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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1442-1449. doi: 10.3934/mbe.2020074
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Research article

Global stability analysis of a viral infection model in a critical case

  • 1.
    College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
  • 2.
    Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China
  • Received: 05 October 2019 Accepted: 17 November 2019 Published: 28 November 2019

  • Recently, it has been proved that for the diffusive viral infection model with cell-to-cell infection, the virus-free steady state E0 is globally attractive when the basic reproduction number R0 < 1, and the virus is uniformly persistent if R0 > 1. However, the global stability analysis in the critical case of R0 = 1 is not given due to a technical difficulty. For the diffusive viral infection model including a single equation with diffusion term, global stability analysis in the critical case has been performed by constructing Lyapunov functions. Unfortunately, this method is not applicable for two or more equations with diffusion terms, which was left it as an open problem. The present study is devoted to solving this open problem and shows that E0 is globally asymptotically stable when R0 = 1 for three equations with diffusion terms by means of Gronwall's inequality, comparison theorem and the properties of semigroup.

    Citation: Wei Wang, Xiulan Lai. Global stability analysis of a viral infection model in a critical case[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1442-1449. doi: 10.3934/mbe.2020074

    Related Papers:

  • Recently, it has been proved that for the diffusive viral infection model with cell-to-cell infection, the virus-free steady state E0 is globally attractive when the basic reproduction number R0 < 1, and the virus is uniformly persistent if R0 > 1. However, the global stability analysis in the critical case of R0 = 1 is not given due to a technical difficulty. For the diffusive viral infection model including a single equation with diffusion term, global stability analysis in the critical case has been performed by constructing Lyapunov functions. Unfortunately, this method is not applicable for two or more equations with diffusion terms, which was left it as an open problem. The present study is devoted to solving this open problem and shows that E0 is globally asymptotically stable when R0 = 1 for three equations with diffusion terms by means of Gronwall's inequality, comparison theorem and the properties of semigroup.


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    [1] W. Mothes, N. M. Sherer, J. Jin, P. Zhong, Virus cell-to-cell transmission, J. Virol., 17 (2010), 8360-8368.
    [2] S. Iwami, J. Takeuchi, S. Nakaoka, F. Mammano, F. Clavel, H. Inaba, et al., Cell-to-cell infection by HIV contributes over half of virus infection, eLife, 23 (2015), e08150.
    [3] L. Agosto, P. Uchil, W. Mothes, Hiv cell-to-cell transmission: Effects on pathogenesis and antiretroviral therapy, Trends Microbiol., 23 (2015), 289-295.
    [4] K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation, 7 (2019), 21.
    [5] X. Ren, Y. Tian, L. Liu, X. Liu, A reaction-diffusion within-host HIV model with cell-to-cell transmission, J. Math. Biol., 76 (2018), 1831-1872.
    [6] J. Wang, J. Yang, T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564.
    [7] R. Cui, K. Lam, Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Eq., 263 (2017), 2343-2373.
    [8] P. Magal, G. Webb, Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity, 31 (2018), 5589-5614.
    [9] Y. Wu, X. Zou, Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates, J. Differ. Eq., 264 (2018), 4989-5024.
    [10] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
    [11] H.R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
    [12] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, CRC Press, New York, 1985.
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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