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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), 7659-7667. doi: 10.1128/JVI.01799-08.

[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), 1129-1133.

[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), 6971-6976. doi: 10.1073/pnas.94.13.6971.

[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), 455-493. doi: 10.1137/S0036139996312703.

[5]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[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), 288-308. doi: 10.1016/j.camwa.2014.06.002.

[7]

A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete. Contin. Dyn. Syst. B., 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251.

[8]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. R. Soc. Edinb. A., 139 (2009), 459-482. doi: 10.1017/S0308210507000455.

[9]

A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multi-group age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8.

[10]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012.

[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), 200-209. doi: 10.1007/PL00008847.

[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), 3442-3449.

[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), 392-417. doi: 10.1093/imamat/hxq009.

[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), 281-288. doi: 10.1016/j.jtbi.2004.04.015.

[15]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[16]

K. Hattaf and N. Yousfi, Global stability for reaction-diffusion equations in biology, Comp. Math. Appl., 66 (2013), 1488-1497. doi: 10.1016/j.camwa.2013.08.023.

[17]

Health Care Stumbling in RI's Hepatitis Figh, The Jakarta Post, 2011-01-13.

[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), 7247-7251. doi: 10.1073/pnas.93.14.7247.

[19]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, in: Pitman Res. Notes Math. Ser., vol. 247, Longman Scientific & Technical, Harlow, 1991.

[20]

D. Ho and Y. Huang, The HIV-1 vaccine race, Cell, 110 (2002), 135-138. doi: 10.1016/S0092-8674(02)00832-2.

[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), 817-842. doi: 10.1007/s10884-011-9224-3.

[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), 25-38. doi: 10.1137/110826588.

[23]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A., 115 (1927), 700-721.

[24]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: II, Proc. R. Soc. Lond. B., 138 (1932), 55-83.

[25]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: III, Proc. R. Soc. Lond. B., 141 (1933), 94-112.

[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), p286.

[27]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[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), 2434-2448. doi: 10.1137/090779322.

[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), 18-35. doi: 10.1016/j.nonrwa.2015.01.001.

[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), 1109-1140. doi: 10.1080/00036810903208122.

[31]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[32]

M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission, J. Math. Anal. Appl., 408 (2013), 225-246. doi: 10.1016/j.jmaa.2013.05.064.

[33]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A.M.S., 321 (1990), 1-44. doi: 10.2307/2001590.

[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), 64-78. doi: 10.1016/j.nonrwa.2015.05.003.

[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), 267-288. doi: 10.3934/mbe.2004.1.267.

[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), 4398-4402. doi: 10.1073/pnas.93.9.4398.

[37]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-i dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[38]

O. Pornillos, J. E. Garrus and W. I. Sundquist, Mechanisms of enveloped RNA virus budding, Trends Cell Biol., 12 (2002), 569-579. doi: 10.1016/S0962-8924(02)02402-9.

[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), 1509-1530. doi: 10.1137/10079690X.

[40]

R. Redlinger, Existence theorem for semilinear parabolic systems with functionals, Nonlinear Anal., 8 (1984), 667-682. doi: 10.1016/0362-546X(84)90011-7.

[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), 163-168.

[42]

R. M. Ribeiro, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microb. Infect., 4 (2002), 829-835. doi: 10.1016/S1286-4579(02)01603-9.

[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), 731-756. doi: 10.1137/060663945.

[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), 1841-1853. doi: 10.1098/rspa.2001.0789.

[45]

X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. doi: 10.1016/j.jmaa.2006.06.064.

[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), 1447-1479. doi: 10.1137/0153068.

[47]

H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447. doi: 10.1016/0022-247X(90)90074-P.

[48]

K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95. doi: 10.1016/j.mbs.2007.05.004.

[49]

K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36-44. doi: 10.1016/j.jtbi.2007.11.007.

[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), 289-313. doi: 10.1016/j.jmaa.2015.06.040.

[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), 613-616. doi: 10.7326/0003-4819-101-5-613.

[52]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[53]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 449-509. doi: 10.1016/j.jtbi.2009.01.001.

[54]

Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Anal. RWA., 15 (2014), 118-139. doi: 10.1016/j.nonrwa.2013.06.005.

[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), 3121-3139. doi: 10.1137/090777645.

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), 7659-7667. doi: 10.1128/JVI.01799-08.

[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), 1129-1133.

[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), 6971-6976. doi: 10.1073/pnas.94.13.6971.

[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), 455-493. doi: 10.1137/S0036139996312703.

[5]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[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), 288-308. doi: 10.1016/j.camwa.2014.06.002.

[7]

A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete. Contin. Dyn. Syst. B., 7 (2007), 251-273. doi: 10.3934/dcdsb.2007.7.251.

[8]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. R. Soc. Edinb. A., 139 (2009), 459-482. doi: 10.1017/S0308210507000455.

[9]

A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multi-group age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331. doi: 10.1007/s00205-008-0203-8.

[10]

A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012.

[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), 200-209. doi: 10.1007/PL00008847.

[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), 3442-3449.

[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), 392-417. doi: 10.1093/imamat/hxq009.

[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), 281-288. doi: 10.1016/j.jtbi.2004.04.015.

[15]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[16]

K. Hattaf and N. Yousfi, Global stability for reaction-diffusion equations in biology, Comp. Math. Appl., 66 (2013), 1488-1497. doi: 10.1016/j.camwa.2013.08.023.

[17]

Health Care Stumbling in RI's Hepatitis Figh, The Jakarta Post, 2011-01-13.

[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), 7247-7251. doi: 10.1073/pnas.93.14.7247.

[19]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, in: Pitman Res. Notes Math. Ser., vol. 247, Longman Scientific & Technical, Harlow, 1991.

[20]

D. Ho and Y. Huang, The HIV-1 vaccine race, Cell, 110 (2002), 135-138. doi: 10.1016/S0092-8674(02)00832-2.

[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), 817-842. doi: 10.1007/s10884-011-9224-3.

[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), 25-38. doi: 10.1137/110826588.

[23]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A., 115 (1927), 700-721.

[24]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: II, Proc. R. Soc. Lond. B., 138 (1932), 55-83.

[25]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: III, Proc. R. Soc. Lond. B., 141 (1933), 94-112.

[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), p286.

[27]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[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), 2434-2448. doi: 10.1137/090779322.

[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), 18-35. doi: 10.1016/j.nonrwa.2015.01.001.

[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), 1109-1140. doi: 10.1080/00036810903208122.

[31]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[32]

M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission, J. Math. Anal. Appl., 408 (2013), 225-246. doi: 10.1016/j.jmaa.2013.05.064.

[33]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A.M.S., 321 (1990), 1-44. doi: 10.2307/2001590.

[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), 64-78. doi: 10.1016/j.nonrwa.2015.05.003.

[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), 267-288. doi: 10.3934/mbe.2004.1.267.

[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), 4398-4402. doi: 10.1073/pnas.93.9.4398.

[37]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-i dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[38]

O. Pornillos, J. E. Garrus and W. I. Sundquist, Mechanisms of enveloped RNA virus budding, Trends Cell Biol., 12 (2002), 569-579. doi: 10.1016/S0962-8924(02)02402-9.

[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), 1509-1530. doi: 10.1137/10079690X.

[40]

R. Redlinger, Existence theorem for semilinear parabolic systems with functionals, Nonlinear Anal., 8 (1984), 667-682. doi: 10.1016/0362-546X(84)90011-7.

[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), 163-168.

[42]

R. M. Ribeiro, A. Lo and A. S. Perelson, Dynamics of hepatitis B virus infection, Microb. Infect., 4 (2002), 829-835. doi: 10.1016/S1286-4579(02)01603-9.

[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), 731-756. doi: 10.1137/060663945.

[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), 1841-1853. doi: 10.1098/rspa.2001.0789.

[45]

X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297. doi: 10.1016/j.jmaa.2006.06.064.

[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), 1447-1479. doi: 10.1137/0153068.

[47]

H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447. doi: 10.1016/0022-247X(90)90074-P.

[48]

K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95. doi: 10.1016/j.mbs.2007.05.004.

[49]

K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36-44. doi: 10.1016/j.jtbi.2007.11.007.

[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), 289-313. doi: 10.1016/j.jmaa.2015.06.040.

[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), 613-616. doi: 10.7326/0003-4819-101-5-613.

[52]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[53]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 449-509. doi: 10.1016/j.jtbi.2009.01.001.

[54]

Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Anal. RWA., 15 (2014), 118-139. doi: 10.1016/j.nonrwa.2013.06.005.

[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), 3121-3139. doi: 10.1137/090777645.

[1]

A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251

[2]

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