Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China

Tailei Zhang; Hui Li; Na Xie; Wenhui Fu; Kai Wang; Xiongjie Ding; Tailei Zhang; Hui Li; Na Xie; Wenhui Fu; Kai Wang; Xiongjie Ding

doi:10.3934/mbe.2020092

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1757-1775. doi: 10.3934/mbe.2020092

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Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China

1.
School of Science, Chang’an University, Xi’an, 710064, China
2.
Central Laboratory of Xinjiang Medical University, Urumqi, China
3.
Xinjiang Center for Disease Control and Prevention, Urumqi, China
4.
Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, China

Received: 29 March 2019 Accepted: 08 November 2019 Published: 13 December 2019

The incubation period for Hepatitis B virus (HBV) within the human is epidemiologically significant because it is typically of long duration (1.5~6 months) and the disease transmission possibility may be increased due to more contact from the patients in this period. In this paper, we investigate an SEICRV epidemic model with time delay to research the transmission dynamics of Hepatitis B disease. The basic reproductive number ${\mathcal R}_0$ is derived and can determine the dynamics of the model. The disease-free equilibrium is globally asymptotically stable if ${\mathcal R}_0 < 1$ and unstable if ${\mathcal R}_0>1$. As ${\mathcal R}_0>1$, the model admits a unique endemic equilibrium which is locally asymptotically stable. The endemic equilibrium is globally asymptotically stable when the vertical transmission is ignored. Numerically, we study the Hepatitis B transmission case in Xinjiang, China. Using the Hepatitis B data from Xinjiang, the basic reproductive number is estimated as 1.47 (95% CI: 1.34–1.50). By the end of 2028, the cumulative number of Hepatitis B cases in Xinjiang will be estimated about 700,000 if there is no more effective preventive measures. The sensitivity analysis of ${\mathcal R}_0$ in terms of parameters indicates prevention and treatment for chronic patients are key measures in controlling the spread of Hepatitis B in Xinjiang.
- Hepatitis B virus,
- transmission dynamics,
- Lyapunov functional,
- model application,
- sensitivity analysis
Citation: Tailei Zhang, Hui Li, Na Xie, Wenhui Fu, Kai Wang, Xiongjie Ding. Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1757-1775. doi: 10.3934/mbe.2020092

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Abstract

The incubation period for Hepatitis B virus (HBV) within the human is epidemiologically significant because it is typically of long duration (1.5~6 months) and the disease transmission possibility may be increased due to more contact from the patients in this period. In this paper, we investigate an SEICRV epidemic model with time delay to research the transmission dynamics of Hepatitis B disease. The basic reproductive number ${\mathcal R}_0$ is derived and can determine the dynamics of the model. The disease-free equilibrium is globally asymptotically stable if ${\mathcal R}_0 < 1$ and unstable if ${\mathcal R}_0>1$. As ${\mathcal R}_0>1$, the model admits a unique endemic equilibrium which is locally asymptotically stable. The endemic equilibrium is globally asymptotically stable when the vertical transmission is ignored. Numerically, we study the Hepatitis B transmission case in Xinjiang, China. Using the Hepatitis B data from Xinjiang, the basic reproductive number is estimated as 1.47 (95% CI: 1.34–1.50). By the end of 2028, the cumulative number of Hepatitis B cases in Xinjiang will be estimated about 700,000 if there is no more effective preventive measures. The sensitivity analysis of ${\mathcal R}_0$ in terms of parameters indicates prevention and treatment for chronic patients are key measures in controlling the spread of Hepatitis B in Xinjiang.

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