocyte cell death in human immunodeficiency virus infection and AIDS, J. Gen. Virol., 84 (2003), 1649-1661.  doi: 10.1099/vir.0.19110-0.

[2]

R. A. ArnaoutM. A. Nowak and D. Wodarz, HIV-1 dynamics revisited: Biphasic decay by cytotoxic T lymphocyte killing?, Proc. R. Soc. Lond. B, 267 (2000), 1347-1354.  doi: 10.1098/rspb.2000.1149.

[3]

G. J. ButlerS. B. Hsu and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM J. Appl. Math., 45 (1983), 435-449.  doi: 10.1137/0145025.

[4]

J. M. Conway and R. M. Ribeiro, Modeling the immune response to HIV infection, Curr. Opin. Syst. Biol., 12 (2018), 61-69. 

[5]

J. M. Conway and A. S. Perelson, Post-treatment control of HIV infection, Proc. Natl. Acad. Sci. USA, 112 (2015), 5467-5472.  doi: 10.1073/pnas.1419162112.

[6]

T. DumrongpokaphanY. LenburyR. Ouncharoen and Y. S. Xu, An intracellular delay-differential equation model of the HIV infection and immune control, Math. Model. Nat. Phenom. Epidemiol., 2 (2007), 75-99.  doi: 10.1051/mmnp:2008012.

[7]

H. I. FreedmanM. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600.  doi: 10.1007/BF02218848.

[8]

Y. Gao, P. F. McKay and J. F. S. Mann, Advances in HIV-1 vaccine development, Viruses, 10 (2018), 167. doi: 10.3390/v10040167.

[9]

J. C. Gea-BanaclocheS. A. Migueles and L. Martino, Maintenance of large numbers of virus-specific CD8 + T cells in HIV-infected progressors and long-term nonprogressors, J. Immunol., 165 (2000), 1082-1092.  doi: 10.4049/jimmunol.165.2.1082.

[10]

B. S. Goh, Global stability in many-species systems, Amer. Natur., 111 (1997), 135-143.  doi: 10.1086/283144.

[11]

H. Gomez-AcevedoM. Y. Li and S. Jacobson, Multi-stability In a model for CTL response to HTLV-1 infection and its implications to HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696.  doi: 10.1007/s11538-009-9465-z.

[12]

R. A. GrutersC. A. van Baalen and A. D. M. E. Osterhaus, The advantage of early recognition of HIV-infected cells by cytotoxic T-lymphocytes, Vaccine, 20 (2002), 2011-2015.  doi: 10.1016/S0264-410X(02)00089-0.

[13]

H. GuoM. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. 

[14]

H. GuoM. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.  doi: 10.1090/S0002-9939-08-09341-6.

[15]

J. K. Hale, Ordinary Differential Equations, John Wiley & Sons, New York, 1969.

[16]

S. M. HammerM. E. Sobieszczyk and H. Janes, Efficacy trial of a DNA/rAd5 HIV-1 preventive vaccine, N. Engl. J. Med., 369 (2013), 2083-2092.  doi: 10.1056/NEJMoa1310566.

[17]

S. B. Hsu, On global stability of a predator-prey systems, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.

[18]

S. ImlachS. McBreen and T. Shirafuji, Activated peripheral CD8 lymphocytes express CD4 in vivo and are targets for infection by human immunodeficiency virus type 1, J. Virol., 75 (2001), 11555-11564.  doi: 10.1128/JVI.75.23.11555-11564.2001.

[19]

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[20]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.  doi: 10.3934/mbe.2004.1.57.

[21]

J. P. LaSalle, The Stability of Dynamical System, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.

[22]

J. Lang and M. Y. Li, Stable and transient periodic oscillations in a mathematical model for CTL response to HTLV-I infection, J. Math. Biol., 65 (2012), 181-199.  doi: 10.1007/s00285-011-0455-z.

[23]

M. Y. LiJ. R. GraefL. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.

[24]

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.  doi: 10.1016/j.jde.2009.09.003.

[25]

M. Y. LiZ. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.  doi: 10.1016/j.jmaa.2009.09.017.

[26]

M. Y. Li and L. Wang, Backward bifurcation in a mathematical model for HIV infection in vivo with anti-retroviral treatment, Nonl. Anal. RWA, 17 (2014), 147-160.  doi: 10.1016/j.nonrwa.2013.11.002.

[27]

J. LinR. Xu and X. Tian, Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses, Math. Biosci. Engin., 16 (2018), 292-319.  doi: 10.3934/mbe.2019015.

[28]

S. A. MiguelesA. C. Laborico and W. L Shupert, HIV-specific CD8+ T cell proliferation is coupled to perforin expression and is maintained in nonprogressors, Nature Immunol., 3 (2002), 1061-1068.  doi: 10.1038/ni845.

[29]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8.

[30]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74.

[31] M. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. 
[32]

G. S. Ogg and X. Jin, Quantitation of HIV-1-specific cytotoxic T lymphocytes and plasma load of viral RNA, Science, 279 (1998), 2103-2106.  doi: 10.1126/science.279.5359.2103.

[33]

A. S. PerelsonD. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD$4^+$ T cells, Math. Biosci., 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.

[34]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.

[35]

A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96. doi: 10.1186/1741-7007-11-96.

[36]

L. B. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011.

[37]

H. ShuL. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280-1302.  doi: 10.1137/120896463.

[38]

V. Simon and D. D. Ho, HIV-1 dynamics in vivo: Implications for therapy, Nat. Rev. Microbiol., 1 (2003), 181-190. 

[39]

G. D. Tomarasa and B. F. Haynes, HIV-1-specific antibody responses during acute and chronic HIV-1 infection, Curr. Opin. HIV AIDS, 4 (2009), 373-379.  doi: 10.1097/COH.0b013e32832f00c0.

[40]

J. Overbaugh and L. Morris, The Antibody Response against HIV-1, Cold Spring Harb. Perspect. Med., 2 (2012), a007039. doi: 10.1101/cshperspect.a007039.

[41]

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

[42]

Y. WangY. C. ZhouF. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. 

[43]

J. Weber, The pathogenesis of HIV-1 infection, Br. Med. Bull., 58 (2001), 61-72.  doi: 10.1093/bmb/58.1.61.

[44]

D. Wodarz and M. A. Nowak, Correlates of cytotoxic t-lymphocyte-mediated virus control: Implications for immunosuppressive infections and their treatment, Phil. Trans. R. Soc. Lond. B, 355 (2000), 1059-1070.  doi: 10.1098/rstb.2000.0643.

[45]

H. Y. Zhu and X. F. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, DCDS B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.

Figure 1.  Transfer-infection diagram of model (2). Solid lines indicate cell transfer and dotted lines indicated virus-cell or cell-cell interaction
Figure 2.  Graphs of functions $ F(v) $ and $ G(v) $ and a geometric demonstration of existence of chronic-infection equilibria $ P_1 $ and $ P_2 $
Figure 3.  Effects of CTL response parameter $ c $ on HIV-1 viral load and CD4 count
Figure 4.  Effects of CTL killing rate $ \alpha $ on HIV-1 viral load and CD4 count
Figure 5.  Effects of viral budding number $ N $ on HIV-1 viral load and CD4 count
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