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Viral infection dynamics in a spatial heterogeneous environment with cell-free and cell-to-cell transmissions

  • Received: 29 December 2019 Accepted: 19 February 2020 Published: 02 March 2020
  • In this paper, we investigate a diffusive viral infection model in a spatial heterogeneous environment with two types of infection mechanisms and distinct dispersal rates for the susceptible and infected target cells. After establishing well-posedness of the model system, we identify the basic reproduction number R0 and explore the properties of R0 when the dispersal rate for infected target cells varies from zero to infinity. Moreover, we demonstrate that the basic reproduction number is a threshold parameter: the infection and virus will be cleared out if R0 ≤ 1, while if R0 > 1, the infection will persist and the model system admits at least one positive (chronic infection) steady state. For the special case when all model parameters are spatial homogeneous, this chronic infection steady state is unique and globally asymptotically stable.

    Citation: Zongwei Ma, Hongying Shu. Viral infection dynamics in a spatial heterogeneous environment with cell-free and cell-to-cell transmissions[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2569-2591. doi: 10.3934/mbe.2020141

    Related Papers:

  • In this paper, we investigate a diffusive viral infection model in a spatial heterogeneous environment with two types of infection mechanisms and distinct dispersal rates for the susceptible and infected target cells. After establishing well-posedness of the model system, we identify the basic reproduction number R0 and explore the properties of R0 when the dispersal rate for infected target cells varies from zero to infinity. Moreover, we demonstrate that the basic reproduction number is a threshold parameter: the infection and virus will be cleared out if R0 ≤ 1, while if R0 > 1, the infection will persist and the model system admits at least one positive (chronic infection) steady state. For the special case when all model parameters are spatial homogeneous, this chronic infection steady state is unique and globally asymptotically stable.


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    [1] S. Bonhoeffer, R. M. May, G. M. Shaw, M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976.
    [2] 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.
    [3] A. S. Perelson, D. E. Kirschner, R. de Boer, Dynamics of HIV infection of CD4 T cells, Math. Biosci., 114 (1993), 81-125.
    [4] A. S. Perelson, P. W. Nelson, Mathematical analysis of HIV-I dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
    [5] A. S. Perelson, A. U. Neumann, M. Markowitz, M. J. Leonard, D. D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.
    [6] 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.
    [7] H. Shu, L. Wang, J. Watmough, Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model, J. Math. Biol., 68 (2014), 477-503.
    [8] H. Shu, Y. Chen, L. Wang, Impacts of the cell-free and cell-to-cell infection modes on viral dynamics, J. Dyn. Diff. Equat., 30 (2018), 1817-1836.
    [9] N. M. Dixit, M. Markowitz, D. D. Ho, A. S. Perelson, 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.
    [10] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas, Viral dynamics in hepatitis B virusinfection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.
    [11] F. Wang, Y. Huang, X. Zou, Global dynamics of a PDE in-host viral model, Appl. Anal., 93 (2014), 2312-2329.
    [12] Y. Wu, X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differential Equations, 264 (2018), 4989-5024.
    [13] N. Martin, Q. Sattentau, Cell-to-cell HIV-1 spread and its implications for immune evasion, Curr. Opin. HIV AIDS, 4 (2009), 143-149.
    [14] Q. Sattentau, Avoiding the void: cell-to-cell spread of human viruses, Nat. Rev. Microbiol., 6 (2008), 28-41.
    [15] W. Hübner, G. P. McNerney, P. Chen, B. M. Dale, R. E. Gordan, F. Y. S. Chuang, et al., Quantitative 3D video microscopy of HIV transfer across T cell virological synapses, Science, 323 (2009), 1743-1747.
    [16] H. L. Smith, Monotone Dynamical Systems: an introduction to the theory of competitive and cooperative systems, American Mathematical Society, Providence, 1995.
    [17] A. Pazy, Semigroups of linear operators and application to partial differential equations, Springer, Berlin, 1983.
    [18] R. Jr. Martin, H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. AMS, 321 (1990), 1-44.
    [19] C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum, New York, 1992.
    [20] M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Am. Math. Soc., 11 (1984), 1-64.
    [21] M. H. Protter, H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
    [22] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
    [23] W. Wang, X-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.
    [24] I. D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1983.
    [25] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, 1988.
    [26] K.-J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.
    [27] X.-Q. Zhao, Dynamical systems in population biology, Second edition, CMS Books in Mathematics, Springer, Cham, 2017.
    [28] L. J. S. Allen, B. M. Bolker, Y. Lou, A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst., 21 (2008), 1-20.
    [29] H. L. Smith, X-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.
    [30] H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
    [31] P. Magal, X-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.
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