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Mathematical Biosciences and Engineering

2016, Volume 13, Issue 5: 935-968. doi: 10.3934/mbe.2016024
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Dynamics of a diffusive age-structured HBV model with saturating incidence

  • 1.
    School of Management, University of Shanghai for Science and Technology, Shanghai 200093
  • 2.
    College of Science, Shanghai University for Science and Technology, Shanghai 200093
  • 3.
    Department of Mathematics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038
  • Received: 01 October 2015 Accepted: 29 June 2018 Published: 01 July 2016

  • MSC : Primary: 35C07, 35K57, 35J61; Secondary: 37B25, 92D30.

  • In this paper, we propose and investigate an age-structured hepatitis B virus (HBV) model with saturating incidence and spatial diffusion where the viral contamination process is described by the age-since-infection. We first analyze the well-posedness of the initial-boundary values problem of the model in the bounded domain $\Omega\subset\mathbb{R}^n$ and obtain an explicit formula for the basic reproductive number $R_0$ of the model. Then we investigate the global behavior of the model in terms of $R_0$: if $R_0\leq1$, then the uninfected steady state is globally asymptotically stable, whereas if $R_0>1$, then the infected steady state is globally asymptotically stable. In addition, when $R_0>1$, by constructing a suitable Lyapunov-like functional decreasing along the travelling waves to show their convergence towards two steady states as $t$ tends to $\pm\infty$, we prove the existence of traveling wave solutions. Numerical simulations are provided to illustrate the theoretical results.

    Citation: Xichao Duan, Sanling Yuan, Kaifa Wang. Dynamics of a diffusive age-structured HBV model with saturating incidence[J]. Mathematical Biosciences and Engineering, 2016, 13(5): 935-968. doi: 10.3934/mbe.2016024

    Related Papers:

  • In this paper, we propose and investigate an age-structured hepatitis B virus (HBV) model with saturating incidence and spatial diffusion where the viral contamination process is described by the age-since-infection. We first analyze the well-posedness of the initial-boundary values problem of the model in the bounded domain $\Omega\subset\mathbb{R}^n$ and obtain an explicit formula for the basic reproductive number $R_0$ of the model. Then we investigate the global behavior of the model in terms of $R_0$: if $R_0\leq1$, then the uninfected steady state is globally asymptotically stable, whereas if $R_0>1$, then the infected steady state is globally asymptotically stable. In addition, when $R_0>1$, by constructing a suitable Lyapunov-like functional decreasing along the travelling waves to show their convergence towards two steady states as $t$ tends to $\pm\infty$, we prove the existence of traveling wave solutions. Numerical simulations are provided to illustrate the theoretical results.


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    [1] J. Virol., 83 (2009), 7659-7667.
    [2] Lancet., 2 (1981), 1129-1133.
    [3] Proc. Natl. Acad. Sci. USA., 94 (1997), 6971-6976.
    [4] SIAM J. Appl. Math., 59 (1998), 455-493.
    [5] Springer-Verlag, Berlin, 1985.
    [6] Comp. Math. Appl., 68 (2014), 288-308.
    [7] Discrete. Contin. Dyn. Syst. B., 7 (2007), 251-273.
    [8] Proc. R. Soc. Edinb. A., 139 (2009), 459-482.
    [9] Arch. Ration. Mech. Anal., 195 (2010), 311-331.
    [10] Nonlinearity, 24 (2011), 2891-2911.
    [11] Oecologia 122 (2000), 200-209.
    [12] J. Immunol., 145 (1990), 3442-3449.
    [13] IMA J. Appl. Math., 75 (2010), 392-417.
    [14] J. Theor. Biol., 229 (2004), 281-288.
    [15] Springer-Verlag, New York, 1993.
    [16] Comp. Math. Appl., 66 (2013), 1488-1497.
    [17] 2011-01-13.
    [18] Proc. Natl. Acad. Sci. USA., 93 (1996), 7247-7251.
    [19] in: Pitman Res. Notes Math. Ser., vol. 247, Longman Scientific & Technical, Harlow, 1991.
    [20] Cell, 110 (2002), 135-138.
    [21] J. Dynam. Differential Equations, 23 (2011), 817-842.
    [22] SIAM. J. Appl. Math., 72 (2012), 25-38.
    [23] Proc. R. Soc. Lond. A., 115 (1927), 700-721.
    [24] Proc. R. Soc. Lond. B., 138 (1932), 55-83.
    [25] Proc. R. Soc. Lond. B., 141 (1933), 94-112.
    [26] Central African Republic. BMC Infectious Diseases, 13 (2013), p286.
    [27] Bull. Math. Biol., 72 (2010), 1492-1505.
    [28] SIAM J. Appl. Math., 70 (2010), 2434-2448.
    [29] Nonlinear Anal. RWA., 24 (2015), 18-35.
    [30] Appl. Anal., 89 (2010), 1109-1140.
    [31] SIAM J. Math. Anal., 37 (2005), 251-275.
    [32] J. Math. Anal. Appl., 408 (2013), 225-246.
    [33] Trans. of A.M.S., 321 (1990), 1-44.
    [34] Nonlinear Anal. RWA., 25 (2015), 64-78.
    [35] Math. Biosci. Eng., 1 (2004), 267-288.
    [36] Proc. Natl. Acad. Sci. USA., 93 (1996), 4398-4402.
    [37] SIAM Rev., 41 (1999), 3-44.
    [38] Trends Cell Biol., 12 (2002), 569-579.
    [39] SIAM J. Appl. Math., 71 (2011), 1509-1530.
    [40] Nonlinear Anal., 8 (1984), 667-682.
    [41] AIDS, 21 (2007), 163-168.
    [42] Microb. Infect., 4 (2002), 829-835.
    [43] SIAM J. Appl. Math., 67 (2007), 731-756.
    [44] Proc. R. Soc. Lond. A., 457 (2001), 1841-1853.
    [45] J. Math. Anal. Appl., 329 (2007), 281-297.
    [46] SIAM J. Appl. Math., 53 (1993), 1447-1479.
    [47] J. Math. Anal. Appl., 152 (1990), 416-447.
    [48] Math. Biosci., 210 (2007), 78-95.
    [49] J. Theor. Biol., 253 (2008), 36-44.
    [50] J. Math. Anal. Appl., 432 (2015), 289-313.
    [51] Ann. Intern. Med., 101 (1984), 613-616.
    [52] Springer, New York, 1996.
    [53] J. Theor. Biol., 257 (2009), 449-509.
    [54] Nonlinear Anal. RWA., 15 (2014), 118-139.
    [55] SIAM J. Appl. Math., 70 (2010), 3121-3139.
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