June 2018, 15(3): 569-594. doi: 10.3934/mbe.2018026

Analysis of an HIV infection model incorporating latency age and infection age

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

Received  February 06, 2017 Accepted  May 25, 2017 Published  December 2017

There is a growing interest to understand impacts of latent infection age and infection age on viral infection dynamics by using ordinary and partial differential equations. On one hand, activation of latently infected cells needs specificity antigen, and latently infected CD4+ T cells are often heterogeneous, which depends on how frequently they encountered antigens, how much time they need to be preferentially activated and quickly removed from the reservoir. On the other hand, infection age plays an important role in modeling the death rate and virus production rate of infected cells. By rigorous analysis for the model, this paper is devoted to the global dynamics of an HIV infection model subject to latency age and infection age from theoretical point of view, where the model formulation, basic reproduction number computation, and rigorous mathematical analysis, such as relative compactness and persistence of the solution semiflow, and existence of a global attractor are involved. By constructing Lyapunov functions, the global dynamics of a threshold type is established. The method developed here is applicable to broader contexts of investigating viral infection subject to age structure.

Citation: Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934/mbe.2018026
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]

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

[3]

C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656. doi: 10.1007/s002850050069.

[4]

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

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, 1988.

[6]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monogr. CNR, vol. 7, Giadini Editori e Stampator, Pisa, 1994.

[7]

H. Kim and A. S. Perelson, Viral and latent reservoir persistence in HIV-1-infected patients on therapy, PLoS Comput. Biol., 10 (2006), e135.

[8]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322.

[9]

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.

[10]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Bio., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[11]

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.

[12]

M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size, Math. Biosci., 182 (2003), 1-25. doi: 10.1016/S0025-5564(02)00184-0.

[13]

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.

[14]

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

[15]

P. W. NelsonM. A. GilchristD. CoombsJ. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allow for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267.

[16]

Y. Nakata, Global dynamics of a viral infection model with a latent period and beddington-DeAngelis response, Nonlinear Anal. TMA, 74 (2011), 2929-2940. doi: 10.1016/j.na.2010.12.030.

[17]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27. doi: 10.1016/j.jmaa.2010.08.025.

[18]

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.

[19]

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. doi: 10.1371/journal.pcbi.1000533.

[20]

L. Rong and A. S. Perelson, Asymmetric division of activated latently infected cells may explain the decay kinetics of the HIV-1 latent reservoir and intermittent viral blips, Math. Biosci., 217 (2009), 77-87. doi: 10.1016/j.mbs.2008.10.006.

[21]

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

[22]

M. C. StrainH. F. GunthardD. V. HavlirC. C. IgnacioD. M. SmithA. J. Leigh-BrownT. R. MacaranasR. Y. LamO. A. DalyM. FischerM. OpravilH. LevineL. BachelerC. A. SpinaD. D. Richman and J. K. Wong, Heterogeneous clearance rates of long-lived lymphocytes infected with HIV: Intrinsic stability predicts lifelong persistence, Proc. Natl. Acad. Sci. USA,, 191 (2005), 1410-1418. doi: 10.1073/pnas.0736332100.

[23]

M. C. StrainS. J. LittleE. S. DaarD. V. HavlirH. F. GünthardR. Y. LamO. A. DalyJ. NguyenC. C. IgnacioC. A. SpinaD. D. Richman and J. K. Wong, Effect of treatment, during primary infection, on establishment and clearance of cellular reservoirs ofHIV-1, J. Infect. Dis., 191 (2005), 1410-1418.

[24]

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.

[25]

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

[26]

H. L. Smith, Mathematics in Population Biology, Princeton University Press, 2003.

[27]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Dif. Int. Eqs., 3 (1990), 1035-1066.

[28]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801. doi: 10.1016/j.jde.2011.01.007.

[29]

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.

[30]

V. Volterra, Leçns Sur la Thérie Mathématique de la Lutte Pour la vie, Éditions Jacques Gabay, Sceaux, 1990.

[31]

J. Wang and S. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression, Commun. Nonlinear Sci. Numer. Simulat., 20 (2015), 263-272. doi: 10.1016/j.cnsns.2014.04.027.

[32]

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.

[33]

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.

[34]

J. WangG. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300. doi: 10.1093/imammb/dqr009.

[35]

J. Wang and L. Guan, Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Cont. Dyn. Sys. B, 17 (2012), 297-302. doi: 10.3934/dcdsb.2012.17.297.

[36]

J. WangJ. PangT. Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298-316. doi: 10.1016/j.amc.2014.05.015.

[37]

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

[38]

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.

[39]

X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Meth. Appl. Sci., 36 (2013), 125-142. doi: 10.1002/mma.2576.

[40]

X. WangS. TangX. Song and L. Rong, Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dyna., 11 (2017), 455-483. doi: 10.1080/17513758.2016.1242784.

[41]

D. Wodarz, -Hepatitis c virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750. doi: 10.1099/vir.0.19118-0.

[42]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York and Basel, 1985.

[43]

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

[44]

Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Cont. Dyn. Sys. B, 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401.

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]

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

[3]

C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656. doi: 10.1007/s002850050069.

[4]

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

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, 1988.

[6]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monogr. CNR, vol. 7, Giadini Editori e Stampator, Pisa, 1994.

[7]

H. Kim and A. S. Perelson, Viral and latent reservoir persistence in HIV-1-infected patients on therapy, PLoS Comput. Biol., 10 (2006), e135.

[8]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322.

[9]

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.

[10]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Bio., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[11]

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.

[12]

M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size, Math. Biosci., 182 (2003), 1-25. doi: 10.1016/S0025-5564(02)00184-0.

[13]

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.

[14]

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

[15]

P. W. NelsonM. A. GilchristD. CoombsJ. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allow for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267.

[16]

Y. Nakata, Global dynamics of a viral infection model with a latent period and beddington-DeAngelis response, Nonlinear Anal. TMA, 74 (2011), 2929-2940. doi: 10.1016/j.na.2010.12.030.

[17]

Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, J. Math. Anal. Appl., 375 (2011), 14-27. doi: 10.1016/j.jmaa.2010.08.025.

[18]

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.

[19]

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. doi: 10.1371/journal.pcbi.1000533.

[20]

L. Rong and A. S. Perelson, Asymmetric division of activated latently infected cells may explain the decay kinetics of the HIV-1 latent reservoir and intermittent viral blips, Math. Biosci., 217 (2009), 77-87. doi: 10.1016/j.mbs.2008.10.006.

[21]

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

[22]

M. C. StrainH. F. GunthardD. V. HavlirC. C. IgnacioD. M. SmithA. J. Leigh-BrownT. R. MacaranasR. Y. LamO. A. DalyM. FischerM. OpravilH. LevineL. BachelerC. A. SpinaD. D. Richman and J. K. Wong, Heterogeneous clearance rates of long-lived lymphocytes infected with HIV: Intrinsic stability predicts lifelong persistence, Proc. Natl. Acad. Sci. USA,, 191 (2005), 1410-1418. doi: 10.1073/pnas.0736332100.

[23]

M. C. StrainS. J. LittleE. S. DaarD. V. HavlirH. F. GünthardR. Y. LamO. A. DalyJ. NguyenC. C. IgnacioC. A. SpinaD. D. Richman and J. K. Wong, Effect of treatment, during primary infection, on establishment and clearance of cellular reservoirs ofHIV-1, J. Infect. Dis., 191 (2005), 1410-1418.

[24]

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.

[25]

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

[26]

H. L. Smith, Mathematics in Population Biology, Princeton University Press, 2003.

[27]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Dif. Int. Eqs., 3 (1990), 1035-1066.

[28]

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011), 3772-3801. doi: 10.1016/j.jde.2011.01.007.

[29]

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.

[30]

V. Volterra, Leçns Sur la Thérie Mathématique de la Lutte Pour la vie, Éditions Jacques Gabay, Sceaux, 1990.

[31]

J. Wang and S. Liu, The stability analysis of a general viral infection model with distributed delays and multi-staged infected progression, Commun. Nonlinear Sci. Numer. Simulat., 20 (2015), 263-272. doi: 10.1016/j.cnsns.2014.04.027.

[32]

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.

[33]

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.

[34]

J. WangG. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300. doi: 10.1093/imammb/dqr009.

[35]

J. Wang and L. Guan, Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Cont. Dyn. Sys. B, 17 (2012), 297-302. doi: 10.3934/dcdsb.2012.17.297.

[36]

J. WangJ. PangT. Kuniya and Y. Enatsu, Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays, Appl. Math. Comput., 241 (2014), 298-316. doi: 10.1016/j.amc.2014.05.015.

[37]

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

[38]

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.

[39]

X. Wang and S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Meth. Appl. Sci., 36 (2013), 125-142. doi: 10.1002/mma.2576.

[40]

X. WangS. TangX. Song and L. Rong, Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dyna., 11 (2017), 455-483. doi: 10.1080/17513758.2016.1242784.

[41]

D. Wodarz, -Hepatitis c virus dynamics and pathology: The role of CTL and antibody responses, J. Gen. Virol., 84 (2003), 1743-1750. doi: 10.1099/vir.0.19118-0.

[42]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York and Basel, 1985.

[43]

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

[44]

Y. Yan and W. Wang, Global stability of a five-dimensional model with immune responses and delay, Discrete Cont. Dyn. Sys. B, 17 (2012), 401-416. doi: 10.3934/dcdsb.2012.17.401.

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