# American Institute of Mathematical Sciences

December  2018, 15(6): 1291-1313. doi: 10.3934/mbe.2018060

## Dynamical analysis for a hepatitis B transmission model with immigration and infection age

 1 School of Science, Xi'an University of Technology, Xi'an 710048, China 2 Department of Mathematics and Statistics, The University of Ottawa, 585 King Edward Ave, Ottawa, ON K1N 6N5, Canada 3 Department of Mathematics and Faculty of Medicine, The University of Ottawa, 585 King Edward Ave, Ottawa, ON K1N 6N5, Canada

* Corresponding author: Robert Smith?

Received  July 14, 2017 Accepted  July 11, 2018 Published  September 2018

Fund Project: SZ was supported by the National Science Foundation of China (Grant numbers 11501443, 11571275 and 11701445). RS? is supported by an NSERC Discovery Grant

Hepatitis B virus (HBV) is responsible for an estimated 378 million infections worldwide and 620, 000 deaths annually. Safe and effective vaccination programs have been available for decades, but coverage is limited due to economic and social factors. We investigate the effect of immigration and infection age on HBV transmission dynamics, incorporating age-dependent immigration flow and vertical transmission. The mathematical model can be used to describe HBV transmission in highly endemic regions with vertical transmission and migration of infected HBV individuals. Due to the effects of immigration, there is no disease-free equilibrium or reproduction number. We show that the unique endemic equilibrium exists only when immigration into the infective class is measurable. The smoothness and attractiveness of the solution semiflow are analyzed, and boundedness and uniform persistence are determined. Global stability of the unique endemic equilibrium is shown by a Lyapunov functional for a special case.

Citation: Suxia Zhang, Hongbin Guo, Robert Smith?. Dynamical analysis for a hepatitis B transmission model with immigration and infection age. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1291-1313. doi: 10.3934/mbe.2018060
##### References:
 [1] World Health Organization, 2008, Hepatitis B. World Health Organization Fact Sheet N°204, Available from http://www.who.int/mediacentre/factsheets/fs204/en/index.html [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. [3] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [4] D. Candotti, O. Opare-Sem and H. Rezvan, 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. [5] P. Dény and F. Zoulim, Hepatitis B virus: from diagnosis to treatment, Pathol Biol., 58 (2010), 245-253. [6] W. Edmunds, G. Medley and D. Nokes, 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. [7] 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. [8] E. Franco, B. Bagnato and M. G. Marino, Hepatitis B: Epidemiology and prevention in developing countries, World J. Hepatol., 4 (2012), 74-80. [9] 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. [10] L. Gross, A Broken Trust: Lessons from the Vaccine-Autism Wars, PLoS Biol., 7 (2009), e1000114. doi: 10.1371/journal.pbio.1000114. [11] H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogenous populations, Discrete Contin. Dyn. Syst. Ser B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. [12] H. Guo and M. Y. Li, Global stability of the endemic equilibrium of a tuberculosis model with immigration and treatment, Canad. Appl. Math. Quart., 19 (2012), 1-17. [13] 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. [14] M. Kane, Global programme for control of hepatitis B infection, Vaccine, 13 (1995), S47-S49. [15] P. Magal, C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [16] E. E. Mast, J. W. Ward and H. B Vaccine, Vaccines (S. Plotkin, W. Orenstein & P. Offit), 5th edition, WB Saunders Company, (2008), 205–242. [17] C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. [18] C. McCluskey, Global stability for an SEI model of infectious disease with age structure and immigration of infecteds, Math. Biosci. Eng., 13 (2016), 381-400. doi: 10.3934/mbe.2015008. [19] G. Medley, N. Lindop, W. Edmunds and D. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nature Medicine, 7 (2001), 619-624. doi: 10.1038/87953. [20] Y. Mekonnen, R. Jegou and R. A. Coutinho, Demographic impact of AIDS in a low-fertility urban African setting: Projection for Addis Ababa, Ethiopia, J. Health Popul. Nutr., 20 (2002), 120-129. [21] S. K. Parker, B. Schwartz, J. Todd and L. K. Pickering, Thimerosal-Containing Vaccines and Autistic Spectrum Disorder: A Critical Review of Published Original Data, Pediatrics, 114 (2004), 793-804. [22] L. Rong, Z. Feng and A. 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. [23] H. Smith and H. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 2011. [24] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York, 1985. [25] W. W. Williams, P.-J. Lu and A. O'Halloran, Vaccination Coverage Among Adults, Excluding Influenza Vaccination -- United States, 2013, Morbidity and Mortality Weekly Report, 64 (2015), 95-102. [26] S. Zhang and X. Xu, A mathematical model for hepatitis B with infection-age structure, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1329-1346. doi: 10.3934/dcdsb.2016.21.1329. [27] 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. [28] X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. [29] 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] World Health Organization, 2008, Hepatitis B. World Health Organization Fact Sheet N°204, Available from http://www.who.int/mediacentre/factsheets/fs204/en/index.html [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. [3] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [4] D. Candotti, O. Opare-Sem and H. Rezvan, 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. [5] P. Dény and F. Zoulim, Hepatitis B virus: from diagnosis to treatment, Pathol Biol., 58 (2010), 245-253. [6] W. Edmunds, G. Medley and D. Nokes, 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. [7] 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. [8] E. Franco, B. Bagnato and M. G. Marino, Hepatitis B: Epidemiology and prevention in developing countries, World J. Hepatol., 4 (2012), 74-80. [9] 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. [10] L. Gross, A Broken Trust: Lessons from the Vaccine-Autism Wars, PLoS Biol., 7 (2009), e1000114. doi: 10.1371/journal.pbio.1000114. [11] H. Guo and M. Y. Li, Impacts of migration and immigration on disease transmission dynamics in heterogenous populations, Discrete Contin. Dyn. Syst. Ser B, 17 (2012), 2413-2430. doi: 10.3934/dcdsb.2012.17.2413. [12] H. Guo and M. Y. Li, Global stability of the endemic equilibrium of a tuberculosis model with immigration and treatment, Canad. Appl. Math. Quart., 19 (2012), 1-17. [13] 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. [14] M. Kane, Global programme for control of hepatitis B infection, Vaccine, 13 (1995), S47-S49. [15] P. Magal, C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140. doi: 10.1080/00036810903208122. [16] E. E. Mast, J. W. Ward and H. B Vaccine, Vaccines (S. Plotkin, W. Orenstein & P. Offit), 5th edition, WB Saunders Company, (2008), 205–242. [17] C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819. [18] C. McCluskey, Global stability for an SEI model of infectious disease with age structure and immigration of infecteds, Math. Biosci. Eng., 13 (2016), 381-400. doi: 10.3934/mbe.2015008. [19] G. Medley, N. Lindop, W. Edmunds and D. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nature Medicine, 7 (2001), 619-624. doi: 10.1038/87953. [20] Y. Mekonnen, R. Jegou and R. A. Coutinho, Demographic impact of AIDS in a low-fertility urban African setting: Projection for Addis Ababa, Ethiopia, J. Health Popul. Nutr., 20 (2002), 120-129. [21] S. K. Parker, B. Schwartz, J. Todd and L. K. Pickering, Thimerosal-Containing Vaccines and Autistic Spectrum Disorder: A Critical Review of Published Original Data, Pediatrics, 114 (2004), 793-804. [22] L. Rong, Z. Feng and A. 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. [23] H. Smith and H. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 2011. [24] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York, 1985. [25] W. W. Williams, P.-J. Lu and A. O'Halloran, Vaccination Coverage Among Adults, Excluding Influenza Vaccination -- United States, 2013, Morbidity and Mortality Weekly Report, 64 (2015), 95-102. [26] S. Zhang and X. Xu, A mathematical model for hepatitis B with infection-age structure, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1329-1346. doi: 10.3934/dcdsb.2016.21.1329. [27] 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. [28] X. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. [29] 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.
Flow diagram of the age-structured HBV transmission model (1)
Definitions of parameters used in model (1)
 Symbol Definition $\Lambda_S$ rate of recruitment into the susceptible compartment, including unsuccessfully immunized birth and immigration $\Lambda_k$ immigration rate into class $k$ ($k=E, R$) $\Lambda_j(a)$ age-dependent immigration rate into class $j$ ($j=i, c$) $\mu_k$ per capital death rate for class $k$ ($k=S, E, R$) $\mu_j(a)$ age-dependent death rate for class $j$ ($j=i, c$) $b$ birth rate $\omega$ proportion of newborns who are unsuccessfully immunized $\sigma$ transfer rate from exposed to acute infection $p$ vaccination rate $\alpha$ degree of infectiousness of carriers relative to acute infections ($\alpha>0$) $\beta(a)$ age-dependent transmission coefficient $v(a)$ age-dependent rate of children born to carrier mothers who become HBV carriers $\gamma_1(a)$ age-dependent transfer rate from acute to immunized or carrier class $\gamma_2(a)$ age-dependent transfer rate from carrier to immunized class $q(a)$ age-dependent progression from acute infection to carrier class $\theta(a)$ age-dependent HBV-induced death rate
 Symbol Definition $\Lambda_S$ rate of recruitment into the susceptible compartment, including unsuccessfully immunized birth and immigration $\Lambda_k$ immigration rate into class $k$ ($k=E, R$) $\Lambda_j(a)$ age-dependent immigration rate into class $j$ ($j=i, c$) $\mu_k$ per capital death rate for class $k$ ($k=S, E, R$) $\mu_j(a)$ age-dependent death rate for class $j$ ($j=i, c$) $b$ birth rate $\omega$ proportion of newborns who are unsuccessfully immunized $\sigma$ transfer rate from exposed to acute infection $p$ vaccination rate $\alpha$ degree of infectiousness of carriers relative to acute infections ($\alpha>0$) $\beta(a)$ age-dependent transmission coefficient $v(a)$ age-dependent rate of children born to carrier mothers who become HBV carriers $\gamma_1(a)$ age-dependent transfer rate from acute to immunized or carrier class $\gamma_2(a)$ age-dependent transfer rate from carrier to immunized class $q(a)$ age-dependent progression from acute infection to carrier class $\theta(a)$ age-dependent HBV-induced death rate
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