doi: 10.3934/dcdsb.2021123

Mathematical analysis of an age-structured HIV model with intracellular delay

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

* Corresponding author: Xianlong Fu, Email: xlfu@math.ecnu.edu.cn

Received  November 2020 Revised  February 2021 Published  April 2021

Fund Project: This work is supported by NSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000)

In this paper, we study an age-structured HIV model with intracellular delay, logistic growth and antiretrowviral therapy. We first rewrite the model as an abstract non-densely defined Cauchy problem and obtain the existence of the unique positive steady state. Then through the linearization arguments we investigate the asymptotic behavior of steady states by determining the distribution of eigenvalues. We obtain successfully the globally asymptotic stability for the null equilibrium and (locally) asymptotic stability for the positive equilibrium respectively. Moreover, we also prove that Hopf bifurcations occur around the positive equilibrium under some conditions. In addition, we address the persistence of the semi-flow by showing the existence of a global attractor. Finally, some numerical examples are provided to illustrate the main results.

Citation: Yuan Yuan, Xianlong Fu. Mathematical analysis of an age-structured HIV model with intracellular delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021123
References:
[1]

E. Avila-ValesN. Chan-Ch'lG. E. Garc'la-Almeida and C. Vargas-De-Leon, Stability and Hopf bifurcation in a delayed viral infection model with mitosis transmission, Appl. Math. Comp., 259 (2015), 293-312.  doi: 10.1016/j.amc.2015.02.053.  Google Scholar

[2]

S. ChenJ. Shi and J. Wei, Time delay-induced instabilities and hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1-38.  doi: 10.1007/s00332-012-9138-1.  Google Scholar

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A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, in Infinite Dimensional Dynamical Systems, Springer, New York, 2013,353–390. doi: 10.1007/978-1-4614-4523-4_14.  Google Scholar

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Z. GuoH. Huo and H. Xiang, Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth, J. Biol. Dyn., 13 (2019), 362-384.  doi: 10.1080/17513758.2019.1602171.  Google Scholar

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Mathematical Surveys and Monographs, vol. 25, AMS, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[6] B. HassardD. N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcaton, London Math. Soc. Lect. Note Ser., vol. 41, Cambridge Univ. Press, Cambridge, 1981.   Google Scholar
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A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

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

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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

[10]

M. Y. Li and H. Shu, Joint effects of mitosis and intracellular delay on viral dynamics: Two-parameter bifurcation analysis, J. Math. Biol., 64 (2012), 1005-1020.  doi: 10.1007/s00285-011-0436-2.  Google Scholar

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M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7.  Google Scholar

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P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

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P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0.  Google Scholar

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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.  doi: 10.3934/mbe.2004.1.267.  Google Scholar

[15]

P. W. Nelson and A. 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.  Google Scholar

[16]

M. A. NowakS. BonhoefferG.M. Shaw and R.M. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203-217.  doi: 10.1006/jtbi.1996.0307.  Google Scholar

[17] M. A. Nowak and R. May, Virus Dynamics, Oxford University Press, Oxford, 2000.  doi: 10.1016/S0168-1702(01)00293-3.  Google Scholar
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A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

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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

[20]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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), 1582-1586.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[21]

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

[22]

H. R. Thieme, Convergence results and a Poincar$\acute{e}$-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[23]

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

[24] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.  doi: 10.1515/9780691187655.  Google Scholar
[25]

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

[26]

Y. WangK. Liu and Y. Lou, An age-structrued within-host HIV model with T-cell competition, Nonl. Anal.(RWA), 38 (2017), 1-20.  doi: 10.1016/j.nonrwa.2017.04.002.  Google Scholar

[27]

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

[28]

Y. WangY. ZhouJ. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosc., 219 (2009), 104-112.  doi: 10.1016/j.mbs.2009.03.003.  Google Scholar

[29]

J. XuY. Geng and Y. Zhou, Global dynamics for an age-structured HIV virus infection model with cellular infection and antiretroviral therapy, Appl. Math. Comput., 305 (2017), 62-83.  doi: 10.1016/j.amc.2017.01.064.  Google Scholar

[30]

D. Yan and X. Fu, Analysis of an age-structured HIV infection model with logistic target-cell growth and antiretroviral therapy, IMA J. Appl. Math., 83 (2018), 1037-1065.  doi: 10.1093/imamat/hxy034.  Google Scholar

show all references

References:
[1]

E. Avila-ValesN. Chan-Ch'lG. E. Garc'la-Almeida and C. Vargas-De-Leon, Stability and Hopf bifurcation in a delayed viral infection model with mitosis transmission, Appl. Math. Comp., 259 (2015), 293-312.  doi: 10.1016/j.amc.2015.02.053.  Google Scholar

[2]

S. ChenJ. Shi and J. Wei, Time delay-induced instabilities and hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1-38.  doi: 10.1007/s00332-012-9138-1.  Google Scholar

[3]

A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, in Infinite Dimensional Dynamical Systems, Springer, New York, 2013,353–390. doi: 10.1007/978-1-4614-4523-4_14.  Google Scholar

[4]

Z. GuoH. Huo and H. Xiang, Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth, J. Biol. Dyn., 13 (2019), 362-384.  doi: 10.1080/17513758.2019.1602171.  Google Scholar

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Mathematical Surveys and Monographs, vol. 25, AMS, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[6] B. HassardD. N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcaton, London Math. Soc. Lect. Note Ser., vol. 41, Cambridge Univ. Press, Cambridge, 1981.   Google Scholar
[7]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[8]

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

[9]

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

[10]

M. Y. Li and H. Shu, Joint effects of mitosis and intracellular delay on viral dynamics: Two-parameter bifurcation analysis, J. Math. Biol., 64 (2012), 1005-1020.  doi: 10.1007/s00285-011-0436-2.  Google Scholar

[11]

M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7.  Google Scholar

[12]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[13]

P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0.  Google Scholar

[14]

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.  doi: 10.3934/mbe.2004.1.267.  Google Scholar

[15]

P. W. Nelson and A. 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.  Google Scholar

[16]

M. A. NowakS. BonhoefferG.M. Shaw and R.M. May, Anti-viral drug treatment: Dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203-217.  doi: 10.1006/jtbi.1996.0307.  Google Scholar

[17] M. A. Nowak and R. May, Virus Dynamics, Oxford University Press, Oxford, 2000.  doi: 10.1016/S0168-1702(01)00293-3.  Google Scholar
[18]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[19]

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

[20]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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), 1582-1586.  doi: 10.1126/science.271.5255.1582.  Google Scholar

[21]

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

[22]

H. R. Thieme, Convergence results and a Poincar$\acute{e}$-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267.  Google Scholar

[23]

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

[24] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.  doi: 10.1515/9780691187655.  Google Scholar
[25]

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

[26]

Y. WangK. Liu and Y. Lou, An age-structrued within-host HIV model with T-cell competition, Nonl. Anal.(RWA), 38 (2017), 1-20.  doi: 10.1016/j.nonrwa.2017.04.002.  Google Scholar

[27]

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

[28]

Y. WangY. ZhouJ. Wu and J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosc., 219 (2009), 104-112.  doi: 10.1016/j.mbs.2009.03.003.  Google Scholar

[29]

J. XuY. Geng and Y. Zhou, Global dynamics for an age-structured HIV virus infection model with cellular infection and antiretroviral therapy, Appl. Math. Comput., 305 (2017), 62-83.  doi: 10.1016/j.amc.2017.01.064.  Google Scholar

[30]

D. Yan and X. Fu, Analysis of an age-structured HIV infection model with logistic target-cell growth and antiretroviral therapy, IMA J. Appl. Math., 83 (2018), 1037-1065.  doi: 10.1093/imamat/hxy034.  Google Scholar

$ V_{I} $" represents the infectious viral">Figure 1.  Solutions of the system (3) go to the disease-free steady state, where "T" represents the uninfected T cells, "I" represents the infected T cells, "$ V_{I} $" represents the infectious viral
Figure 2.  The infection is persistent
Figure 3.  Solutions of the system (7) go to the positive steady state when $ \tau_1 = \tau_2 = 0.5 $
Figure 4.  The solutions oscillate periodically for $ \tau_1 = \tau_2 = 1 $
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