2016, 13(5): 887-909. doi: 10.3934/mbe.2016022

Immune response in virus model structured by cell infection-age

1. 

Mathematics Department, University of Louisiana at Lafayette, Lafayette, LA 70504, United States

Received  October 2015 Revised  April 2016 Published  July 2016

This paper concerns modeling the coupled within-host population dynamics of virus and CTL (Cytotoxic T Lymphocyte) immune response. There is substantial evidence that the CTL immune response plays a crucial role in controlling HIV in infected patients. Recent experimental studies have demonstrated that certain CTL variants can recognize HIV infected cells early in the infected cell lifecycle before viral production, while other CTLs only detect viral proteins (epitopes) presented on the surface of infected cells after viral production. The kinetics of epitope presentation and immune recognition can impact the efficacy of the immune response. We extend previous virus models to include cell infection-age structure in the infected cell compartment and immune response killing/activation rates of a PDE-ODE system. We characterize solutions to our system utilizing semigroup theory, determine equilibria and reproduction numbers, and prove stability and persistence results. Numerical simulations show that ``early immune recognition'' precipitates both enhanced viral control and sustained oscillations via a Hopf bifurcation. In addition to inducing oscillatory dynamics, considering immune process rates to be functions of cell infection-age can also lead to coexistence of multiple distinct immune effector populations.
Citation: Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022
References:
[1]

A. Akram and R. D. Inman, Immunodominance: A pivotal principle in host response to viral infections,, Clinical Immunology, 143 (2012), 99.  doi: 10.1016/j.clim.2012.01.015.  Google Scholar

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C. L. Althaus and R. J. De Boer, Implications of ctl-mediated killing of hiv-infected cells during the non-productive stage of infection,, PLoS One, 6 (2011).  doi: 10.1371/journal.pone.0016468.  Google Scholar

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A. Balamurugan, A. Ali, J. Boucau, S. Le Gall, H. L. Ng and O. O. Yang, Hiv-1 gag cytotoxic t lymphocyte epitopes vary in presentation kinetics relative to hla class i downregulation,, Journal of Virology, 87 (2013), 8726.  doi: 10.1128/JVI.01040-13.  Google Scholar

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H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in hiv infection dynamics,, Mathematical Biosciences, 183 (2003), 63.  doi: 10.1016/S0025-5564(02)00218-3.  Google Scholar

[6]

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

[7]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

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R. W. Buckheit III, R. F. Siliciano and J. N. Blankson, Primary cd8+ t cells from elite suppressors effectively eliminate non-productively hiv-1 infected resting and activated cd4+ t cells,, Retrovirology, 10 (2013), 1.   Google Scholar

[9]

B. Buonomo and C. Vargas-De-León, Global stability for an hiv-1 infection model including an eclipse stage of infected cells,, Journal of Mathematical Analysis and Applications, 385 (2012), 709.  doi: 10.1016/j.jmaa.2011.07.006.  Google Scholar

[10]

D. Y. Chen, A. Balamurugan, H. L. Ng, W. G. Cumberland and O. O. Yang, Epitope targeting and viral inoculum are determinants of nef-mediated immune evasion of hiv-1 from cytotoxic t lymphocytes,, Blood, 120 (2012), 100.  doi: 10.1182/blood-2012-02-409870.  Google Scholar

[11]

R. V Culshaw and S. Ruan, A delay-differential equation model of hiv infection of cd4+ t-cells,, Mathematical Biosciences, 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

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R. J. De Boer, Which of our modeling predictions are robust,, PLoS Comput. Biol., 8 (2012).  doi: 10.1371/journal.pcbi.1002593.  Google Scholar

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R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections,, SIAM Journal on Applied Mathematics, 73 (2013), 572.  doi: 10.1137/120890351.  Google Scholar

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A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined cauchy problems,, Journal of Mathematical Analysis and applications, 341 (2008), 501.  doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar

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M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate,, Journal of Theoretical Biology, 229 (2004), 281.  doi: 10.1016/j.jtbi.2004.04.015.  Google Scholar

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H. N. Kløverpris, R. P. Payne, J. B. Sacha, J. T. Rasaiyaah, F. Chen, M. Takiguchi, O. O. Yang, G. J. Towers, P. Goulder and J. G. Prado, Early antigen presentation of protective hiv-1 kf11gag and kk10gag epitopes from incoming viral particles facilitates rapid recognition of infected cells by specific cd8+ t cells,, Journal of Virology, 87 (2013), 2628.   Google Scholar

[20]

X. Lai and X. Zou, Dynamics of evolutionary competition between budding and lytic viral release strategies,, Mathematical Biosciences and Engineering: MBE, 11 (2014), 1091.  doi: 10.3934/mbe.2014.11.1091.  Google Scholar

[21]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bulletin of Mathematical Biology, 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[22]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[23]

P. Magal, Compact attractors for time periodic age-structured population models,, Electronic Journal of Differential Equations, 2001 (2001), 1.   Google Scholar

[24]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM Journal on Mathematical Analysis, 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar

[25]

P. W. Nelson, M. A. Gilchrist, D. Coombs, J. 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.  doi: 10.3934/mbe.2004.1.267.  Google Scholar

[26]

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

[27]

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

[28]

M. A. Nowak, R. M. May and K. Sigmund, Immune responses against multiple epitopes,, Journal of Theoretical Biology, 175 (1995), 325.  doi: 10.1006/jtbi.1995.0146.  Google Scholar

[29]

R. P. Payne, H. Kløverpris, J. B. Sacha, Z. Brumme, C. Brumme, S. Buus, S. Sims, S. Hickling, L. Riddell, F. Chen, et al, Efficacious early antiviral activity of hiv gag-and pol-specific hla-b* 2705-restricted cd8+ t cells,, Journal of Virology, 84 (2010), 10543.  doi: 10.1128/JVI.00793-10.  Google Scholar

[30]

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

[31]

A. S. Perelson, A. U. Neumann, M. Markowitz, J. 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.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[32]

L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured hiv-1 dynamics with combination antiretroviral therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731.  doi: 10.1137/060663945.  Google Scholar

[33]

L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host hiv-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,, Journal of Theoretical Biology, 247 (2007), 804.  doi: 10.1016/j.jtbi.2007.04.014.  Google Scholar

[34]

J. B. Sacha, C. Chung, E. G. Rakasz, S. P. Spencer, A. K. Jonas, A. T. Bean, W. Lee, B. J. Burwitz, J. J. Stephany, J. T. Loffredo, et al, Gag-specific cd8+ t lymphocytes recognize infected cells before aids-virus integration and viral protein expression,, The Journal of Immunology, 178 (2007), 2746.  doi: 10.4049/jimmunol.178.5.2746.  Google Scholar

[35]

H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and ctl immune responses,, SIAM Journal on Applied Mathematics, 73 (2013), 1280.  doi: 10.1137/120896463.  Google Scholar

[36]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, volume 57., Springer Science & Business Media, (2011).  doi: 10.1007/978-1-4419-7646-8.  Google Scholar

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H. L. Smith and P. De Leenheer, Virus dynamics: A global analysis,, SIAM Journal on Applied Mathematics, 63 (2003), 1313.  doi: 10.1137/S0036139902406905.  Google Scholar

[38]

X. Song, S. Wang and J. Dong, Stability properties and hopf bifurcation of a delayed viral infection model with lytic immune response,, Journal of Mathematical Analysis and Applications, 373 (2011), 345.  doi: 10.1016/j.jmaa.2010.04.010.  Google Scholar

[39]

H. R. Thieme, Integrated semigroups and integrated solutions to abstract cauchy problems,, Journal of Mathematical Analysis and Applications, 152 (1990), 416.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[40]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, Advances in Mathematical Population Dynamics-Molecules, (1997), 691.   Google Scholar

[41]

H. R. Thieme et al, Semiflows generated by lipschitz perturbations of non-densely defined operators,, Differential and Integral Equations, 3 (1990), 1035.   Google Scholar

[42]

B. D. Walker and G. Y. Xu, Unravelling the mechanisms of durable control of hiv-1,, Nature Reviews Immunology, 13 (2013), 487.  doi: 10.1038/nri3478.  Google Scholar

[43]

K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response,, Physica D: Nonlinear Phenomena, 226 (2007), 197.  doi: 10.1016/j.physd.2006.12.001.  Google Scholar

[44]

Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with ctl immune response incorporating antiretroviral therapy,, Journal of Mathematical Biology, 67 (2013), 901.  doi: 10.1007/s00285-012-0580-3.  Google Scholar

[45]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, CRC Press, (1985).   Google Scholar

[46]

D. Wodarz, Ecological and evolutionary principles in immunology,, Ecology Letters, 9 (2006), 694.  doi: 10.1111/j.1461-0248.2006.00921.x.  Google Scholar

[47]

S. Zhou, Z. Hu, W. Ma and F. Liao, Dynamics analysis of an hiv infection model including infected cells in an eclipse stage,, Journal of Applied Mathematics, (2013).   Google Scholar

show all references

References:
[1]

A. Akram and R. D. Inman, Immunodominance: A pivotal principle in host response to viral infections,, Clinical Immunology, 143 (2012), 99.  doi: 10.1016/j.clim.2012.01.015.  Google Scholar

[2]

C. L. Althaus and R. J. De Boer, Implications of ctl-mediated killing of hiv-infected cells during the non-productive stage of infection,, PLoS One, 6 (2011).  doi: 10.1371/journal.pone.0016468.  Google Scholar

[3]

C. L. Althaus, A. S. De Vos and R. J. De Boer, Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, r0,, Journal of Virology, 83 (2009), 7659.  doi: 10.1128/JVI.01799-08.  Google Scholar

[4]

A. Balamurugan, A. Ali, J. Boucau, S. Le Gall, H. L. Ng and O. O. Yang, Hiv-1 gag cytotoxic t lymphocyte epitopes vary in presentation kinetics relative to hla class i downregulation,, Journal of Virology, 87 (2013), 8726.  doi: 10.1128/JVI.01040-13.  Google Scholar

[5]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in hiv infection dynamics,, Mathematical Biosciences, 183 (2003), 63.  doi: 10.1016/S0025-5564(02)00218-3.  Google Scholar

[6]

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

[7]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999.  doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[8]

R. W. Buckheit III, R. F. Siliciano and J. N. Blankson, Primary cd8+ t cells from elite suppressors effectively eliminate non-productively hiv-1 infected resting and activated cd4+ t cells,, Retrovirology, 10 (2013), 1.   Google Scholar

[9]

B. Buonomo and C. Vargas-De-León, Global stability for an hiv-1 infection model including an eclipse stage of infected cells,, Journal of Mathematical Analysis and Applications, 385 (2012), 709.  doi: 10.1016/j.jmaa.2011.07.006.  Google Scholar

[10]

D. Y. Chen, A. Balamurugan, H. L. Ng, W. G. Cumberland and O. O. Yang, Epitope targeting and viral inoculum are determinants of nef-mediated immune evasion of hiv-1 from cytotoxic t lymphocytes,, Blood, 120 (2012), 100.  doi: 10.1182/blood-2012-02-409870.  Google Scholar

[11]

R. V Culshaw and S. Ruan, A delay-differential equation model of hiv infection of cd4+ t-cells,, Mathematical Biosciences, 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[12]

R. J. De Boer, Which of our modeling predictions are robust,, PLoS Comput. Biol., 8 (2012).  doi: 10.1371/journal.pcbi.1002593.  Google Scholar

[13]

R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections,, SIAM Journal on Applied Mathematics, 73 (2013), 572.  doi: 10.1137/120890351.  Google Scholar

[14]

A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined cauchy problems,, Journal of Mathematical Analysis and applications, 341 (2008), 501.  doi: 10.1016/j.jmaa.2007.09.074.  Google Scholar

[15]

M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate,, Journal of Theoretical Biology, 229 (2004), 281.  doi: 10.1016/j.jtbi.2004.04.015.  Google Scholar

[16]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Analysis, 20 (1989), 388.  doi: 10.1137/0520025.  Google Scholar

[17]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured hiv infection model,, SIAM Journal on Applied Mathematics, 72 (2012), 25.  doi: 10.1137/110826588.  Google Scholar

[18]

W. Kastenmuller, G. Gasteiger, J. H. Gronau, R. Baier, R. Ljapoci, D. H. Busch and I. Drexler, Cross-competition of cd8+ t cells shapes the immunodominance hierarchy during boost vaccination,, The Journal of Experimental Medicine, 204 (2007), 2187.  doi: 10.1084/jem.20070489.  Google Scholar

[19]

H. N. Kløverpris, R. P. Payne, J. B. Sacha, J. T. Rasaiyaah, F. Chen, M. Takiguchi, O. O. Yang, G. J. Towers, P. Goulder and J. G. Prado, Early antigen presentation of protective hiv-1 kf11gag and kk10gag epitopes from incoming viral particles facilitates rapid recognition of infected cells by specific cd8+ t cells,, Journal of Virology, 87 (2013), 2628.   Google Scholar

[20]

X. Lai and X. Zou, Dynamics of evolutionary competition between budding and lytic viral release strategies,, Mathematical Biosciences and Engineering: MBE, 11 (2014), 1091.  doi: 10.3934/mbe.2014.11.1091.  Google Scholar

[21]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bulletin of Mathematical Biology, 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[22]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109.  doi: 10.1080/00036810903208122.  Google Scholar

[23]

P. Magal, Compact attractors for time periodic age-structured population models,, Electronic Journal of Differential Equations, 2001 (2001), 1.   Google Scholar

[24]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM Journal on Mathematical Analysis, 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar

[25]

P. W. Nelson, M. A. Gilchrist, D. Coombs, J. 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.  doi: 10.3934/mbe.2004.1.267.  Google Scholar

[26]

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

[27]

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

[28]

M. A. Nowak, R. M. May and K. Sigmund, Immune responses against multiple epitopes,, Journal of Theoretical Biology, 175 (1995), 325.  doi: 10.1006/jtbi.1995.0146.  Google Scholar

[29]

R. P. Payne, H. Kløverpris, J. B. Sacha, Z. Brumme, C. Brumme, S. Buus, S. Sims, S. Hickling, L. Riddell, F. Chen, et al, Efficacious early antiviral activity of hiv gag-and pol-specific hla-b* 2705-restricted cd8+ t cells,, Journal of Virology, 84 (2010), 10543.  doi: 10.1128/JVI.00793-10.  Google Scholar

[30]

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

[31]

A. S. Perelson, A. U. Neumann, M. Markowitz, J. 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.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[32]

L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured hiv-1 dynamics with combination antiretroviral therapy,, SIAM Journal on Applied Mathematics, 67 (2007), 731.  doi: 10.1137/060663945.  Google Scholar

[33]

L. Rong, M. A. Gilchrist, Z. Feng and A. S. Perelson, Modeling within-host hiv-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,, Journal of Theoretical Biology, 247 (2007), 804.  doi: 10.1016/j.jtbi.2007.04.014.  Google Scholar

[34]

J. B. Sacha, C. Chung, E. G. Rakasz, S. P. Spencer, A. K. Jonas, A. T. Bean, W. Lee, B. J. Burwitz, J. J. Stephany, J. T. Loffredo, et al, Gag-specific cd8+ t lymphocytes recognize infected cells before aids-virus integration and viral protein expression,, The Journal of Immunology, 178 (2007), 2746.  doi: 10.4049/jimmunol.178.5.2746.  Google Scholar

[35]

H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and ctl immune responses,, SIAM Journal on Applied Mathematics, 73 (2013), 1280.  doi: 10.1137/120896463.  Google Scholar

[36]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, volume 57., Springer Science & Business Media, (2011).  doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[37]

H. L. Smith and P. De Leenheer, Virus dynamics: A global analysis,, SIAM Journal on Applied Mathematics, 63 (2003), 1313.  doi: 10.1137/S0036139902406905.  Google Scholar

[38]

X. Song, S. Wang and J. Dong, Stability properties and hopf bifurcation of a delayed viral infection model with lytic immune response,, Journal of Mathematical Analysis and Applications, 373 (2011), 345.  doi: 10.1016/j.jmaa.2010.04.010.  Google Scholar

[39]

H. R. Thieme, Integrated semigroups and integrated solutions to abstract cauchy problems,, Journal of Mathematical Analysis and Applications, 152 (1990), 416.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar

[40]

H. R. Thieme, Quasi-compact semigroups via bounded perturbation,, Advances in Mathematical Population Dynamics-Molecules, (1997), 691.   Google Scholar

[41]

H. R. Thieme et al, Semiflows generated by lipschitz perturbations of non-densely defined operators,, Differential and Integral Equations, 3 (1990), 1035.   Google Scholar

[42]

B. D. Walker and G. Y. Xu, Unravelling the mechanisms of durable control of hiv-1,, Nature Reviews Immunology, 13 (2013), 487.  doi: 10.1038/nri3478.  Google Scholar

[43]

K. Wang, W. Wang, H. Pang and X. Liu, Complex dynamic behavior in a viral model with delayed immune response,, Physica D: Nonlinear Phenomena, 226 (2007), 197.  doi: 10.1016/j.physd.2006.12.001.  Google Scholar

[44]

Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with ctl immune response incorporating antiretroviral therapy,, Journal of Mathematical Biology, 67 (2013), 901.  doi: 10.1007/s00285-012-0580-3.  Google Scholar

[45]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, CRC Press, (1985).   Google Scholar

[46]

D. Wodarz, Ecological and evolutionary principles in immunology,, Ecology Letters, 9 (2006), 694.  doi: 10.1111/j.1461-0248.2006.00921.x.  Google Scholar

[47]

S. Zhou, Z. Hu, W. Ma and F. Liao, Dynamics analysis of an hiv infection model including infected cells in an eclipse stage,, Journal of Applied Mathematics, (2013).   Google Scholar

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