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 Google Scholar

[2]

F. BrauerZ. 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

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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.  Google Scholar

[4]

D. CandottiO. 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.   Google Scholar

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P. Dény and F. Zoulim, Hepatitis B virus: from diagnosis to treatment, Pathol Biol., 58 (2010), 245-253.   Google Scholar

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W. EdmundsG. 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.  Google Scholar

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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

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E. FrancoB. Bagnato and M. G. Marino, Hepatitis B: Epidemiology and prevention in developing countries, World J. Hepatol., 4 (2012), 74-80.   Google Scholar

[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.  Google Scholar

[10]

L. Gross, A Broken Trust: Lessons from the Vaccine-Autism Wars, PLoS Biol., 7 (2009), e1000114.  doi: 10.1371/journal.pbio.1000114.  Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[13]

G. HuangX. 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

[14]

M. Kane, Global programme for control of hepatitis B infection, Vaccine, 13 (1995), S47-S49.   Google Scholar

[15]

P. MagalC. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. MedleyN. LindopW. Edmunds and D. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nature Medicine, 7 (2001), 619-624.  doi: 10.1038/87953.  Google Scholar

[20]

Y. MekonnenR. 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.   Google Scholar

[21]

S. K. ParkerB. SchwartzJ. Todd and L. K. Pickering, Thimerosal-Containing Vaccines and Autistic Spectrum Disorder: A Critical Review of Published Original Data, Pediatrics, 114 (2004), 793-804.   Google Scholar

[22]

L. RongZ. 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.  Google Scholar

[23]

H. Smith and H. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 2011.  Google Scholar

[24]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[25]

W. W. WilliamsP.-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.   Google Scholar

[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.  Google Scholar

[27]

S. ZhaoZ. 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

[28]

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

[29]

L. ZouS. 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.  Google Scholar

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 Google Scholar

[2]

F. BrauerZ. 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]

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.  Google Scholar

[4]

D. CandottiO. 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.   Google Scholar

[5]

P. Dény and F. Zoulim, Hepatitis B virus: from diagnosis to treatment, Pathol Biol., 58 (2010), 245-253.   Google Scholar

[6]

W. EdmundsG. 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.  Google Scholar

[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.  Google Scholar

[8]

E. FrancoB. Bagnato and M. G. Marino, Hepatitis B: Epidemiology and prevention in developing countries, World J. Hepatol., 4 (2012), 74-80.   Google Scholar

[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.  Google Scholar

[10]

L. Gross, A Broken Trust: Lessons from the Vaccine-Autism Wars, PLoS Biol., 7 (2009), e1000114.  doi: 10.1371/journal.pbio.1000114.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

G. HuangX. 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

[14]

M. Kane, Global programme for control of hepatitis B infection, Vaccine, 13 (1995), S47-S49.   Google Scholar

[15]

P. MagalC. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

G. MedleyN. LindopW. Edmunds and D. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nature Medicine, 7 (2001), 619-624.  doi: 10.1038/87953.  Google Scholar

[20]

Y. MekonnenR. 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.   Google Scholar

[21]

S. K. ParkerB. SchwartzJ. Todd and L. K. Pickering, Thimerosal-Containing Vaccines and Autistic Spectrum Disorder: A Critical Review of Published Original Data, Pediatrics, 114 (2004), 793-804.   Google Scholar

[22]

L. RongZ. 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.  Google Scholar

[23]

H. Smith and H. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, 2011.  Google Scholar

[24]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York, 1985.  Google Scholar

[25]

W. W. WilliamsP.-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.   Google Scholar

[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.  Google Scholar

[27]

S. ZhaoZ. 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

[28]

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

[29]

L. ZouS. 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.  Google Scholar

Figure 1.  Flow diagram of the age-structured HBV transmission model (1)
Table 1.  Definitions of parameters used in model (1)
SymbolDefinition
$\Lambda_S$ 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
SymbolDefinition
$\Lambda_S$ 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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