Global dynamics of a delayed diffusive virus infection model with cell-mediated immunity and cell-to-cell transmission

Chunyang Qin; Yuming Chen; Xia Wang; Chunyang Qin; Yuming Chen; Xia Wang

doi:10.3934/mbe.2020257

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 4678-4705. doi: 10.3934/mbe.2020257

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Global dynamics of a delayed diffusive virus infection model with cell-mediated immunity and cell-to-cell transmission

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

Received: 09 May 2020 Accepted: 03 July 2020 Published: 08 July 2020

In this paper, we propose and analyze a delayed diffusive viral dynamic model incorporating cell-mediated immunity and both cell-free and cell-to-cell transmission. After discussing the well-posedness, we provide some preliminary results on solutions. Then we study the existence and uniqueness of homogeneous steady states, which turned out to be completely determined by the basic reproduction number of infection R₀ and the basic reproduction number of immunity R₁. Note that when R₁ is defined, it is necessary that R₀ > 1. The main result is a threefold dynamics. Roughly speaking, when R₀ < 1 the infection-free steady state is globally asymptotically stable; when R₁ ≤ 1 < R₀ the immunity-free infected steady state is globally asymptotically stable; when R₁ > 1 the infected-immune steady state is globally asymptotically stable. The approaches are linearization technique and the Lyapunov functional method. The theoretical results are also illustrated with numerical simulations.
- cell-mediated immunity,
- cell-to-cell transmission,
- spatial heterogeneity,
- delay,
- global stability
Citation: Chunyang Qin, Yuming Chen, Xia Wang. Global dynamics of a delayed diffusive virus infection model with cell-mediated immunity and cell-to-cell transmission[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 4678-4705. doi: 10.3934/mbe.2020257

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Abstract

In this paper, we propose and analyze a delayed diffusive viral dynamic model incorporating cell-mediated immunity and both cell-free and cell-to-cell transmission. After discussing the well-posedness, we provide some preliminary results on solutions. Then we study the existence and uniqueness of homogeneous steady states, which turned out to be completely determined by the basic reproduction number of infection R₀ and the basic reproduction number of immunity R₁. Note that when R₁ is defined, it is necessary that R₀ > 1. The main result is a threefold dynamics. Roughly speaking, when R₀ < 1 the infection-free steady state is globally asymptotically stable; when R₁ ≤ 1 < R₀ the immunity-free infected steady state is globally asymptotically stable; when R₁ > 1 the infected-immune steady state is globally asymptotically stable. The approaches are linearization technique and the Lyapunov functional method. The theoretical results are also illustrated with numerical simulations.

References

[1]	M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas, H. McDade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.
[2]	M. A. Nowak, R. M. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University Press, 2000.
[3]	J. Guedj, A. U. Neumann, Understanding hepatitis C viral dynamics with direct-acting antiviral agents due to the interplay between intracellular replication and cellular infection dynamics, J. Theoret. Biol., 267 (2010), 330-340.
[4]	L. Rong, A. S. Pereslon, Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents, Math. Biosci., 245 (2013), 22-30.
[5]	R. V. Culshaw, S. Ruan, G. Webb, A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, J. Math. Biol., 46 (2003), 425-444.
[6]	P. K. Srivastava, M. Banerjee, P. Chandra, A primary infection model for HIV and immune response with two discrete time delays, Differ. Equ. Dyn. Syst., 18 (2010), 385-399.
[7]	X. Wang, X. Song, S. Tang, L. Rong, Dynamics of an HIV model with multiple infection stages and treatment with different drug classes, Bull. Math. Biol., 78 (2016), 322-349.
[8]	X. Wang, G. Mink, D. Lin, X. Song, L. Rong, Influence of raltegravir intensification on viral load and 2-LTR dynamics in HIV patients on suppressive antiretroviral therapy, J. Theoret. Biol., 416 (2017), 16-27.
[9]	M. Y. Li, H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448.
[10]	M. Y. Li, H. Shu, Global dynamics of a mathematical model for HTLV-I infection of CD4⁺ T cells with delayed CTL response, Nonlinear Anal. Real World Appl., 13 (2012), 1080-1092.
[11]	H. Zhao, S. Liu, A mathematical model of HTLV-I infection with nonlinear incidence and two time delays, Commun. Math. Biol. Neurosci., 2016 (2016).
[12]	M. A. Nowak, C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.
[13]	K. Allali, S. Harroudi, 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.
[14]	X. Wang, Y. Tao, Lyapunov function and global properties of virus dynamics with CTL immune response, Int. J. Biomath., 1 (2008), 443-448.
[15]	J. Li, K. Men, Y. Yang, D. Li, Dynamical analysis on a chronic hepatitis C virus infection model with immune response, J. Theoret. Biol., 365 (2015), 337-346.
[16]	A. M. Elaiw, A. A. Raezah, K. Hattaf, Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response, Int. J. Biomath., 10 (2017), 1750070.
[17]	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.
[18]	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.
[19]	Y. Liu, 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.
[20]	H. Zhu, 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.
[21]	X. Wang, S. Liu, A class of delayed viral models with saturation infection rate and immune response, Math. Methods Appl. Sci., 36 (2013), 125-142.
[22]	B. Li, Y. Chen, X. Lu, S. Liu, A delayed HIV-1 model with virus waning term, Math. Biosci. Eng., 13 (2016), 135-157.
[23]	O. T. Fackler, T. T. Murooka, A. Imle, T. R. Mempel, Adding new dimensions: towards an integrative understanding of HIV-1 spread, Nat. Rev. Microbiol., 12 (2014), 563-574.
[24]	K. Wang, W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.
[25]	R. Xu, Z. Ma, An HBV model with diffusion and time delay, J. Theoret. Biol., 257 (2009), 499-509.
[26]	C. C. McCluskey, Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78.
[27]	C. M. Brauner, D. Jolly, L. Lorenzi, R. Thiebaut, Heterogeneous viral environment in an HIV spatial model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 545-572.
[28]	Y. Zhang, Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Anal. Real World Appl., 15 (2014), 118-139.
[29]	Y. Yang, Y. Xu, Global stability of a diffusive and delayed virus dynamics model with BeddingtonDeAngelis incidence function and CTL immune response, Comput. Math. Appl., 71 (2016), 922-930.
[30]	W. Hübner, G. P. McNerney, P. Chen, B. M. Dale1, 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.
[31]	P. Zhong, L. M. Agosto, J. B. Munro, W. Mothes, Cell-to-cell transmission of viruses, Curr. Opin. Virol., 3 (2013), 44-50.
[32]	N. Martin, Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion, Curr. Opin. HIV AIDS, 4 (2009), 143-149.
[33]	C. Zhang, S. Zhou, E. Groppelli, P. Pellegrino, I. Williams, P. Borrow, Hybrid spreading mechanisms and T cell activation shape the dynamics of HIV-1 infection, PLoS Comput. Biol., 11 (2015), e1004179.
[34]	A. Sigal, J. T. Kim, A. B. Balazs, E. Dekel, A. Mayo, R. Milo, et al., Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy, Nature, 477 (2011), 95-98.
[35]	X. Lai, 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.
[36]	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.
[37]	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. Dyn., 11 (2017), 455-483.
[38]	H. Shu, Y. Chen, L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, J. Dyn. Differ. Equ., 30 (2018), 1817-1836.
[39]	A. Debadatta, B. Nandadulal, Analysis and computation of multi-pathways and multi-delays HIV-1 infection model, Appl. Math. Model., 54 (2018), 517-536.
[40]	X. Song, A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.
[41]	X. Wang, Y. Tao, X. Song, Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response, Nonlinear Dynam., 66 (2011), 825-830.
[42]	H. Sun, J. Wang, Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay, Comput. Math. Appl., 77 (2019), 284-301.
[43]	R. H. Martin, H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
[44]	Y. Gao, J. Wang, Threshold dynamics of a delayed nonlocal reaction-diffusion HIV infection model with both cell-free and cell-to-cell transmissions, J. Math. Anal. Appl., 488 (2020), 124047.
[45]	Y. Lou, X. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
[46]	M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, 1984.
[47]	J. Wu, Theory and applications of partial functional differential equations, Springer, New York, 1996.
[48]	J. K. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, Providence, 1988.
[49]	W. Wang, X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.
[50]	J. K. Hale, S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag, 1993.
[51]	J. LaSalle, S. Lefschetz, Stability by Lyapunov's direct method, with applications, in Mathematics in Science and Engineering, Academic Press, 1961.
[52]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
[53]	H. L. Smith, X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.
[54]	P. W. Nelson, J. D. Murray, A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215.
[55]	K. A. Pawelek, S. Liu, F. Pahlevani, 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.
[56]	H. McCallum, N. Barlow, J. Hone, How should pathogen transmission be modelled? Trends Ecol. Evol., 16 (2001), 295-300.

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