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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[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.

[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.

[41]

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

[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.

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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