A mathematical model for hepatitis B with infection-age structure

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June 2016, 21(4): 1329-1346. doi: 10.3934/dcdsb.2016.21.1329

A mathematical model for hepatitis B with infection-age structure

1.	School of Science, Xi'an University of Technology, Xi'an 710048, China, China

Received February 2015 Revised November 2015 Published March 2016

Abstract
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A model with age of infection is formulated to study the possible effects of variable infectivity on HBV transmission dynamics. The stability of equilibria and persistence of the model are analyzed. The results show that if the basic reproductive number $\mathcal{R}_0<1$, then the disease-free equilibrium is globally asymptotically stable. For $\mathcal{R}_0>1$, the disease is uniformly persistent, and a Lyapunov function is used to show that the unique endemic equilibrium is globally stable in a special case.

Keywords: stability, uniform persistence., age of infection, Mathematical model, hepatitis B.

Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 35B35, 37B2.

Citation: Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329

References:

[1]	, Hepatitis B. World Health Organization Fact Sheet N$^\circ$204, World Health Organization,, 2008. Available from: , ().
[2]	F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model,, Math. Biosci. Eng., 10 (2013), 1335. doi: 10.3934/mbe.2013.10.1335.
[3]	D. Candotti, O. Opare-Sem, H. Rezvan, F. Sarkodie and J. P. Allain, Molecular and serological characterization of hepatitis B virus in deferred Ghanaian blood donors with and without elevated alanine aminotransferase,, J. Viral. Hepat., 13 (2006), 715. doi: 10.1111/j.1365-2893.2006.00741.x.
[4]	W. J. Edmunds, G. F. Medley, D. J. Nokes, A. J. Hall and H. C. Whittle, The influence of age on the development of the hepatitis B carrier state,, Proc. R. Soc. Lond. B., 253 (1993), 197. doi: 10.1098/rspb.1993.0102.
[5]	A. Franceschetti and A. Pugliese, Threshold behaviour of a SIR epidemic model with age structure and immigration,, J. Math. Biol., 57 (2008), 1. doi: 10.1007/s00285-007-0143-1.
[6]	D. Ganem and A. M. Prince, Hepatitis B virus infection-natural history and clinical consequences,, N. Engl. J. Med., 350 (2004), 1118. doi: 10.1056/NEJMra031087.
[7]	J. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977).
[8]	J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025.
[9]	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.
[10]	H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Stud., 1 (1988), 49. doi: 10.1080/08898488809525260.
[11]	M. Kane, Global programme for control of hepatitis B infection,, Vaccine, 13 (1995). doi: 10.1016/0264-410X(95)80050-N.
[12]	G. F. Medley, N. A. Lindop, W. J. Edmunds and D. J. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control,, Nat. Med., 7 (2001), 619. doi: 10.1038/87953.
[13]	C. O'Leary, Z. Hong, F. Zhang, M. Dawood, G. Smart, K. Kaita and J. Wu, A Mathematical model to study the effect of hepatitis B virus vaccine and antivirus treatment among the Canadian Inuit population,, Eur. J. Clin. Microbiol. Infect. Dis., 29 (2010), 63. doi: 10.1007/s10096-009-0821-6.
[14]	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.
[15]	P. Magal and X. Q. Zhao, Global attractors in uniformlu persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173.
[16]	J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand,, J. Theor. Biol., 269 (2011), 266. doi: 10.1016/j.jtbi.2010.10.028.
[17]	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.
[18]	H. R. Thieme, Persistence under relaxed point-dissipativity(with application to an endemic model),, SIAM J. Appl. Math., 24 (1993), 407. doi: 10.1137/0524026.
[19]	H. R. Thieme, Semiflows generated by Lipschitz perturbation of non-densely defined operators,, Diff. Integr. Equs., 3 (1990), 1035.
[20]	S. Thornley, C. Bullen and M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy,, J. Theor. Biol., 254 (2008), 599. doi: 10.1016/j.jtbi.2008.06.022.
[21]	S. X. Zhang and Y. C. Zhou, The analysis and application of an HBV model,, Appl. Math. Model., 36 (2012), 1302. doi: 10.1016/j.apm.2011.07.087.
[22]	X. Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1.
[23]	S. Zhao, Z. Xu and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China,, Int. J. epidemiol., 29 (1994), 744. doi: 10.1093/ije/29.4.744.
[24]	L. Zou, S. G. Ruan and W. N. Zhang, An age-structured model for the transmission dynamics of hepatitis B,, SIAM J. Appl. Math., 70 (2010), 3121. doi: 10.1137/090777645.
[25]	L. Zou, W. N. Zhang and S. G. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China,, J. Theor. Biol., 262 (2010), 330. doi: 10.1016/j.jtbi.2009.09.035.

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References:

[1]	, Hepatitis B. World Health Organization Fact Sheet N$^\circ$204, World Health Organization,, 2008. Available from: , ().
[2]	F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model,, Math. Biosci. Eng., 10 (2013), 1335. doi: 10.3934/mbe.2013.10.1335.
[3]	D. Candotti, O. Opare-Sem, H. Rezvan, F. Sarkodie and J. P. Allain, Molecular and serological characterization of hepatitis B virus in deferred Ghanaian blood donors with and without elevated alanine aminotransferase,, J. Viral. Hepat., 13 (2006), 715. doi: 10.1111/j.1365-2893.2006.00741.x.
[4]	W. J. Edmunds, G. F. Medley, D. J. Nokes, A. J. Hall and H. C. Whittle, The influence of age on the development of the hepatitis B carrier state,, Proc. R. Soc. Lond. B., 253 (1993), 197. doi: 10.1098/rspb.1993.0102.
[5]	A. Franceschetti and A. Pugliese, Threshold behaviour of a SIR epidemic model with age structure and immigration,, J. Math. Biol., 57 (2008), 1. doi: 10.1007/s00285-007-0143-1.
[6]	D. Ganem and A. M. Prince, Hepatitis B virus infection-natural history and clinical consequences,, N. Engl. J. Med., 350 (2004), 1118. doi: 10.1056/NEJMra031087.
[7]	J. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977).
[8]	J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025.
[9]	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.
[10]	H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process,, Math. Popul. Stud., 1 (1988), 49. doi: 10.1080/08898488809525260.
[11]	M. Kane, Global programme for control of hepatitis B infection,, Vaccine, 13 (1995). doi: 10.1016/0264-410X(95)80050-N.
[12]	G. F. Medley, N. A. Lindop, W. J. Edmunds and D. J. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control,, Nat. Med., 7 (2001), 619. doi: 10.1038/87953.
[13]	C. O'Leary, Z. Hong, F. Zhang, M. Dawood, G. Smart, K. Kaita and J. Wu, A Mathematical model to study the effect of hepatitis B virus vaccine and antivirus treatment among the Canadian Inuit population,, Eur. J. Clin. Microbiol. Infect. Dis., 29 (2010), 63. doi: 10.1007/s10096-009-0821-6.
[14]	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.
[15]	P. Magal and X. Q. Zhao, Global attractors in uniformlu persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173.
[16]	J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand,, J. Theor. Biol., 269 (2011), 266. doi: 10.1016/j.jtbi.2010.10.028.
[17]	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.
[18]	H. R. Thieme, Persistence under relaxed point-dissipativity(with application to an endemic model),, SIAM J. Appl. Math., 24 (1993), 407. doi: 10.1137/0524026.
[19]	H. R. Thieme, Semiflows generated by Lipschitz perturbation of non-densely defined operators,, Diff. Integr. Equs., 3 (1990), 1035.
[20]	S. Thornley, C. Bullen and M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy,, J. Theor. Biol., 254 (2008), 599. doi: 10.1016/j.jtbi.2008.06.022.
[21]	S. X. Zhang and Y. C. Zhou, The analysis and application of an HBV model,, Appl. Math. Model., 36 (2012), 1302. doi: 10.1016/j.apm.2011.07.087.
[22]	X. Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1.
[23]	S. Zhao, Z. Xu and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China,, Int. J. epidemiol., 29 (1994), 744. doi: 10.1093/ije/29.4.744.
[24]	L. Zou, S. G. Ruan and W. N. Zhang, An age-structured model for the transmission dynamics of hepatitis B,, SIAM J. Appl. Math., 70 (2010), 3121. doi: 10.1137/090777645.
[25]	L. Zou, W. N. Zhang and S. G. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China,, J. Theor. Biol., 262 (2010), 330. doi: 10.1016/j.jtbi.2009.09.035.

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