Stability and Hopf bifurcation of an HIV infection model with two time delays

1.
School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China
2.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Academic Editor: Xiang-Sheng Wang

Received: 04 September 2022 Revised: 25 October 2022 Accepted: 30 October 2022 Published: 09 November 2022

This work focuses on an HIV infection model with intracellular delay and immune response delay, in which the former delay refers to the time it takes for healthy cells to become infectious after infection, and the latter delay refers to the time when immune cells are activated and induced by infected cells. By investigating the properties of the associated characteristic equation, we derive sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model. Based on normal form theory and center manifold theorem, the stability and the direction of the Hopf bifurcating periodic solutions are studied. The results reveal that the intracellular delay cannot affect the stability of the immunity-present equilibrium, but the immune response delay can destabilize the stable immunity-present equilibrium through the Hopf bifurcation. Numerical simulations are provided to support the theoretical results.
- HIV infection model,
- intracellular delay,
- immune response delay,
- stability analysis,
- Hopf bifurcation
Citation: Yu Yang, Gang Huang, Yueping Dong. Stability and Hopf bifurcation of an HIV infection model with two time delays[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 1938-1959. doi: 10.3934/mbe.2023089

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Abstract

This work focuses on an HIV infection model with intracellular delay and immune response delay, in which the former delay refers to the time it takes for healthy cells to become infectious after infection, and the latter delay refers to the time when immune cells are activated and induced by infected cells. By investigating the properties of the associated characteristic equation, we derive sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model. Based on normal form theory and center manifold theorem, the stability and the direction of the Hopf bifurcating periodic solutions are studied. The results reveal that the intracellular delay cannot affect the stability of the immunity-present equilibrium, but the immune response delay can destabilize the stable immunity-present equilibrium through the Hopf bifurcation. Numerical simulations are provided to support the theoretical results.

References

[1]	E. A. Hernandez-Vargas, R. H. Middleton, Modeling the three stages in HIV infection, J. Theor. Biol., 320 (2013), 33–40. https://doi.org/10.1016/j.jtbi.2012.11.028 doi: 10.1016/j.jtbi.2012.11.028
[2]	K. A. Lythgoe, L. Pellis, C. Fraser, Is HIV short-sighted? Insights from a multistrain nested model, Evolution, 67 (2013), 2769–2782. https://doi.org/10.1111/evo.12166 doi: 10.1111/evo.12166
[3]	M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79. https://doi.org/10.1126/science.272.5258.74 doi: 10.1126/science.272.5258.74
[4]	M. A. Nowak, R. M. May, K. Sigmund, Immune responses against multiple epitopes, J. Theor. Biol., 175 (1995), 325–353. https://doi.org/10.1006/jtbi.1995.0146 doi: 10.1006/jtbi.1995.0146
[5]	M. A. Nowak, R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000.
[6]	R. A. Arnaout, M. A. Nowak, D. Wodarz, HIV-1 dynamics revisited: biphasic decay by cytotoxic T lymphocyte killing, Proc. Biol. Sci., 267 (2000), 1347–1354. https://doi.org/10.1098/rspb.2000.1149 doi: 10.1098/rspb.2000.1149
[7]	X. Lai, X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth, J. Math. Anal. Appl., 426 (2015), 563–584. https://doi.org/10.1016/j.jmaa.2014.10.086 doi: 10.1016/j.jmaa.2014.10.086
[8]	Y. Yang, L. Zou, 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. https://doi.org/10.1016/j.mbs.2015.05.001 doi: 10.1016/j.mbs.2015.05.001
[9]	A. V. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, M. A. Nowak, Viral dynamics in vivo: limitations on estimations on intracellular delay and virus decay, Proc. Natl. Acad. Sci., 93 (1996), 7247–7251. https://doi.org/10.1073/pnas.93.14.7247 doi: 10.1073/pnas.93.14.7247
[10]	G. Huang, Y. Takeuchi, W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693–2708. https://doi.org/10.1137/090780821 doi: 10.1137/090780821
[11]	Y. Jiang, T. Zhang, Global stability of a cytokine-enhanced viral infection model with nonlinear incidence rate and time delays, Appl. Math. Lett., 132 (2022), 108110. https://doi.org/10.1016/j.aml.2022.108110 doi: 10.1016/j.aml.2022.108110
[12]	J. Wang, H. Shi, L. Xu, L. Zang, Hopf bifurcation and chaos of tumor-Lymphatic model with two time delays, Chaos Solitons Fractals, 157 (2022), 111922. https://doi.org/10.1016/j.chaos.2022.111922 doi: 10.1016/j.chaos.2022.111922
[13]	K. Wang, W. Wang, X. Liu, Global stability in a viral infection model with lytic and nonlytic immune responses, Comput. Math. Appl., 51 (2006), 1593–1610. https://doi.org/10.1016/j.camwa.2005.07.020 doi: 10.1016/j.camwa.2005.07.020
[14]	P. Borrow, A. Tishon, S. Lee, J. Xu, I. S. Grewal, M. B. Oldstone, et al., CD40L-deficient mice show deficits in antiviral immunity and have an impaired memory CD8+ CTL response, J. Exp. Med., 183 (1996), 2129–2142. https://doi.org/10.1084/jem.183.5.2129 doi: 10.1084/jem.183.5.2129
[15]	R. V. Culshaw, S. Ruan, R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545–562. https://doi.org/10.1007/s00285-003-0245-3 doi: 10.1007/s00285-003-0245-3
[16]	A. R. Thomsen, A. Nansen, J. P. Christensen, S. O. Andreasen, O. Marker, CD40 ligand is pivotal to efficient control of virus replication in mice infected with lymphocytic choriomeningitis virus, J. Immunol., 161 (1998), 4583–4590.
[17]	H. Zhu, Y. Luo, M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Comput. Math. Appl., 62 (2011), 3091–3102. https://doi.org/10.1016/j.camwa.2011.08.022 doi: 10.1016/j.camwa.2011.08.022
[18]	G. Huang, Y. Takeuchi, A. Korobeinikov, HIV evolution and progression of the infection to AIDS, J. Theor. Biol., 307 (2012), 149–159. https://doi.org/10.1016/j.jtbi.2012.05.013 doi: 10.1016/j.jtbi.2012.05.013
[19]	A. M. Elaiw, A. A. Raezah, Stability of general virus dynamics models with both cellular and viral infections and delays, Math. Methods Appl. Sci., 40 (2017), 5863–5880. https://doi.org/10.1002/mma.4436 doi: 10.1002/mma.4436
[20]	G. Huang, H. Yokoi, Y. Takeuchi, T. Kajiwara, T. Sasaki, Impact of intracellular delay, immune activation delay andnonlinear incidence on viral dynamics, Japan. J. Indust. Appl. Math., 28 (2011), 383–411. https://doi.org/10.1007/s13160-011-0045-x doi: 10.1007/s13160-011-0045-x
[21]	M. L. Mann Manyombe, J. Mbang, G. Chendjou, Stability and Hopf bifurcation of a CTL-inclusive HIV-1 infection model with both viral and cellular infections, and three delays, Chaos Solitons Fractals, 144 (2021), 110695. https://doi.org/10.1016/j.chaos.2021.110695 doi: 10.1016/j.chaos.2021.110695
[22]	H. Miao, Z. Teng, X. Abdurahman, Stability and Hopf bifurcation for five-dimensional virus infection model with Beddington-DeAngelis incidence and three delays, J. Biol. Dyn., 12 (2018), 146–170. https://doi.org/10.1080/17513758.2017.1408861 doi: 10.1080/17513758.2017.1408861
[23]	H. Miao, Z. Teng, C. Kang, Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays, Discrete Contin. Dyn. Syst. - B, 22 (2017), 2365–2387. https://doi.org/10.3934/dcdsb.2017121 doi: 10.3934/dcdsb.2017121
[24]	H. Shu, L. Wang, J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73 (2013), 1280–1302. https://doi.org/10.1137/120896463 doi: 10.1137/120896463
[25]	J. Wang, C. Qin, Y. Chen, X. Wang, Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays, Math. Biosci. Eng., 16 (2019), 2587–2612. https://doi.org/10.3934/mbe.2019130 doi: 10.3934/mbe.2019130
[26]	J. Xu, 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. https://doi.org/10.3934/mbe.2015006 doi: 10.3934/mbe.2015006
[27]	K. Wang, W. Wang, H. Pang, X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197–208. https://doi.org/10.1016/j.physd.2006.12.001 doi: 10.1016/j.physd.2006.12.001
[28]	J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
[29]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Mathematical Surveys and Monographs, Providence, Rhode Island, 41 (1995). https://doi.org/10.1090/surv/041
[30]	Y. Tian, Y. Yuan, Effect of time delays in an HIV virotherapy model with nonlinear incidence, Proc. Math. Phys. Eng. Sci., 472 (2016), 20150626. https://doi.org/10.1098/rspa.2015.0626 doi: 10.1098/rspa.2015.0626
[31]	S. Ruan, J. Wei, On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J. Math. Appl. Med. Biol., 18 (2001), 41–52.
[32]	M. Y. Li, H. Shu, Multiple stable periodic oscillations in a mathematical model of CTL response to HTLV-I infection, Bull. Math. Biol., 73 (2011), 1774–1793. https://doi.org/10.1007/s11538-010-9591-7 doi: 10.1007/s11538-010-9591-7
[33]	B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
[34]	Q. An, E. Beretta, Y. Kuang, C. Wang, H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differ. Equations, 266 (2019), 7073–7100. https://doi.org/10.1016/j.jde.2018.11.025 doi: 10.1016/j.jde.2018.11.025
[35]	X. Lin, H. Wang, Stability analysis of delay differential equations with two discrete delays, Can. Appl. Math. Q., 20 (2012), 519–533. Available from: https://www.math.ualberta.ca/hwang/TwoDelayCAMQ.pdf.
[36]	P. Wu, Z. He, A. Khan, Dynamical analysis and optimal control of an age-since infection HIV model at individuals and population levels, Appl. Math. Modell., 106 (2022), 325–342. https://doi.org/10.1016/j.apm.2022.02.008 doi: 10.1016/j.apm.2022.02.008
[37]	P. Wu, H. Zhao, Mathematical analysis of an age-structured HIV/AIDS epidemic model with HAART and spatial diffusion, Nonlinear Anal. Real World Appl., 60 (2021), 103289. https://doi.org/10.1016/j.nonrwa.2021.103289 doi: 10.1016/j.nonrwa.2021.103289
[38]	R. Xu, C. Song, Dynamics of an HIV infection model with virus diffusion and latently infected cell activation, Nonlinear Anal. Real World Appl., 67 (2022), 103618. https://doi.org/10.1016/j.nonrwa.2022.103618 doi: 10.1016/j.nonrwa.2022.103618

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