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Lyapunov functional for virus infection model with diffusion and state-dependent delays

  • Received: 17 November 2018 Accepted: 20 December 2018 Published: 30 January 2019
  • In this paper, a virus dynamics model with di usion, state-dependent delays and a general nonlinear functional response is investigated. At first, the dynamical system is constructed on a nonlinear metric space. Then the stability of the interior equilibrium is established by using a novel Lyapunov functional. Further, the proposed algorithm has been extended to the model with logistic growth rate.

    Citation: Jitai Liang, Junjie Wei. Lyapunov functional for virus infection model with diffusion and state-dependent delays[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 947-966. doi: 10.3934/mbe.2019044

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  • In this paper, a virus dynamics model with di usion, state-dependent delays and a general nonlinear functional response is investigated. At first, the dynamical system is constructed on a nonlinear metric space. Then the stability of the interior equilibrium is established by using a novel Lyapunov functional. Further, the proposed algorithm has been extended to the model with logistic growth rate.


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    [1] V. Doceul, M. Hollinshead, L. V. D. Linden and G. L. Smith, Repulsion of superinfecting virions: A mechanism for rapid virus spread, Science, 327 (2010): 873–876.
    [2] A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington–Deangelis functional response, Math. Method. Appl. Sci., 36 (2013): 383–394.
    [3] S. Guo, W. Ma and X. Q. Zhao, Global dynamics of a time-delayed Microorganism Flocculation model with saturated functional responses, J. Dyn. Differ. Equ., 30 (2018): 1247–1271.
    [4] K. Hattaf and N. Yousfi, A generalized HBV model with diffusion and two delays, Comput. Math. Appl., 69 (2015): 31–40.
    [5] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981.
    [6] G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington–Deangelis functional response, Appl. Math. Lett., 24 (2011): 1199–1203.
    [7] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010): 2693–2708.
    [8] T. Krisztin and A. Rezounenko, Parabolic partial differential equations with discrete statedependent delay: classical solutions and solution manifold, J. Differ. Equ., 260 (2016): 4454– 4472.
    [9] X. Lai and X. Zou, Repulsion effect on superinfecting virions by infected cells, B. Math. Biol., 76 (2014): 2806–2833.
    [10] Y. Lv, R. Yuan and Y. Pei, Smoothness of semiflows for parabolic partial differential equations with state-dependent delay, J. Differ. Equ., 260 (2016): 6201–6231.
    [11] J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functionaldi fferential equations with multiple state-dependent time lags, Topol. Method. Nonl. An., 3 (1994): 101–162.
    [12] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, T. Am. Math. Soc., 321 (1990): 1–44.
    [13] C. C. Mccluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real, 25 (2015): 64–78.
    [14] H. Miao, X. Abdurahman, Z. Teng and L. Zhang, Dynamical analysis of a delayed reactiondi ffusion virus infection model with logistic growth and humoral immune impairment, Chaos Soliton. Fract., 110 (2018): 280–291.
    [15] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, Berlin, 1983.
    [16] A. Rezounenko, Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses, Discrete Cont. Dyn. S. -B, 22 (2017): 1547–1563.
    [17] A. Rezounenko, Viral infection model with diffusion and state-dependent delay: stability of classical solutions, preprint, 2017. arXiv:1706.08620.
    [18] A. Rezounenko and P. Zagalak, Non-local PDEs with discrete state-dependent delays: wellposedness in a metric space, Discrete Cont. Dyn. S., 33 (2013): 819–835.
    [19] Y. Tian and X. Liu, Global dynamics of a virus dynamical model with general incidence rate and cure rate, Nonlinear Anal. Real, 16 (2014): 17–26.
    [20] J. Wang, J. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-tocell transmission, J. Math. Anal. Appl., 444 (2016): 1542–1564.
    [21] K.Wang andW.Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007): 78–95.
    [22] S.Wang, J. Zhang, F. Xu and X. Song, Dynamics of virus infection models with density-dependent diffusion, Comput. Math. Appl., 74 (2017): 2403–2422.
    [23] W.Wang andW. Ma, Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections, Discrete Cont. Dyn. S.-B, 23 (2018): 3213– 3235.
    [24] W. Wang, W. Ma and X. Lai, Repulsion effect on superinfecting virions by infected cells for virus infection dynamic model with absorption effect and chemotaxis, Nonlinear Anal. Real, 33 (2017): 253–283.
    [25] W. Wang and T. Zhang, Caspase-1-mediated pyroptosis of the predominance for driving CD4+ T cells death: A nonlocal spatial mathematical model, B Math. Biol., 80 (2018): 540–582.
    [26] Z. Wang, Qualitative analysis for a predator-prey system with nonlinear saturated functional response. J. Biomath., 22 (2007): 215–218.
    [27] J. Wu, Theory and applications of partial functional differential equations, Springer, New York, 1996.
    [28] W. Xia, S. Tang, X. Song and 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.
    [29] Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington–Deangelis response, Nonlinear Anal. Real, 15 (2014): 118–139.
    [30] X. Zhou and J. Cui, Global stability of the viral dynamics with Crowley-Martin functional response, B. Korean Math. Soc., 48 (2011): 555–574.
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