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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

Suxia Zhang 1, and  Xiaxia Xu 1,

1. 

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

Received  February 2015 Revised  November 2015 Published  March 2016

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: , ().   Google Scholar

[2]

F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335.  Google Scholar

[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), 715-724. doi: 10.1111/j.1365-2893.2006.00741.x.  Google Scholar

[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), 197-201. doi: 10.1098/rspb.1993.0102.  Google Scholar

[5]

A. Franceschetti and A. Pugliese, Threshold behaviour of a SIR epidemic model with age structure and immigration, J. Math. Biol., 57 (2008), 1-27. doi: 10.1007/s00285-007-0143-1.  Google Scholar

[6]

D. Ganem and A. M. Prince, Hepatitis B virus infection-natural history and clinical consequences, N. Engl. J. Med., 350 (2004), 1118-1129. doi: 10.1056/NEJMra031087.  Google Scholar

[7]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[8]

J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.  Google Scholar

[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), 25-38. doi: 10.1137/110826588.  Google Scholar

[10]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Stud., 1 (1988), 49-77. doi: 10.1080/08898488809525260.  Google Scholar

[11]

M. Kane, Global programme for control of hepatitis B infection, Vaccine, 13 (1995), S47-S49. doi: 10.1016/0264-410X(95)80050-N.  Google Scholar

[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), 619-624. doi: 10.1038/87953.  Google Scholar

[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), 63-72. doi: 10.1007/s10096-009-0821-6.  Google Scholar

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

[15]

P. Magal and X. Q. Zhao, Global attractors in uniformlu persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.  Google Scholar

[16]

J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol., 269 (2011), 266-272. doi: 10.1016/j.jtbi.2010.10.028.  Google Scholar

[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), 731-756. doi: 10.1137/060663945.  Google Scholar

[18]

H. R. Thieme, Persistence under relaxed point-dissipativity(with application to an endemic model), SIAM J. Appl. Math., 24 (1993), 407-435. doi: 10.1137/0524026.  Google Scholar

[19]

H. R. Thieme, Semiflows generated by Lipschitz perturbation of non-densely defined operators, Diff. Integr. Equs., 3 (1990), 1035-1066.  Google Scholar

[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), 599-603. doi: 10.1016/j.jtbi.2008.06.022.  Google Scholar

[21]

S. X. Zhang and Y. C. Zhou, The analysis and application of an HBV model, Appl. Math. Model., 36 (2012), 1302-1312. doi: 10.1016/j.apm.2011.07.087.  Google Scholar

[22]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[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), 744-752. doi: 10.1093/ije/29.4.744.  Google Scholar

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

[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), 330-338. doi: 10.1016/j.jtbi.2009.09.035.  Google Scholar

show all references

References:
[1]

, Hepatitis B. World Health Organization Fact Sheet N$^\circ$204, World Health Organization,, 2008. Available from: , ().   Google Scholar

[2]

F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335.  Google Scholar

[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), 715-724. doi: 10.1111/j.1365-2893.2006.00741.x.  Google Scholar

[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), 197-201. doi: 10.1098/rspb.1993.0102.  Google Scholar

[5]

A. Franceschetti and A. Pugliese, Threshold behaviour of a SIR epidemic model with age structure and immigration, J. Math. Biol., 57 (2008), 1-27. doi: 10.1007/s00285-007-0143-1.  Google Scholar

[6]

D. Ganem and A. M. Prince, Hepatitis B virus infection-natural history and clinical consequences, N. Engl. J. Med., 350 (2004), 1118-1129. doi: 10.1056/NEJMra031087.  Google Scholar

[7]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[8]

J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.  Google Scholar

[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), 25-38. doi: 10.1137/110826588.  Google Scholar

[10]

H. Inaba, A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Stud., 1 (1988), 49-77. doi: 10.1080/08898488809525260.  Google Scholar

[11]

M. Kane, Global programme for control of hepatitis B infection, Vaccine, 13 (1995), S47-S49. doi: 10.1016/0264-410X(95)80050-N.  Google Scholar

[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), 619-624. doi: 10.1038/87953.  Google Scholar

[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), 63-72. doi: 10.1007/s10096-009-0821-6.  Google Scholar

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

[15]

P. Magal and X. Q. Zhao, Global attractors in uniformlu persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.  Google Scholar

[16]

J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol., 269 (2011), 266-272. doi: 10.1016/j.jtbi.2010.10.028.  Google Scholar

[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), 731-756. doi: 10.1137/060663945.  Google Scholar

[18]

H. R. Thieme, Persistence under relaxed point-dissipativity(with application to an endemic model), SIAM J. Appl. Math., 24 (1993), 407-435. doi: 10.1137/0524026.  Google Scholar

[19]

H. R. Thieme, Semiflows generated by Lipschitz perturbation of non-densely defined operators, Diff. Integr. Equs., 3 (1990), 1035-1066.  Google Scholar

[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), 599-603. doi: 10.1016/j.jtbi.2008.06.022.  Google Scholar

[21]

S. X. Zhang and Y. C. Zhou, The analysis and application of an HBV model, Appl. Math. Model., 36 (2012), 1302-1312. doi: 10.1016/j.apm.2011.07.087.  Google Scholar

[22]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[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), 744-752. doi: 10.1093/ije/29.4.744.  Google Scholar

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

[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), 330-338. doi: 10.1016/j.jtbi.2009.09.035.  Google Scholar

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