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Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays

1 School of Mathematics and Statistics, Xinyang Normal University, Xinyang, Henan, 464000, P. R. China
2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
3 College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453000, P. R. China

Special Issues: Modeling and Complex Dynamics of Populations

In this paper, we investigate a delayed HIV-1 infection model with immune response. Though a logistic growth is incorporated in the growth of the target cells, our focus is on the effect of delays on the infection dynamics. We first study the existence of steady states, which depends on the basic reproduction number $R_0$. When $R_0\le 1$, there is only the infection-free steady state, which is globally asymptotically stable if $R_0<1 when="" r_0="">1$, besides the unstable infection-free steady state, there is a unique infected steady state. We then study the local stability of the infected steady state and local Hopf bifurcation at it. The theoretical analysis indicates that the dynamics scenario is complicated. For example, there can be three sequences of critical values, stability switches and double Hopf bifurcation can occur. Some of the theoretical results are also illustrated with numerical simulations.
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Keywords HIV-1 infection; Hopf bifurcation; time delay; CTL immune response; stability

Citation: Juan Wang, Chunyang Qin, Yuming Chen, Xia Wang. Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays. Mathematical Biosciences and Engineering, 2019, 16(4): 2587-2612. doi: 10.3934/mbe.2019130


  • 1. A. L. Cunningham, H. Donaghy, A. N. Harman, et al., Manipulation of dendritic cell function by viruses, Curr. Opin. Microbiol., 13 (2010), 524–529.
  • 2. A. S. Perelson and R. M. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), 96.
  • 3. R. M. Riberio and A. S. Perelson, The analysis of HIV dynamics using mathematical modeling, in AIDS and other manisfestation of HIV infection, (Edited by G.P. Wormser), San Diego, Elsevier, (2004), 905–912.
  • 4. P. De Leenheer and H. L. Smith, Virus dynamics: a global analysis, SIAM J. Appl. Math., 63 (2003), 1313–1327.
  • 5. A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Math. Methods Appl. Sci., 36 (2013), 383–394.
  • 6. 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.
  • 7. K. A. Pawelek, S. Liu, F. Pahlevani, et al., A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98–109.
  • 8. A. S. Perelson, D. E. Kirschner and R. de Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81–125.
  • 9. A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3–44.
  • 10. P. K. Srivastava, M. Banerjee and P. Chandra, A primary infection model for HIV and immune response with two discrete time delays, Differ. Equ. Dyn. Syst., 18 (2010), 385–399.
  • 11. P. K. Srivastava and P. Chandra, Hopf bifurcation and periodic solutions in a dynamical model for HIV and immune response, Differ. Equ. Dyn. Syst., 16 (2008), 77–100.
  • 12. H. Wang, R. Xu, Z. Wang, et al., Global dynamics of a class of HIV-1 infection models with latently infected cells, Nonlinear Anal. Model. Control, 20 (2015), 21–37.
  • 13. X. Wang, Y. Lou and X. Song, Age-structured within-host HIV dynamics with multiple target cells, Stud. Appl. Math., 138 (2017), 43–76.
  • 14. X. Wang, G. Mink, D. Lin, et al., Influence of raltegravir intensification on viral load and 2-LTR dynamics in HIV patients on suppressive antiretroviral therapy, J. Theor. Biol., 416 (2017), 16–27.
  • 15. X. Wang, X. Song, S. Tang, et al., Dynamics of an HIV model with multiple infection stages and treatment with different drug classes, Bull. Math. Biol., 78 (2016), 322–349.
  • 16. Y. Wang, Y. Zhou, F. Brauer, et al., Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901–934.
  • 17. N. M. Dixit, M. Markowitz, D. D. Ho, et al., Estimates of intracellular delay and average drug efficacy from viral load data of HIV-infected individuals under antiretroviral therapy, Antivir. Ther., 9 (2004), 237–246.
  • 18. A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, et al., Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proc. Nat. Acad. Sci. USA, 93 (1996), 7247– 7251.
  • 19. K. Allali, S. Harroudi and D. F. M. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Math. Comput. Sci., 12 (2018), 111–127.
  • 20. M. S. Ciupe, B. L. Bivort, D. M. Bortz, et al., Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1–27.
  • 21. R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T cells, ath. Biosci., 165 (2000), 27–39.
  • 22. R. V. Culshaw, S. Ruan and G. Webb, A mathematical model of cell-to-cell HIV-1 that include a time delay, J. Math. Biol., 46 (2003), 425–444.
  • 23. B. Li, Y. Chen, X. Lu, et al., A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135–157.
  • 24. Y. Liu and C. Wu, Global dynamics for an HIV infection model with Crowley-Martin functional response and two distributed delays, J. Syst. Sci. Complex., 31 (2018), 385–395.
  • 25. Y. Wang, Y. Zhou, J. Wu, et al., Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104–112.
  • 26. H. Zhu and X. Zou, Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511–524.
  • 27. N. Tarfulea, A. Blink, E. Nelson, et al., A CTL-inclusive mathematical model for antiretroviral treatment of HIV infection, Int. J. Biomath., 4 (2011), 1–22.
  • 28. 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.
  • 29. J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer- Verlag, New York, 1993.
  • 30. H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188–211.
  • 31. J. LaSalle and S. Lefschetz, Stability by Liapunov's Direct Method, with Applications, Academic Press, New York, 1961.
  • 32. E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144–1165.
  • 33. X. Yan andW. Li, Stability and bifurcation in a simplified four-neural BAM network with multiple delays, Discrete Dyn. Nat. Soc., 2006 (2006), 1–29.
  • 34. M. Y. Li and H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774–1793.
  • 35. M. Y. Li, X. Lin and H. Wang, Global Hopf branches in a delayed model for immune response to HTLV-1 infections: coexistence of multiple limit cycles, Can. Appl. Math. Q., 20 (2012), 39–50.
  • 36. A. Debadatta and B. Nandadulal, Analysis and computation of multi-pathways and multi-delays HIV-1 infection model, Appl. Math. Model., 54 (2018), 517–536.
  • 37. B. M. Adams, H. T. Banks, M. Davidian, et al., HIV dynamics: modeling, data analysis, and optimal treatment protocols, J. Comput. Appl. Math., 184 (2005), 10–49.


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