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April  2022, 27(4): 2077-2106. 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 2022 Early access  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 and Continuous Dynamical Systems - B, 2022, 27 (4) : 2077-2106. doi: 10.3934/dcdsb.2021123
##### References:
 [1] E. Avila-Vales, N. Chan-Ch'l, G. 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. [2] S. Chen, J. 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. [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. [4] Z. Guo, H. 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. [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. [6] B. Hassard, D. 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. [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. [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. [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. [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. [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. [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. [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. [14] 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), 267-288.  doi: 10.3934/mbe.2004.1.267. [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. [16] M. A. Nowak, S. Bonhoeffer, G.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. [17] M. A. Nowak and R. May, Virus Dynamics, Oxford University Press, Oxford, 2000.  doi: 10.1016/S0168-1702(01)00293-3. [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. [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. [20] 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), 1582-1586.  doi: 10.1126/science.271.5255.1582. [21] L. Rong, Z. 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. [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. [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. [24] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.  doi: 10.1515/9780691187655. [25] J. Wang, J. 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. [26] Y. Wang, K. 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. [27] J. Wang, R. 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. [28] Y. Wang, Y. Zhou, J. 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. [29] J. Xu, Y. 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. [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.

show all references

##### References:
 [1] E. Avila-Vales, N. Chan-Ch'l, G. 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. [2] S. Chen, J. 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. [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. [4] Z. Guo, H. 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. [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. [6] B. Hassard, D. 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. [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. [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. [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. [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. [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. [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. [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. [14] 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), 267-288.  doi: 10.3934/mbe.2004.1.267. [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. [16] M. A. Nowak, S. Bonhoeffer, G.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. [17] M. A. Nowak and R. May, Virus Dynamics, Oxford University Press, Oxford, 2000.  doi: 10.1016/S0168-1702(01)00293-3. [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. [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. [20] 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), 1582-1586.  doi: 10.1126/science.271.5255.1582. [21] L. Rong, Z. 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. [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. [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. [24] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.  doi: 10.1515/9780691187655. [25] J. Wang, J. 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. [26] Y. Wang, K. 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. [27] J. Wang, R. 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. [28] Y. Wang, Y. Zhou, J. 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. [29] J. Xu, Y. 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. [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.
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
The infection is persistent
Solutions of the system (7) go to the positive steady state when $\tau_1 = \tau_2 = 0.5$
The solutions oscillate periodically for $\tau_1 = \tau_2 = 1$
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