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2016, 13(5): 935-968. doi: 10.3934/mbe.2016024

## Dynamics of a diffusive age-structured HBV model with saturating incidence

 1 School of Management, University of Shanghai for Science and Technology, Shanghai 200093, China 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, China

Received  October 2015 Revised  April 2016 Published  July 2016

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. Mathematical Biosciences & Engineering, 2016, 13 (5) : 935-968. doi: 10.3934/mbe.2016024
##### References:
 [1] C. L. Althaus, A. S. De Vos and R. J. De Boer, Reassessing the Human Immunodeficiency Virus Type 1 life cycle through age-structured modeling: Life span of infected cells, viral generation time, and basic reproductive number, $R_0$,, J. Virol., 83 (2009), 7659. doi: 10.1128/JVI.01799-08. Google Scholar [2] R. P. Beasley, C. C. Lin, K. Y. Wang, F. J. Hsieh, L. Y. Hwang, C. E. Stevens, T. S. Sun and W. Szmuness, Hepatocellular carcinoma and hepatitis B virus,, Lancet., 2 (1981), 1129. Google Scholar [3] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy,, Proc. Natl. Acad. Sci. USA., 94 (1997), 6971. doi: 10.1073/pnas.94.13.6971. Google Scholar [4] J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,, SIAM J. Appl. Math., 59 (1998), 455. doi: 10.1137/S0036139996312703. Google Scholar [5] K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar [6] X. Duan, S. Yuan, Z. Qiu and J. Ma, Global stability of an SVEIR epidemic model with ages of vaccination and latency,, Comp. Math. Appl., 68 (2014), 288. doi: 10.1016/j.camwa.2014.06.002. Google Scholar [7] A. Ducrot, Travelling wave solutions for a scalar age-structured equation,, Discrete. Contin. Dyn. Syst. B., 7 (2007), 251. doi: 10.3934/dcdsb.2007.7.251. Google Scholar [8] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. R. Soc. Edinb. A., 139 (2009), 459. doi: 10.1017/S0308210507000455. Google Scholar [9] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multi-group age-structured epidemic models,, Arch. Ration. Mech. Anal., 195 (2010), 311. doi: 10.1007/s00205-008-0203-8. Google Scholar [10] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies,, Nonlinearity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. Google Scholar [11] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna,, Oecologia 122 (2000), 122 (2000), 200. doi: 10.1007/PL00008847. Google Scholar [12] C. Ferrari, A. Penna, A. Bertoletti, A. Valli, A. D. Antoni, T. Giuberti, A. Cavalli, M. A. Petit and F. Fiaccadori, Cellular immune response to hepatitis B virus encoded antigens in acute and chronic hepatitis B virus infection,, J. Immunol., 145 (1990), 3442. Google Scholar [13] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay,, IMA J. Appl. Math., 75 (2010), 392. doi: 10.1093/imamat/hxq009. Google Scholar [14] M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate,, J. Theor. Biol., 229 (2004), 281. doi: 10.1016/j.jtbi.2004.04.015. Google Scholar [15] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [16] K. Hattaf and N. Yousfi, Global stability for reaction-diffusion equations in biology,, Comp. Math. Appl., 66 (2013), 1488. doi: 10.1016/j.camwa.2013.08.023. Google Scholar [17] Health Care Stumbling in RI's Hepatitis Figh, The Jakarta Post,, 2011-01-13., (): 2011. Google Scholar [18] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay,, Proc. Natl. Acad. Sci. USA., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. Google Scholar [19] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, in: Pitman Res. Notes Math. Ser., (1991). Google Scholar [20] D. Ho and Y. Huang, The HIV-1 vaccine race,, Cell, 110 (2002), 135. doi: 10.1016/S0092-8674(02)00832-2. Google Scholar [21] S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone,, J. Dynam. Differential Equations, 23 (2011), 817. doi: 10.1007/s10884-011-9224-3. Google Scholar [22] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM. J. Appl. Math., 72 (2012), 25. doi: 10.1137/110826588. Google Scholar [23] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. Lond. A., 115 (1927), 700. Google Scholar [24] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: II,, Proc. R. Soc. Lond. B., 138 (1932), 55. Google Scholar [25] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: III,, Proc. R. Soc. Lond. B., 141 (1933), 94. Google Scholar [26] N. P. Komas, U. Vickos, J. M. Hübschen, A. Béré, A. Manirakiza and C. P. Muller et al., Cross-sectional study of hepatitis B virus infection in rural communities,, Central African Republic. BMC Infectious Diseases, 13 (2013). Google Scholar [27] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x. Google Scholar [28] M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434. doi: 10.1137/090779322. Google Scholar [29] L. Liu, J. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse,, Nonlinear Anal. RWA., 24 (2015), 18. doi: 10.1016/j.nonrwa.2015.01.001. Google Scholar [30] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. Google Scholar [31] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. Google Scholar [32] M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission,, J. Math. Anal. Appl., 408 (2013), 225. doi: 10.1016/j.jmaa.2013.05.064. Google Scholar [33] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. of A.M.S., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [34] C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay,, Nonlinear Anal. RWA., 25 (2015), 64. doi: 10.1016/j.nonrwa.2015.05.003. Google Scholar [35] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells,, Math. Biosci. Eng., 1 (2004), 267. doi: 10.3934/mbe.2004.1.267. Google Scholar [36] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme and H. C. Thomas, Viral dynamics in hepatitis B virus infection,, Proc. Natl. Acad. Sci. USA., 93 (1996), 4398. doi: 10.1073/pnas.93.9.4398. Google Scholar [37] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-i dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [38] O. Pornillos, J. E. Garrus and W. I. Sundquist, Mechanisms of enveloped RNA virus budding,, Trends Cell Biol., 12 (2002), 569. doi: 10.1016/S0962-8924(02)02402-9. Google Scholar [39] R. Qesmi, S. Elsaadany, J. M. Heffernan and J. Wu, A hepatitis B and C virus model with age since infection that exhibits backward bifurcation,, SIAM J. Appl. Math., 71 (2011), 1509. doi: 10.1137/10079690X. Google Scholar [40] R. Redlinger, Existence theorem for semilinear parabolic systems with functionals,, Nonlinear Anal., 8 (1984), 667. doi: 10.1016/0362-546X(84)90011-7. Google Scholar [41] C. Reilly, S. Wietgrefe, G. Sedgewick and A. Haase, Determination of simmian immunodeficiency virus production by infected activated and resting cells,, AIDS, 21 (2007), 163. Google Scholar [42] R. M. Ribeiro, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection,, Microb. Infect., 4 (2002), 829. doi: 10.1016/S1286-4579(02)01603-9. Google Scholar [43] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM J. Appl. Math., 67 (2007), 731. doi: 10.1137/060663945. Google Scholar [44] J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure: I. Travelling wavefronts on unbounded domains,, Proc. R. Soc. Lond. A., 457 (2001), 1841. doi: 10.1098/rspa.2001.0789. Google Scholar [45] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281. doi: 10.1016/j.jmaa.2006.06.064. Google Scholar [46] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. Google Scholar [47] H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems,, J. Math. Anal. Appl., 152 (1990), 416. doi: 10.1016/0022-247X(90)90074-P. Google Scholar [48] K. Wang and W. Wang, Propagation of HBV with spatial dependence,, Math. Biosci., 210 (2007), 78. doi: 10.1016/j.mbs.2007.05.004. Google Scholar [49] K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay,, J. Theor. Biol., 253 (2008), 36. doi: 10.1016/j.jtbi.2007.11.007. Google Scholar [50] J. Wang, R. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate,, J. Math. Anal. Appl., 432 (2015), 289. doi: 10.1016/j.jmaa.2015.06.040. Google Scholar [51] J. I. Weissberg, L. L. Andres, C. I. Smith, S. Weick, J. E. Nichols, G. Garcia, W. S. Robinson, T. C. Merigan and P. B. Gregory, Survival in chronic hepatitis B,, Ann. Intern. Med., 101 (1984), 613. doi: 10.7326/0003-4819-101-5-613. Google Scholar [52] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (1996). doi: 10.1007/978-1-4612-4050-1. Google Scholar [53] R. Xu and Z. Ma, An HBV model with diffusion and time delay,, J. Theor. Biol., 257 (2009), 449. doi: 10.1016/j.jtbi.2009.01.001. Google Scholar [54] Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response,, Nonlinear Anal. RWA., 15 (2014), 118. doi: 10.1016/j.nonrwa.2013.06.005. Google Scholar [55] L. Zou, S. Ruan and W. Zhang, An age-structured model for the transmission dynamics of hepatitis B,, SIAM J. Appl. Math., 70 (2010), 3121. doi: 10.1137/090777645. Google Scholar

show all references

##### References:
 [1] C. L. Althaus, A. S. De Vos and R. J. De Boer, Reassessing the Human Immunodeficiency Virus Type 1 life cycle through age-structured modeling: Life span of infected cells, viral generation time, and basic reproductive number, $R_0$,, J. Virol., 83 (2009), 7659. doi: 10.1128/JVI.01799-08. Google Scholar [2] R. P. Beasley, C. C. Lin, K. Y. Wang, F. J. Hsieh, L. Y. Hwang, C. E. Stevens, T. S. Sun and W. Szmuness, Hepatocellular carcinoma and hepatitis B virus,, Lancet., 2 (1981), 1129. Google Scholar [3] S. Bonhoeffer, R. M. May, G. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy,, Proc. Natl. Acad. Sci. USA., 94 (1997), 6971. doi: 10.1073/pnas.94.13.6971. Google Scholar [4] J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,, SIAM J. Appl. Math., 59 (1998), 455. doi: 10.1137/S0036139996312703. Google Scholar [5] K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar [6] X. Duan, S. Yuan, Z. Qiu and J. Ma, Global stability of an SVEIR epidemic model with ages of vaccination and latency,, Comp. Math. Appl., 68 (2014), 288. doi: 10.1016/j.camwa.2014.06.002. Google Scholar [7] A. Ducrot, Travelling wave solutions for a scalar age-structured equation,, Discrete. Contin. Dyn. Syst. B., 7 (2007), 251. doi: 10.3934/dcdsb.2007.7.251. Google Scholar [8] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. R. Soc. Edinb. A., 139 (2009), 459. doi: 10.1017/S0308210507000455. Google Scholar [9] A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multi-group age-structured epidemic models,, Arch. Ration. Mech. Anal., 195 (2010), 311. doi: 10.1007/s00205-008-0203-8. Google Scholar [10] A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies,, Nonlinearity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. Google Scholar [11] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna,, Oecologia 122 (2000), 122 (2000), 200. doi: 10.1007/PL00008847. Google Scholar [12] C. Ferrari, A. Penna, A. Bertoletti, A. Valli, A. D. Antoni, T. Giuberti, A. Cavalli, M. A. Petit and F. Fiaccadori, Cellular immune response to hepatitis B virus encoded antigens in acute and chronic hepatitis B virus infection,, J. Immunol., 145 (1990), 3442. Google Scholar [13] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay,, IMA J. Appl. Math., 75 (2010), 392. doi: 10.1093/imamat/hxq009. Google Scholar [14] M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate,, J. Theor. Biol., 229 (2004), 281. doi: 10.1016/j.jtbi.2004.04.015. Google Scholar [15] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [16] K. Hattaf and N. Yousfi, Global stability for reaction-diffusion equations in biology,, Comp. Math. Appl., 66 (2013), 1488. doi: 10.1016/j.camwa.2013.08.023. Google Scholar [17] Health Care Stumbling in RI's Hepatitis Figh, The Jakarta Post,, 2011-01-13., (): 2011. Google Scholar [18] A. V. M. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay,, Proc. Natl. Acad. Sci. USA., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. Google Scholar [19] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, in: Pitman Res. Notes Math. Ser., (1991). Google Scholar [20] D. Ho and Y. Huang, The HIV-1 vaccine race,, Cell, 110 (2002), 135. doi: 10.1016/S0092-8674(02)00832-2. Google Scholar [21] S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone,, J. Dynam. Differential Equations, 23 (2011), 817. doi: 10.1007/s10884-011-9224-3. Google Scholar [22] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM. J. Appl. Math., 72 (2012), 25. doi: 10.1137/110826588. Google Scholar [23] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. Lond. A., 115 (1927), 700. Google Scholar [24] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: II,, Proc. R. Soc. Lond. B., 138 (1932), 55. Google Scholar [25] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: III,, Proc. R. Soc. Lond. B., 141 (1933), 94. Google Scholar [26] N. P. Komas, U. Vickos, J. M. Hübschen, A. Béré, A. Manirakiza and C. P. Muller et al., Cross-sectional study of hepatitis B virus infection in rural communities,, Central African Republic. BMC Infectious Diseases, 13 (2013). Google Scholar [27] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x. Google Scholar [28] M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434. doi: 10.1137/090779322. Google Scholar [29] L. Liu, J. Wang and X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse,, Nonlinear Anal. RWA., 24 (2015), 18. doi: 10.1016/j.nonrwa.2015.01.001. Google Scholar [30] P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. Google Scholar [31] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. Google Scholar [32] M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission,, J. Math. Anal. Appl., 408 (2013), 225. doi: 10.1016/j.jmaa.2013.05.064. Google Scholar [33] R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, Trans. of A.M.S., 321 (1990), 1. doi: 10.2307/2001590. Google Scholar [34] C. C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay,, Nonlinear Anal. RWA., 25 (2015), 64. doi: 10.1016/j.nonrwa.2015.05.003. Google Scholar [35] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells,, Math. Biosci. Eng., 1 (2004), 267. doi: 10.3934/mbe.2004.1.267. Google Scholar [36] M. A. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme and H. C. Thomas, Viral dynamics in hepatitis B virus infection,, Proc. Natl. Acad. Sci. USA., 93 (1996), 4398. doi: 10.1073/pnas.93.9.4398. Google Scholar [37] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-i dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar [38] O. Pornillos, J. E. Garrus and W. I. Sundquist, Mechanisms of enveloped RNA virus budding,, Trends Cell Biol., 12 (2002), 569. doi: 10.1016/S0962-8924(02)02402-9. Google Scholar [39] R. Qesmi, S. Elsaadany, J. M. Heffernan and J. Wu, A hepatitis B and C virus model with age since infection that exhibits backward bifurcation,, SIAM J. Appl. Math., 71 (2011), 1509. doi: 10.1137/10079690X. Google Scholar [40] R. Redlinger, Existence theorem for semilinear parabolic systems with functionals,, Nonlinear Anal., 8 (1984), 667. doi: 10.1016/0362-546X(84)90011-7. Google Scholar [41] C. Reilly, S. Wietgrefe, G. Sedgewick and A. Haase, Determination of simmian immunodeficiency virus production by infected activated and resting cells,, AIDS, 21 (2007), 163. Google Scholar [42] R. M. Ribeiro, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection,, Microb. Infect., 4 (2002), 829. doi: 10.1016/S1286-4579(02)01603-9. Google Scholar [43] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM J. Appl. Math., 67 (2007), 731. doi: 10.1137/060663945. Google Scholar [44] J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure: I. Travelling wavefronts on unbounded domains,, Proc. R. Soc. Lond. A., 457 (2001), 1841. doi: 10.1098/rspa.2001.0789. Google Scholar [45] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics,, J. Math. Anal. Appl., 329 (2007), 281. doi: 10.1016/j.jmaa.2006.06.064. Google Scholar [46] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. Google Scholar [47] H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems,, J. Math. Anal. Appl., 152 (1990), 416. doi: 10.1016/0022-247X(90)90074-P. Google Scholar [48] K. Wang and W. Wang, Propagation of HBV with spatial dependence,, Math. Biosci., 210 (2007), 78. doi: 10.1016/j.mbs.2007.05.004. Google Scholar [49] K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay,, J. Theor. Biol., 253 (2008), 36. doi: 10.1016/j.jtbi.2007.11.007. Google Scholar [50] J. Wang, R. Zhang and T. 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Zhang, An age-structured model for the transmission dynamics of hepatitis B,, SIAM J. Appl. Math., 70 (2010), 3121. doi: 10.1137/090777645. Google Scholar
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