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

Suxia Zhang; Hongbin Guo; Robert Smith?; Suxia Zhang; Hongbin Guo; Robert Smith?

doi:10.3934/mbe.2018060

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 6: 1291-1313. doi: 10.3934/mbe.2018060

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

Suxia Zhang ^{1
,},
Hongbin Guo ^{2
,},
Robert Smith? ^{3
,
,}

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

Received: 14 July 2017 Accepted: 11 July 2018 Published: 01 December 2018
MSC : Primary: 92B05; Secondary: 35A99

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.
- Hepatitis B virus,
- infection age,
- immigration age,
- global stability,
- Lyapunov functional
Citation: Suxia Zhang, Hongbin Guo, Robert Smith?. Dynamical analysis for a hepatitis B transmission model with immigration and infection age[J]. Mathematical Biosciences and Engineering, 2018, 15(6): 1291-1313. doi: 10.3934/mbe.2018060

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Abstract

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.

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