Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

Special Issues: Transmission dynamics in infectious diseases

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.
  Article Metrics

Keywords 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. Mathematical Biosciences and Engineering, 2020, 17(2): 1757-1775. doi: 10.3934/mbe.2020092


  • 1. World Health Organization, Hepatitis B, 2018. Available from: https://www.who.int/newsroom/fact-sheets/detail/hepatitis-b.
  • 2. Y. Cui, M. Moriyama, M. Rahman, Analysis of the incidence of hepatitis B and hepatitis C and association with socio-economic factors in various regions in China, Health, 10 (2018), 1210-1220.
  • 3. Z. Sun, L. Ming, X. Zhu, J. Lu, Prevention and Control of Hepatitis B in China, J. Med. Virol., 67 (2002), 447-450.
  • 4. China Center for Disease Control and Prevention, Questions and answers on hepatitis B vaccination, 2013.Available from: http://www.chinacdc.cn/zxdt/201312/t20131230_92034.htm.
  • 5. M. Nowak, S. Bonhoeffer, A. M. Hill, R. Boehme, H. C. Thomas, H. Mcdade, Viral dynamics in hepatitis B virus infection,Proc. Natl. Acad. Sci. U. S. A., 93 (1996), 4398-4402.
  • 6. S. M. Ciupe, R. M. Ribeiro, P. W. Nelson, A. S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection,J. Theor. Biol., 247 (2007), 23-35.
  • 7. R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.
  • 8. P. M. Tchinda, J. J. Tewa, B. Mewoli, S. Bowong, A Theoretical assessment of the effects of distributed delay of the transmission dynamics of hepatitis B,J. Biol. Syst., 23 (2015), 423-455.
  • 9. K. Hattaf, N. Yousfi, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays,Appl. Math. Comput., 221 (2013), 514-521.
  • 10. K. Wang, W. Wang, S. Song,Dynamics of an HBV model with diffusion and delay,J. Theor. Biol., 253 (2008), 36-44.
  • 11. K. Wang, A. Fan, A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Anal.-Real World Appl., 11 (2010), 3131-3138.
  • 12. S. Zhao, Z. Xu, Y. Lu,A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China,Int. J. Epidemiol., 29 (2000), 744-752.
  • 13. L. Zou, S. Ruan, W. Zhang, On the sexual transmission dynamics of hepatitis B virus in China, J. Theor. Biol., 369 (2015), 1-12.
  • 14. S. Zhang, Y. Zhou, Dynamic analysis of a hepatitis B model with three-age-classes, Commun. Nonlinear. Sci., 19 (2014), 2466-2478.
  • 15. S. Zhang, X. Xu, A mathematical model for hepatitis B with infection-age structure, Discrete Contin. Dyn. Syst.-Ser. B, 21 (2016), 1329-1346.
  • 16. J. Pang, J. Cui, X. Zhou, Dynamical behavior of a hepatitis B virus transmission model with vaccination,J. Theor. Biol.,265 (2010), 572-578.
  • 17. L. Zou, S. Ruan, W. Zhang, An age-structured model for the transmission dynamics of hepatitis B,SIAM J. Appl. Math., 70 (2010), 3121-3139.
  • 18. B. O. Emerenini, S. Inyama, Mathematical model and analysis of hepatitis B virus transmission dynamics,F1000 Research,7 (2017).
  • 19. R. Akbari, A. V. Kamyad, A. A. Heydari, Stability analysis of the transmission dynamics of an HBV model,Int. J. Indu. Math., 8 (2016), 119-219.
  • 20. J. Mann, M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol., 269 (2011), 266-272.
  • 21. World Health Organization, Media centre, 2014. Available from: http://www.who.int/mediacentre/factsheets/fs204/en/.
  • 22. X. Q. Zhao,Basic reproduction ratios for periodic compartmental models with time delay,J. Dyn. Differ. Equ., 29 (2017), 67-82.
  • 23. Y. Kuang, Delay differential equations with application to population dynamics, Academic Press, San Diego, 1993.
  • 24. C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete,Nonlinear Anal. RWA 11 (2010), 55-59.
  • 25. C. Vargas-De-León, Stability analysis of a model for HBV infection with cure of infected cells and intracellular delay,Appl. Math. Comput., 219 (2012), 389-398.
  • 26. X. Q. Zhao,Dynamical systems in population biology,Springer, New York, 2017.
  • 27. T. Zhang, K. Wang, X. Zhang, Modeling and analyzing the transmission dynamics of HBV Epidemic in Xinjiang, China,PLoS ONE, 10 (2015), e0138765.
  • 28. S. Zhang and Y. Zhou, The analysis and application of an HBV model, Appl. Math. Model., 36 (2012), 1302-1312.
  • 29. L. Zou, W. Zhang, S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China, J. Theor. Biol., 262 (2010), 330-338.
  • 30. P. Mathurin, C. Mouquet, T. Poynard, C. Sylla, H. Benalia, C. Fretz, et al., Impact of hepatitis B and C virus on kidney transplantation outcome,Hepatology, 29 (1999), 257-263.    
  • 31. S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,J. Theor. Biol., 254 (2008), 178-196.
  • 32. World Health Organization,Global hepatitis report, 2017.
  • 33. J. K. Hale,Theory of functional differential equations,Springer-Verlag, New York, 1977.


Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved