Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment

Ning Bai; Rui Xu; Ning Bai; Rui Xu

doi:10.3934/mbe.2021087

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 2: 1689-1707. doi: 10.3934/mbe.2021087

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Research article

Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment

Ning Bai ,
Rui Xu ^,

Complex Systems Research Center, Shanxi University, Taiyuan 030006, China

Received: 30 November 2020 Accepted: 04 February 2021 Published: 05 February 2021

In this paper, an in-host HIV infection model with latent reservoir, delayed CTL immune response and immune impairment is investigated. By using suitable Lyapunov functions and LaSalle's invariance principle, it is shown that when time delay is equal to zero, the immunity-inactivated reproduction ratio is a threshold determining the global dynamics of the model. By means of the persistence theory for infinite dimensional systems, it is proven that if the immunity-inactivated reproduction ratio is greater than unity, the model is permanent. Choosing time delay as the bifurcation parameter and analyzing the corresponding characteristic equation of the linearized system, the existence of a Hopf bifurcation at the immunity-activated equilibrium is established. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics.
- latent reservoir,
- delayed CTL immune response,
- immune impairment,
- Lyapunov functions,
- Hopf bifurcation
Citation: Ning Bai, Rui Xu. Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1689-1707. doi: 10.3934/mbe.2021087

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Abstract

In this paper, an in-host HIV infection model with latent reservoir, delayed CTL immune response and immune impairment is investigated. By using suitable Lyapunov functions and LaSalle's invariance principle, it is shown that when time delay is equal to zero, the immunity-inactivated reproduction ratio is a threshold determining the global dynamics of the model. By means of the persistence theory for infinite dimensional systems, it is proven that if the immunity-inactivated reproduction ratio is greater than unity, the model is permanent. Choosing time delay as the bifurcation parameter and analyzing the corresponding characteristic equation of the linearized system, the existence of a Hopf bifurcation at the immunity-activated equilibrium is established. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics.

References

[1]	F. J. Palella, K. M. Delaney, A. C. Moorman, M. O. Loveless, J. Fuhrer, G. A. Satten, et al., Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection, N. Engl. J. Med., 338 (1998), 853–860. doi: 10.1056/NEJM199803263381301
[2]	E. L. Murphy, A. C. Collier, L. A. Kalish, S. F. Assmannet, M. F. Para, T. P. Flanigan, et al., Highly active antiretroviral therapy decreases mortality and morbidity in patients with advanced HIV disease, Ann. Int. Med., 135 (2001), 17–26. doi: 10.7326/0003-4819-135-1-200107030-00005
[3]	G. Dornadula, H. Zhang, B. VanUitert, J. Stern, L. Livornese Jr, M. J. Ingerman, et al., Residual HIV-1 RNA in blood plasma of patients taking suppressive highly active antiretroviral therapy, J. Am. Med. Assoc., 282 (1999), 1627–1632. doi: 10.1001/jama.282.17.1627
[4]	T. W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. M. Mican, M. Baseler, et al., Presence of an inducible HIV-1 latent reservoir during highly active antiretroviral therapy, Proc. Nat. Acad. Sci., 94 (1997), 13193–13197. doi: 10.1073/pnas.94.24.13193
[5]	T. W. Chun, D. C. Nickle, J. S. Justement, D. Large, A. Semerjian, M. E. Curlin, et al., HIV-infected individuals receiving effective antiviral therapy for extended periods of time continually replenish their viral reservoir, J. Clin. Invest., 115 (2005), 3250–3255. doi: 10.1172/JCI26197
[6]	T. W. Chun, D. C. Nickle, J. S. Justement, J. H. Meyers, G. Roby, C. W. Hallahan, et al., Persistence of HIV in gut-associated lymphoid tissue despite long-term antiretroviral therapy, J. Infect. Dis., 197 (2008), 714–720. doi: 10.1086/527324
[7]	S. Palmer, A. P. Wiegand, F. Maldarelli, H. Bazmi, J. M. Mican, M. Polis, et al., New real-time reverse transcriptase-initiated PCR assay with single-copy sensitivity for human immunodeficiency virus type 1 RNA in plasma, J. Clin. Microbiol., 41 (2003), 4531–4536. doi: 10.1128/JCM.41.10.4531-4536.2003
[8]	T. W. Chun, D. Engel, S. B. Mizell, L. A. Ehler, A. S. Fauci, Induction of HIV-1 replication in latently infected CD4$^+$ T cells using a combination of cytokines, J. Exp. Med., 188 (1998), 83–91. doi: 10.1084/jem.188.1.83
[9]	A. S. Perelson, P. Essunger, Y. Z. Cao, M. Vesanen, A. Hurley, K. Saksela, et al., Decay characteristics of HIV-1-infected compartments during combination therapy, Nature, 387 (1997), 188–191. doi: 10.1038/387188a0
[10]	V. Müller, J. F. Vigueras-Gómez, S. Bonhoeffer, Decelerating decay of latently infected cells during prolonged therapy for human immunodeficiency virus type 1 infection, J. Virol., 76 (2002), 8963–8965. doi: 10.1128/JVI.76.17.8963-8965.2002
[11]	H. Kim, A. S. Perelson, Viral and latent reservoir persistence in HIV-1-infected patients on therapy, PLoS Comput. Biol., 2 (2006), e135. doi: 10.1371/journal.pcbi.0020135
[12]	L. B. 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), e1000533. doi: 10.1371/journal.pcbi.1000533
[13]	L. B. Rong, 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
[14]	M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74–79. doi: 10.1126/science.272.5258.74
[15]	J. H. Cao, J. Mcnevin, S. Holte, L. Fink, L. Corey, M. J. McElrath, Comprehensive analysis of human immunodeficiency virus type 1 (HIV-1)-specific gamma interferon-secreting CD8$^+$ T cells in primary HIV-1 infection, J. Virol., 77 (2003), 6867–6878. doi: 10.1128/JVI.77.12.6867-6878.2003
[16]	A. A. Canabarro, I. M. Gléria, M. L. Lyra, Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Physica A, 342 (2004), 234–241. doi: 10.1016/j.physa.2004.04.083
[17]	K. F. Wang, W. D. Wang, H. Y. Pang, X. Liu, Complex dynamic behavior in a viral model with delayed immune response, Physica D, 226 (2007), 197–208. doi: 10.1016/j.physd.2006.12.001
[18]	X. H. Tian, R. Xu, Global stability and Hopf bifurcation of an HIV-1 infection model with saturation incidence and delayed CTL immune response, Appl. Math. Comput., 237 (2014), 146–154.
[19]	N. L. Komarova, E. Barnes, P. Klenerman, D. Wodarz, Boosting immunity by antiviral drug therapy: a simple relationship among timing, efficacy, and success, Proc. Nat. Acad. Sci., 100 (2003), 1855–1860. doi: 10.1073/pnas.0337483100
[20]	A. Folgori, E. Spada, M. Pezzanera, L. Ruggeri, A. Mele, A. R. Garbuglia, et al., Early impairment of hepatitis C virus specific T cell proliferation during acute infection leads to failure of viral clearance, Gut, 55 (2006), 1012–1019. doi: 10.1136/gut.2005.080077
[21]	Y. Kuroda, H. Takashima, Impairment of cell-mediated immune responses in HTLV-I-associated myelopathy, J. Neurol. Sci., 100 (1990), 211–216. doi: 10.1016/0022-510X(90)90035-L
[22]	R. R. Regoes, D. Wodarz, M. A. Nowak, Virus dynamics: the effect of target cell limitation and immune responses on virus evolution, J. Theor. Biol., 191 (1998), 451–462. doi: 10.1006/jtbi.1997.0617
[23]	Z. P. Wang, X. N. Liu, A chronic viral infection model with immune impairment, J. Theor. Biol., 249 (2007), 532–542. doi: 10.1016/j.jtbi.2007.08.017
[24]	S. L. Wang, X. Y. Song, Z. H. Ge, Dynamics analysis of a delayed viral infection model with immune impairment, Appl. Math. Model., 35 (2011), 4877–4885. doi: 10.1016/j.apm.2011.03.043
[25]	D. Wodarz, J. P. Christensen, A. R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trends Immunol., 23 (2002), 194–200. doi: 10.1016/S1471-4906(02)02189-0
[26]	J. Hale, L. Verduyn, Introduction to Functional Differential Equations, Springer, New York, 1993.
[27]	O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382.
[28]	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
[29]	J. Hale, Theory of Functional Differential Equations, Springer, New York, 1976.
[30]	J. Hale, P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388–395. doi: 10.1137/0520025
[31]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
[32]	X. Yan, W. Li, Stability and bifurcation in a simplified four-neuron BAM neural network with miltiple delays, Discrete Dyn. Nat. Soc., 2006 (2006), 1–29.
[33]	C. Yan, W. D. Wang, Modeling HIV Dynamics Under Combination Therapy with Inducers and Antibodies, Math. Biosci., 81 (2019), 2625–2648.
[34]	W. Hübner, G. P. McNerney, P. Chen, B. M. Dale, R. E. Gordon, F. Y. S. Chuang, et al., Quantitative 3D video microscopy of HIV transfer across T cell virological synapses, Science, 323 (2009), 1743–1747. doi: 10.1126/science.1167525
[35]	K. M. Law, N. L. Komarova, A. W. Yewdall, R. K. Lee, O. L. Herrera, et al., In vivo HIV-1 cell-to-cell transmission promotes multicopy micro-compartmentalized infection, Cell Rep., 15 (2016), 2771–2783. doi: 10.1016/j.celrep.2016.05.059

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