Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays

Yan Wang; Minmin Lu; Daqing Jiang; Yan Wang; Minmin Lu; Daqing Jiang

doi:10.3934/mbe.2021014

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 274-299. doi: 10.3934/mbe.2021014

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Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays

1.
College of Science, China University of Petroleum (East China), Qingdao 266580, China
2.
Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum (East China), Qingdao 266580, China
3.
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received: 31 August 2020 Accepted: 24 November 2020 Published: 27 November 2020

In this paper, an HIV infection model with latent infection, Beddington-DeAngelis infection function, B-cell immune response and four time delays is formulated. The well-posedness of the model solution is rigorously derived, and the basic reproduction number $\mathcal{R}_0$ and the B-cell immune response reproduction number $\mathcal{R}_1$ are also obtained. By analyzing the modulus of the characteristic equation and constructing suitable Lyapunov functions, we establish the global asymptotic stability of the uninfected and the B-cell-inactivated equilibria for the four time delays, respectively. Hopf bifurcation occurs at the B-cell-activated equilibrium when the model includes the immune delay, and the B-cell-activated equilibrium is globally asymptotically stable if the model does not include it. Numerical simulations indicate that the increase of the latency delay, the cell infection delay and the virus maturation delay can cause the B-cell-activated equilibrium stabilize, while the increase of the immune delay can cause it destabilize.
- latent infection,
- B-cell immune response,
- delay,
- Beddington-DeAngelis function,
- stability,
- Hopf bifurcation
Citation: Yan Wang, Minmin Lu, Daqing Jiang. Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 274-299. doi: 10.3934/mbe.2021014

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Abstract

In this paper, an HIV infection model with latent infection, Beddington-DeAngelis infection function, B-cell immune response and four time delays is formulated. The well-posedness of the model solution is rigorously derived, and the basic reproduction number $\mathcal{R}_0$ and the B-cell immune response reproduction number $\mathcal{R}_1$ are also obtained. By analyzing the modulus of the characteristic equation and constructing suitable Lyapunov functions, we establish the global asymptotic stability of the uninfected and the B-cell-inactivated equilibria for the four time delays, respectively. Hopf bifurcation occurs at the B-cell-activated equilibrium when the model includes the immune delay, and the B-cell-activated equilibrium is globally asymptotically stable if the model does not include it. Numerical simulations indicate that the increase of the latency delay, the cell infection delay and the virus maturation delay can cause the B-cell-activated equilibrium stabilize, while the increase of the immune delay can cause it destabilize.

References

[1]	T. W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. A. M. Mican, M. Baseler, et al., Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proc. Natl. Acad. Sci., 94 (1997), 13193-13197. doi: 10.1073/pnas.94.24.13193
[2]	D. Finzi, J. Blankson, J. D. Siliciano, J. B. Margolick, K. Chadwick, T. Pierson, 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
[3]	A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107
[4]	M. A. Nowak, R. M. May, Virus dynamics: mathematical principles of immunology and virology, Oxford University, Oxford, 2000.
[5]	L. Rong, 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
[6]	L. Rong, 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), 1-18.
[7]	J. A. Deans, S. Cohen, Immunology of malaria, Annu. Rev. Microbiol., 37 (1983), 25-49. doi: 10.1146/annurev.mi.37.100183.000325
[8]	X. Song, A. 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
[9]	D. Li, W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl., 335 (2007), 683-691. doi: 10.1016/j.jmaa.2007.02.006
[10]	G. Huang, W. Ma, Y. Takeuchi, Global properties for virus dynamics model with BeddingtonDeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004
[11]	G. Huang, W. Ma, Y. Takeuchi, Global analysis for delay virus dynamics model with BeddingtonDeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007
[12]	H. Miao, Z. Teng, X. Abdurahman, Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington-DeAngelis incidence and three delays, J. Biol. Dynam., 12 (2018), 146-170. doi: 10.1080/17513758.2017.1408861
[13]	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. doi: 10.1137/120896463
[14]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340. doi: 10.2307/3866
[15]	D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298
[16]	A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May, 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
[17]	H. Liu, J. Zhang, Dynamics of two time delays differential equation model to HIV latent infection, Physica A., 514 (2019), 384-395. doi: 10.1016/j.physa.2018.09.087
[18]	Y. Yang, Y. Dong, Y. Takeuchi, Global dynamics of a latent HIV infection model with general incidence function and multiple delays, Discrete Cont. Dyn. B, 24 (2019), 783-800.
[19]	Y. Wang, M. Lu, J. Liu. Global stability of a delayed virus model with latent infection and Beddington-DeAngelis infection function, Appl. Math. Lett., 107 (2020), 106463. doi: 10.1016/j.aml.2020.106463
[20]	Y. Wang, F. Brauer, J. Wu, 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
[21]	X. Wang, S. Tang, X. Song, L. Rong, Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission, J. Biol. Dynam., 11 (2017), 455-483. doi: 10.1080/17513758.2016.1242784
[22]	Y. Wang, Y. Zhou, J. Wu, J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Math. Biosci., 219 (2009), 104-112. doi: 10.1016/j.mbs.2009.03.003
[23]	A. Alshorman, X. Wang, M. J. Meyer, L. Rong, Analysis of HIV models with two time delays, J. Biol. Dynam., 11 (2017), 40-64. doi: 10.1080/17513758.2016.1148202
[24]	J. Lin, R. Xu, X. Tian, Threshold dynamics of an HIV-1 model with both viral and cellular infections, cell-mediated and humoral immune responses, Math. Biosci. Eng., 16 (2019), 292-319. doi: 10.3934/mbe.2019015
[25]	J. Lin, R. Xu, X. Tian, Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, intracellular delay, and humoral immunity, Appl. Math. Comput., 315 (2017), 516-530.
[26]	H. Xiang, L. Feng, H. Huo, Stability of the virus dynamics model with Beddington-DeAngelis functional response and delays, Appl. Math. Model., 37 (2013), 5414-5423. doi: 10.1016/j.apm.2012.10.033
[27]	X. Song, X. Zhou, X. Zhao, Properties of stability and Hopf bifurcation for a HIV infection model with time delay, Appl. Math. Model., 34 (2010), 1511-1523. doi: 10.1016/j.apm.2009.09.006
[28]	B. Li, Y. Chen, X. Lu, 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
[29]	J. Xu, Y. Geng, S. Zhang, Global stability and Hopf bifurcation in a delayed viral infection model with cell-to-cell transmission and humoral immune response, Int. J. Bifurcat. Chaos, 29 (2019) 1950161. doi: 10.1142/S021812741950161X
[30]	J. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, New York, 1993.
[31]	P. van den Driessche, 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
[32]	J. M. Heffernan, R. J. Smith, L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interf., 2 (2005), 281-293. doi: 10.1098/rsif.2005.0042
[33]	C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014
[34]	J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, 1976.
[35]	J. M. Heffernan, L. M. Wahl, Natural variation in HIV infection: monte carlo estimates that include CD8 effector cells. J. Theor. Biol. 243 (2006), 191-204. doi: 10.1016/j.jtbi.2006.05.032
[36]	Y. Wang, Y. Zhou, F. Brauer, J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934. doi: 10.1007/s00285-012-0580-3
[37]	M. A. Nowak, C. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74

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