American Institute of Mathematical Sciences

February 2019, 24(2): 783-800. doi: 10.3934/dcdsb.2018207

Global dynamics of a latent HIV infection model with general incidence function and multiple delays

 1 School of Science and Technology, Zhejiang International Studies University, Hangzhou 310023, China 2 Faculty of Engineering, Kyushu University, Fukuoka 819-0395, Japan 3 College of Science and Engineering, Aoyama Gakuin University, Kanagawa 252-5258, Japan

* Corresponding author: yangyu@zisu.edu.cn

Received  August 2017 Revised  January 2018 Published  June 2018

Fund Project: The first author was partially supported by National Natural Science Foundation of China (No. 11501519) and the third author was partially supported by the Japan Society Promotion of Science) through the "Grant-in-Aid 26400211"

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

Citation: Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207
References:
 [1] WHO, July 2017, HIV/AIDS. Geneva: World Health Organization, http://www.who.int/mediacentre/factsheets/fs360/en/. [2] A. Alshorman, X. Wang, M. J. Meyer and L. Rong, Analysis of HIV models with two time delays, J. Biol. Dyn., 11 (2017), 40-64. doi: 10.1080/17513758.2016.1148202. [3] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971. [4] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. doi: 10.1006/bulm.2001.0266. [5] S. S. Chen, C. Y. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672. doi: 10.1016/j.jmaa.2016.05.003. [6] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [7] R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214. [8] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [9] N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay, J. Theor. Biol., 226 (2004), 95-109. doi: 10.1016/j.jtbi.2003.09.002. [10] Y. Dong and W. Ma, Global properties for a class of latent HIV infection dynamics model with CTL immune response, Int. J. Wavelets Multiresolut. Inf. Process., 10 (2012), 1250045 (19 pages). doi: 10.1142/S0219691312500452. [11] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209. doi: 10.1007/PL00008847. [12] D. Finzi, J. Blankson and J. D. Siliciano et al., Latent infection of CD4+ T cells provides a mechanism for lifelong persistence of HIV-1, even in patients on effective combination therapy, Nat. Med., 5 (1999), 512–517. doi: 10.1038/8394. [13] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [14] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247. [15] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [16] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [17] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2. [18] 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. [19] B. Li, Y. Chen, X. Lu and S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135-157. doi: 10.3934/mbe.2016.13.135. [20] 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. [21] D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006. [22] M. Markowitz, M. Louie, A. Hurley, E. Sun, M. D. Mascio, A. S. Perelson and D. D. Ho, A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T-cell decay in vivo, J. Virol., 77 (2003), 5037-5038. doi: 10.1128/JVI.77.8.5037-5038.2003. [23] C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. RWA, 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.05.003. [24] C. C. McCluskey, Using Lyapunov functions to construct Lyapunov functionals for delay differential equations, SIAM J. Appl. Dyn. Syst., 14 (2014), 1-24. doi: 10.1137/140971683. [25] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [26] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [27] A. R. McLean and C. J. Bostock, Scrapie infections initiated at varying doses: An analysis of 117 titration experiments, Philos. Trans. R. Soc. Lond. B Biol. Sci., 355 (2000), 1043-1050. doi: 10.1098/rstb.2000.0641. [28] H. Miao, Z. Teng and C. Kang, Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2365-2387. doi: 10.3934/dcdsb.2017121. [29] J. E. Mittler, B. Sulzer, A. U. Neumann and A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5. [30] H. Mohri, S. Bonhoeffer, S. Monard, A. S. Perelson and D. D. Ho, Rapid turnover of T lymphocytes in SIV-infected rhesus macaques, Science, 279 (1998), 1223-1227. doi: 10.1126/science.279.5354.1223. [31] V. Müller, J. F. Vigueras-Gómez 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. [32] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3. [33] P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8. [34] 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. [35] 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. [36] Y. Otani, T. Kajiwara and T. Sasaki, Lyapunov functionals for virus-immune models with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3093-3114. doi: 10.3934/dcdsb.2015.20.3093. [37] Y. Otani, T. Kajiwara and T. Sasaki, Lyapunov functionals for multistrain models with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2011), 507-536. doi: 10.3934/dcdsb.2017025. [38] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [39] 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. [40] A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96-105. doi: 10.1186/1741-7007-11-96. [41] 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. [42] A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen and A. Hurley, et al., Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188–191. doi: 10.1038/387188a0. [43] A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. [44] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. Ser. B, 269 (2002), 271-279. doi: 10.1098/rspb.2001.1816. [45] L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011. [46] 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. [47] 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. [48] R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689. doi: 10.1016/j.amc.2014.06.020. [49] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Socienty, Providence, RI, 1995. [50] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. doi: 10.1016/j.jmaa.2006.06.064. [51] Y. Tian, Y. Bai and P. Yu, Impact of delay on HIV-1 dynamics of fighting a virus with another virus, Math. Biosci. Eng., 11 (2014), 1181-1198. doi: 10.3934/mbe.2014.11.1181. [52] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [53] Y. Wang, F. Brauer, J. Wu and J. M. Heffernan, A delay-dependent model with HIV drug resistance during therapy, J. Math. Anal. Appl., 414 (2014), 514-531. doi: 10.1016/j.jmaa.2013.12.064. [54] L. Wang and M. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells, Math. Biosci., 200 (2006), 44-57. doi: 10.1016/j.mbs.2005.12.026. [55] X. Wang, S. Tang, X. 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. Dyn., 11 (2017), 455-483. doi: 10.1080/17513758.2016.1242784. [56] J. Xu and Y. Zhou, Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay, Math. Biosci. Eng., 13 (2016), 343-367. doi: 10.3934/mbe.2015006. [57] Y. Yang, L. Zou and S. Ruan, Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions, Math. Biosci., 270 (2015), 183-191. doi: 10.1016/j.mbs.2015.05.001.

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References:
 [1] WHO, July 2017, HIV/AIDS. Geneva: World Health Organization, http://www.who.int/mediacentre/factsheets/fs360/en/. [2] A. Alshorman, X. Wang, M. J. Meyer and L. Rong, Analysis of HIV models with two time delays, J. Biol. Dyn., 11 (2017), 40-64. doi: 10.1080/17513758.2016.1148202. [3] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971. [4] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64. doi: 10.1006/bulm.2001.0266. [5] S. S. Chen, C. Y. Cheng and Y. Takeuchi, Stability analysis in delayed within-host viral dynamics with both viral and cellular infections, J. Math. Anal. Appl., 442 (2016), 642-672. doi: 10.1016/j.jmaa.2016.05.003. [6] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7. [7] R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214. [8] P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905. [9] N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: Influence of pharmacokinetics and intracellular delay, J. Theor. Biol., 226 (2004), 95-109. doi: 10.1016/j.jtbi.2003.09.002. [10] Y. Dong and W. Ma, Global properties for a class of latent HIV infection dynamics model with CTL immune response, Int. J. Wavelets Multiresolut. Inf. Process., 10 (2012), 1250045 (19 pages). doi: 10.1142/S0219691312500452. [11] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209. doi: 10.1007/PL00008847. [12] D. Finzi, J. Blankson and J. D. Siliciano et al., Latent infection of CD4+ T cells provides a mechanism for lifelong persistence of HIV-1, even in patients on effective combination therapy, Nat. Med., 5 (1999), 512–517. doi: 10.1038/8394. [13] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [14] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251. doi: 10.1073/pnas.93.14.7247. [15] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [16] G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6. [17] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2. [18] 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. [19] B. Li, Y. Chen, X. Lu and S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135-157. doi: 10.3934/mbe.2016.13.135. [20] 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. [21] D. Li and W. Ma, Asymptotic properties of a HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006. [22] M. Markowitz, M. Louie, A. Hurley, E. Sun, M. D. Mascio, A. S. Perelson and D. D. Ho, A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T-cell decay in vivo, J. Virol., 77 (2003), 5037-5038. doi: 10.1128/JVI.77.8.5037-5038.2003. [23] C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. RWA, 25 (2015), 64-78. doi: 10.1016/j.nonrwa.2015.05.003. [24] C. C. McCluskey, Using Lyapunov functions to construct Lyapunov functionals for delay differential equations, SIAM J. Appl. Dyn. Syst., 14 (2014), 1-24. doi: 10.1137/140971683. [25] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [26] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [27] A. R. McLean and C. J. Bostock, Scrapie infections initiated at varying doses: An analysis of 117 titration experiments, Philos. Trans. R. Soc. Lond. B Biol. Sci., 355 (2000), 1043-1050. doi: 10.1098/rstb.2000.0641. [28] H. Miao, Z. Teng and C. Kang, Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2365-2387. doi: 10.3934/dcdsb.2017121. [29] J. E. Mittler, B. Sulzer, A. U. Neumann and A. S. Perelson, Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152 (1998), 143-163. doi: 10.1016/S0025-5564(98)10027-5. [30] H. Mohri, S. Bonhoeffer, S. Monard, A. S. Perelson and D. D. Ho, Rapid turnover of T lymphocytes in SIV-infected rhesus macaques, Science, 279 (1998), 1223-1227. doi: 10.1126/science.279.5354.1223. [31] V. Müller, J. F. Vigueras-Gómez 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. [32] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215. doi: 10.1016/S0025-5564(99)00055-3. [33] P. Nelson and A. S. Perelson, Mathematica analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8. [34] 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. [35] 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. [36] Y. Otani, T. Kajiwara and T. Sasaki, Lyapunov functionals for virus-immune models with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3093-3114. doi: 10.3934/dcdsb.2015.20.3093. [37] Y. Otani, T. Kajiwara and T. Sasaki, Lyapunov functionals for multistrain models with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2011), 507-536. doi: 10.3934/dcdsb.2017025. [38] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109. doi: 10.1016/j.mbs.2011.11.002. [39] 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. [40] A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biol., 11 (2013), 96-105. doi: 10.1186/1741-7007-11-96. [41] 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. [42] A. S. Perelson, P. Essunger, Y. Cao, M. Vesanen and A. Hurley, et al., Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188–191. doi: 10.1038/387188a0. [43] A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A. [44] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. Ser. B, 269 (2002), 271-279. doi: 10.1098/rspb.2001.1816. [45] L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theor. Biol., 260 (2009), 308-331. doi: 10.1016/j.jtbi.2009.06.011. [46] 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. [47] 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. [48] R. P. Sigdel and C. C. McCluskey, Global stability for an SEI model of infectious disease with immigration, Appl. Math. Comput., 243 (2014), 684-689. doi: 10.1016/j.amc.2014.06.020. [49] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Socienty, Providence, RI, 1995. [50] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. doi: 10.1016/j.jmaa.2006.06.064. [51] Y. Tian, Y. Bai and P. Yu, Impact of delay on HIV-1 dynamics of fighting a virus with another virus, Math. Biosci. Eng., 11 (2014), 1181-1198. doi: 10.3934/mbe.2014.11.1181. [52] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [53] Y. Wang, F. Brauer, J. Wu and J. M. Heffernan, A delay-dependent model with HIV drug resistance during therapy, J. Math. Anal. Appl., 414 (2014), 514-531. doi: 10.1016/j.jmaa.2013.12.064. [54] L. Wang and M. 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The solution converges to the infection-free equilibrium $E_0(10^6, 0, 0, 0)$ when $\mathcal{R}_0 = 0.1781<1$. The parameter values are taken from Data1 in Table 1
The solution converges to the infected equilibrium $E^*(9.0262\times10^5, 68.5219,944.7647, 8.1662\times10^4)$ when $\mathcal{R}_0 = 2.0126>1$. The parameter values are taken from Data2 in Table 1
The function $\mathcal{R}_0(\tau_3)$ with respect to $\tau_3$
Effect of different values of $\tau_3$ on the dynamics of system (12)
The solution converges to $E_0(10^6, 0, 0, 0)$ when $\tau_3$ becomes large. Here we choose $\tau_3 = 80$. The parameter values are taken from Data3 in Table 1
List of parameters
 Parameters Data1 Data2 Data3 Source $s$ (cells ml$^{-1}$ day$^{-1}$) 10$^{4}$ 10$^{4}$ 10$^{4}$ [4] $d_T$ (day$^{-1}$) 0.01 0.01 0.01 [30] $\beta$ (ml virion$^{-1}$ day$^{-1}$) 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ [43] $\alpha_1$ 0.00001 0.00001 0.00001 Assumed $k$ 1.5$\times10^{-6}$ 0.001 0.001 [4,2] $\delta_1$ (day$^{-1}$) 0.05 0.05 0.05 [2] $\delta_L$ (day$^{-1}$) 0.004 0.004 0.004 [4] $\delta_2$ (day$^{-1}$) 0.01 0.01 0.01 [28] $\alpha$ (day$^{-1}$) 0.01 0.01 0.01 [47] $\delta$ (day$^{-1}$) 0.7 1 1 [4,22] $N$ (virions cell$^{-1}$) 100 2000 2000 [4,47] $c$ (day$^{-1}$) 13 23 23 [4,45] $\tau_1$ 0.3 0.3 0.3 [2] $\tau_2$ 0.6 0.6 0.6 [2] $\tau_3$ 0.6 0.6 Assumed [41]
 Parameters Data1 Data2 Data3 Source $s$ (cells ml$^{-1}$ day$^{-1}$) 10$^{4}$ 10$^{4}$ 10$^{4}$ [4] $d_T$ (day$^{-1}$) 0.01 0.01 0.01 [30] $\beta$ (ml virion$^{-1}$ day$^{-1}$) 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ 2.4$\times10^{-8}$ [43] $\alpha_1$ 0.00001 0.00001 0.00001 Assumed $k$ 1.5$\times10^{-6}$ 0.001 0.001 [4,2] $\delta_1$ (day$^{-1}$) 0.05 0.05 0.05 [2] $\delta_L$ (day$^{-1}$) 0.004 0.004 0.004 [4] $\delta_2$ (day$^{-1}$) 0.01 0.01 0.01 [28] $\alpha$ (day$^{-1}$) 0.01 0.01 0.01 [47] $\delta$ (day$^{-1}$) 0.7 1 1 [4,22] $N$ (virions cell$^{-1}$) 100 2000 2000 [4,47] $c$ (day$^{-1}$) 13 23 23 [4,45] $\tau_1$ 0.3 0.3 0.3 [2] $\tau_2$ 0.6 0.6 0.6 [2] $\tau_3$ 0.6 0.6 Assumed [41]
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