December  2017, 22(10): 3721-3747. doi: 10.3934/dcdsb.2017186

Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission

1. 

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

2. 

Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5 Canada

* Corresponding author: Yuming Chen

Received  October 2016 Revised  May 2017 Published  July 2017

Fund Project: The research of Wang is supported partially by the National Natural Science Foundation of China (No. 11226255 and No. 11201128), the Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No. 2014TD005); while that of Chen is supported by NSERC

In this paper, we are concerned with an age-structured HIV infection model incorporating latency and cell-to-cell transmission. The model is a hybrid system consisting of coupled ordinary differential equations and partial differential equations. First, we address the relative compactness and persistence of the solution semi-flow, and the existence of a global attractor. Then, applying the approach of Lyapunov functionals, we establish the global stability of the infection-free equilibrium and the infection equilibrium, which is completely determined by the basic reproduction number.

Citation: Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186
References:
[1]

A. AlshormanC. SamarasingheW. Lu and L. Rong, An HIV model with age-structured latently infected cells, J. Biol. Dyna., 11 (2017), 192-215. doi: 10.1080/17513758.2016.1198835. Google Scholar

[2]

J. N. BlanksonD. Persaud and R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection, Annu. Rev. Med., 53 (2002), 557-593. doi: 10.1146/annurev.med.53.082901.104024. Google Scholar

[3]

C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Anal.: Real World Appl., 22 (2015), 354-372. doi: 10.1016/j.nonrwa.2014.10.004. Google Scholar

[4]

N. ChomontM. El-FarP. Ancuta and L. Trautmann, HIV reservoir size and persistence are driven by T cell survival and homeostatic proliferation, Nat. Med., 15 (2009), 893-900. doi: 10.1038/nm.1972. Google Scholar

[5]

J. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005. Google Scholar

[6]

Z. Feng and L. Rong, The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 71 (2006), 161-179. Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. Google Scholar

[8]

G. HuangX. Liu and Y. Takeuchi, Lyapunov fucntions and global stability for age-structure HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588. Google Scholar

[9]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini, Pisa, 1985.Google Scholar

[10]

H.-D. KwonJ. Lee and S.-D. Yang, Optimal control of an age-structured model of HIV infection, Appl. Math. Comput., 219 (2012), 2766-2779. doi: 10.1016/j.amc.2012.09.003. Google Scholar

[11]

H. Kim and A. S. Perelson, Viral and latent reservoir persistence in HIV-1-infected patients on therapy, PLoS Comput. Biol, 10 (2006), e135. doi: 10.1016/j.amc.2012.09.003. Google Scholar

[12]

H.-D. KwonJ. Lee and M. Yoon, An age-structured model with immune response of HIV infection: Modeling and optimal control approach, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 153-172. doi: 10.3934/dcdsb.2014.19.153. Google Scholar

[13]

X. Lai and X. Zou, Modeling the HIV-1 virus dynamics with both virus-to-cell and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145. Google Scholar

[14]

X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563-584. doi: 10.1016/j.jmaa.2014.10.086. Google Scholar

[15]

P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35. Google Scholar

[16]

D. MazurovA. IlinskayaG. HeideckerP. Lloyd and D. Derse, Quantitative comparison of HTLV-1 and HIV-1 cell-to-cell infection with new replication dependent vectors, PLoS Pathog., 6 (2001), e1000788. doi: 10.1371/journal.ppat.1000788. Google Scholar

[17]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. Google Scholar

[18]

A. Mojaver and H. Kheiri, Mathematical analysis of a class of HIV infection models of CD4$^+$ T-cells with combined antiretroviral therapy, Appl. Math. Comput., 259 (2015), 258-270. doi: 10.1016/j.amc.2015.02.064. Google Scholar

[19]

B. MonelE. BeaumontD. VendramO. SchwartzD. Brand and F. Mammano, HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through Env-mediated fusion pores, J. Virol., 86 (2012), 3924-3933. doi: 10.1128/JVI.06478-11. Google Scholar

[20]

V. MullerJ. F. Vigueras-Gomez and S. Bonhoeffer, Decelerating decay of latently infected cells during prolonged therapy for human immunodeficiency virus type 1 infection, J. Virol., 76 (2002), 8963-8965. doi: 10.1128/JVI.76.17.8963-8965.2002. Google Scholar

[21]

P. W. NelsonM. A. GilchristD. CoombsJ. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. Google Scholar

[22]

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. Google Scholar

[23]

M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. Google Scholar

[24]

A. S. PerelsonD. E. Kirschner and R. De Boer, Dynamics of HIV infectin of CD4$^+$ T cells, Math. Bisci., 114 (1993), 81-125. Google Scholar

[25]

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. Google Scholar

[26]

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. doi: 10.1038/387188a0. Google Scholar

[27]

V. Piguet and Q. Sattentau, Dangerous liaisons at the virological synapse, J. Clin. Invest., 114 (2004), 605-610. doi: 10.1172/JCI22812. Google Scholar

[28]

H. PourbashashS. S. PilyuginC. C. McCluskey and P. De Leenheer, Global dynamics of within host virus models with cell-to-cell transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341. Google Scholar

[29]

L. RongZ. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945. Google Scholar

[30]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc. , Providence, 2011. Google Scholar

[31]

M. C. StrainH. F. Gunthard and D. V. Havlir, Heterogeneous clearance rates of long-lived lymphocytes infected with HIV: Intrinsic stability predicts lifelong persistence, Proc. Natl. Acad. Sci. USA, 100 (2003), 4819-4824. doi: 10.1073/pnas.0736332100. Google Scholar

[32]

M. C. StrainS. J. Little and E. S. Daar, Effect of treatment, during primary infection, on establishment and clearance of cellular reservoirs of HIV-1, J. Infect. Dis., 191 (2005), 1410-1418. doi: 10.1086/428777. Google Scholar

[33]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201. doi: 10.1016/S0025-5564(00)00018-3. Google Scholar

[34]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068. Google Scholar

[35]

J. A. Walker, Dynamical Systems and Evolution Equations, Plenum Press, New York and London, 1980. Google Scholar

[36]

H. WangR. XuZ. Wang and H. Chen, Glbal dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal.: Model. Control, 20 (2015), 21-37. doi: 10.15388/NA.2015.1.2. Google Scholar

[37]

J. WangJ. Lang and X. Zou, Analysis of an age structured HIV infection model with both virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal.: RWA, 34 (2017), 75-96. doi: 10.1016/j.nonrwa.2016.08.001. Google Scholar

[38]

J. WangR. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl., 432 (2015), 289-313. doi: 10.1016/j.jmaa.2015.06.040. Google Scholar

[39]

J. WangR. Zhang and T. Kuniya, Mathematical analysis for an age-structured HIV infection model with saturation infection rate, Electron. J. Differential Equations, 33 (2015), 1-19. Google Scholar

[40]

J. WangR. Zhang and T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321-343. doi: 10.1093/imamat/hxv039. Google Scholar

[41]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc. , New York, 1985. Google Scholar

[42]

Q. Wen and J. Lou, The global dynamics of a model about HIV-11 infection in vivo, Ric. Mat., 58 (2009), 77-90. doi: 10.1007/s11587-009-0048-y. Google Scholar

show all references

References:
[1]

A. AlshormanC. SamarasingheW. Lu and L. Rong, An HIV model with age-structured latently infected cells, J. Biol. Dyna., 11 (2017), 192-215. doi: 10.1080/17513758.2016.1198835. Google Scholar

[2]

J. N. BlanksonD. Persaud and R. F. Siliciano, The challenge of viral reservoirs in HIV-1 infection, Annu. Rev. Med., 53 (2002), 557-593. doi: 10.1146/annurev.med.53.082901.104024. Google Scholar

[3]

C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Anal.: Real World Appl., 22 (2015), 354-372. doi: 10.1016/j.nonrwa.2014.10.004. Google Scholar

[4]

N. ChomontM. El-FarP. Ancuta and L. Trautmann, HIV reservoir size and persistence are driven by T cell survival and homeostatic proliferation, Nat. Med., 15 (2009), 893-900. doi: 10.1038/nm.1972. Google Scholar

[5]

J. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005. Google Scholar

[6]

Z. Feng and L. Rong, The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 71 (2006), 161-179. Google Scholar

[7]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988. Google Scholar

[8]

G. HuangX. Liu and Y. Takeuchi, Lyapunov fucntions and global stability for age-structure HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588. Google Scholar

[9]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini, Pisa, 1985.Google Scholar

[10]

H.-D. KwonJ. Lee and S.-D. Yang, Optimal control of an age-structured model of HIV infection, Appl. Math. Comput., 219 (2012), 2766-2779. doi: 10.1016/j.amc.2012.09.003. Google Scholar

[11]

H. Kim and A. S. Perelson, Viral and latent reservoir persistence in HIV-1-infected patients on therapy, PLoS Comput. Biol, 10 (2006), e135. doi: 10.1016/j.amc.2012.09.003. Google Scholar

[12]

H.-D. KwonJ. Lee and M. Yoon, An age-structured model with immune response of HIV infection: Modeling and optimal control approach, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 153-172. doi: 10.3934/dcdsb.2014.19.153. Google Scholar

[13]

X. Lai and X. Zou, Modeling the HIV-1 virus dynamics with both virus-to-cell and cell-to-cell transmission, SIAM J. Appl. Math., 74 (2014), 898-917. doi: 10.1137/130930145. Google Scholar

[14]

X. Lai and X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563-584. doi: 10.1016/j.jmaa.2014.10.086. Google Scholar

[15]

P. Magal, Compact attractors for time periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35. Google Scholar

[16]

D. MazurovA. IlinskayaG. HeideckerP. Lloyd and D. Derse, Quantitative comparison of HTLV-1 and HIV-1 cell-to-cell infection with new replication dependent vectors, PLoS Pathog., 6 (2001), e1000788. doi: 10.1371/journal.ppat.1000788. Google Scholar

[17]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. Google Scholar

[18]

A. Mojaver and H. Kheiri, Mathematical analysis of a class of HIV infection models of CD4$^+$ T-cells with combined antiretroviral therapy, Appl. Math. Comput., 259 (2015), 258-270. doi: 10.1016/j.amc.2015.02.064. Google Scholar

[19]

B. MonelE. BeaumontD. VendramO. SchwartzD. Brand and F. Mammano, HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through Env-mediated fusion pores, J. Virol., 86 (2012), 3924-3933. doi: 10.1128/JVI.06478-11. Google Scholar

[20]

V. MullerJ. F. Vigueras-Gomez and S. Bonhoeffer, Decelerating decay of latently infected cells during prolonged therapy for human immunodeficiency virus type 1 infection, J. Virol., 76 (2002), 8963-8965. doi: 10.1128/JVI.76.17.8963-8965.2002. Google Scholar

[21]

P. W. NelsonM. A. GilchristD. CoombsJ. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. Google Scholar

[22]

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. Google Scholar

[23]

M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000. Google Scholar

[24]

A. S. PerelsonD. E. Kirschner and R. De Boer, Dynamics of HIV infectin of CD4$^+$ T cells, Math. Bisci., 114 (1993), 81-125. Google Scholar

[25]

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. Google Scholar

[26]

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. doi: 10.1038/387188a0. Google Scholar

[27]

V. Piguet and Q. Sattentau, Dangerous liaisons at the virological synapse, J. Clin. Invest., 114 (2004), 605-610. doi: 10.1172/JCI22812. Google Scholar

[28]

H. PourbashashS. S. PilyuginC. C. McCluskey and P. De Leenheer, Global dynamics of within host virus models with cell-to-cell transmission, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3341-3357. doi: 10.3934/dcdsb.2014.19.3341. Google Scholar

[29]

L. RongZ. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945. Google Scholar

[30]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc. , Providence, 2011. Google Scholar

[31]

M. C. StrainH. F. Gunthard and D. V. Havlir, Heterogeneous clearance rates of long-lived lymphocytes infected with HIV: Intrinsic stability predicts lifelong persistence, Proc. Natl. Acad. Sci. USA, 100 (2003), 4819-4824. doi: 10.1073/pnas.0736332100. Google Scholar

[32]

M. C. StrainS. J. Little and E. S. Daar, Effect of treatment, during primary infection, on establishment and clearance of cellular reservoirs of HIV-1, J. Infect. Dis., 191 (2005), 1410-1418. doi: 10.1086/428777. Google Scholar

[33]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201. doi: 10.1016/S0025-5564(00)00018-3. Google Scholar

[34]

H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447-1479. doi: 10.1137/0153068. Google Scholar

[35]

J. A. Walker, Dynamical Systems and Evolution Equations, Plenum Press, New York and London, 1980. Google Scholar

[36]

H. WangR. XuZ. Wang and H. Chen, Glbal dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal.: Model. Control, 20 (2015), 21-37. doi: 10.15388/NA.2015.1.2. Google Scholar

[37]

J. WangJ. Lang and X. Zou, Analysis of an age structured HIV infection model with both virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal.: RWA, 34 (2017), 75-96. doi: 10.1016/j.nonrwa.2016.08.001. Google Scholar

[38]

J. WangR. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl., 432 (2015), 289-313. doi: 10.1016/j.jmaa.2015.06.040. Google Scholar

[39]

J. WangR. Zhang and T. Kuniya, Mathematical analysis for an age-structured HIV infection model with saturation infection rate, Electron. J. Differential Equations, 33 (2015), 1-19. Google Scholar

[40]

J. WangR. Zhang and T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321-343. doi: 10.1093/imamat/hxv039. Google Scholar

[41]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, Inc. , New York, 1985. Google Scholar

[42]

Q. Wen and J. Lou, The global dynamics of a model about HIV-11 infection in vivo, Ric. Mat., 58 (2009), 77-90. doi: 10.1007/s11587-009-0048-y. Google Scholar

Figure 1.  Transfer diagram for model (1.4)
Figure 2.  The distributions of infected cells and latent cells with parameter (8.3)
Figure 3.  Evolutions of infected cells and latent cells with parameter (8.3) except that h = 1
Figure 4.  Evolutions of infected cells and latent cells with parameter (8.4)
Table 1.  Biological meanings of parameters (1.4) and (1.5)
ParameterMeaning
$h$Constant recruitment rate of uninfected CD4+ T cells
$d$Death rate of uninfected CD4+ T cells
$\beta$Infection rate of CD4+ T cells by infectious virus
$\beta_1$Infection rate of CD4+ T cells by latently infected T cells
$\beta_2$Infection rate of CD4+ T cells by infectious T cells
$q_1(a)$Infectivity of a latently infected T cell with latency age $a$
$q_2(b)$Infectivity of an infectious T cell with infection age $b$
$\theta_1(a)$Sum of death rate and activation rate $\xi(a)$
of latently infected T cells with latency age $a$
$\theta_2(b)$Death rate of infectious T cells with infection age $b$
$p(b)$Viral production rate of an infectious T cell with infection age $b$
$c$Clearance rate of virions
$\xi(a)$Activation rate of latently infected T cells with latency age $a$
ParameterMeaning
$h$Constant recruitment rate of uninfected CD4+ T cells
$d$Death rate of uninfected CD4+ T cells
$\beta$Infection rate of CD4+ T cells by infectious virus
$\beta_1$Infection rate of CD4+ T cells by latently infected T cells
$\beta_2$Infection rate of CD4+ T cells by infectious T cells
$q_1(a)$Infectivity of a latently infected T cell with latency age $a$
$q_2(b)$Infectivity of an infectious T cell with infection age $b$
$\theta_1(a)$Sum of death rate and activation rate $\xi(a)$
of latently infected T cells with latency age $a$
$\theta_2(b)$Death rate of infectious T cells with infection age $b$
$p(b)$Viral production rate of an infectious T cell with infection age $b$
$c$Clearance rate of virions
$\xi(a)$Activation rate of latently infected T cells with latency age $a$
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