Export file:


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


  • Citation Only
  • Citation and Abstract

A model of HBV infection with intervention strategies: dynamics analysis and numerical simulations

School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, P.R. China

In this paper, we analyze the effect of environment noise on the transmission dynamics of a stochastic hepatitis B virus (HBV) infection model with intervention strategies. By using the Markov semigroups theory, we define the stochastic basic reproduction number and find it can be used to govern disease extinction or persistence. When it is less than one, under a mild extra condition, the stochastic system has a disease-free equilibrium and the disease is predicted to die out with probability one. When it is greater than one, under mild extra conditions, the model admits a stationary distribution which means the persistence of the disease. Thus, we observe that larger intensity of noise (resulting in a smaller stochastic basic reproduction number) can suppress the emergence of hepatitis B outbreak. Numerical simulations are also carried out to investigate the influence of information intervention strategies that may change individual behavior and protect the susceptible from infection. Our analysis shows that the environmental noise can greatly a ect the long-term behavior of the system, highlighting the importance of the role of intervention strategies in the control of hepatitis B.
  Article Metrics


1. WHO, Hepatitis B, 2014 (revised Agust 2014). Available from: http://www.who.int/meadiacenter /factsheet/fs204/en/index.html.

2. J.Wu, U.Wang, S. Chang, et al., Impacts of a mass vaccination campaign against pandemic H1N1 2009 influenza in Taiwan: a time-series regression analysis, Int. J. Infect. Dis., 23 (2014), 82–89.

3. Y. Liu and J. Cui, The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath., 1 (2008), 65–74.

4. Y. Cai, Y. Kang, M. Banerjee, et al., A stochastic SIRS epidemic model with infectious force under intervention strategies, J. Differ. Equations, 259 (2015), 7463–7502.

5. Q. Wang, L. Zhao, R. Huang, et al., Interaction of media and disease dynamics and its impact on emerging infection management, Discrete Cont. Dyn.-B, 20 (2014), 215–230.

6. Y. Xiao, T. Zhao and S. Tang, Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. Eng., 10 (2013), 445–461.

7. M. Brinn, K. Carson, A. Esterman, et al., Mass media interventions for preventing smoking in young people, Cochrane DB Syst. Rev., 11 (2010), CD001006.

8. Q. Li, L. Zhou, M. Zhou, et al, Epidemiology of human infections with avian influenza A(H7N9) virus in China, New Engl. J. Med., 370 (2014), 520–532.

9. WHO, Cholera-United Republic of Tanzania, 2018. Available from: http://www.who.int/csr/don/ 12-january-2018-cholera-tanzania/en/.

10. J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equ., 20 (2018), 31–53.

11. J. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky MT J. Math., 38 (2008), 1323–1334.

12. L. Zou, W. Zhang and S. Ruan, Modeling the transmission dynamics and control of hepatitis b virus in China, J. Theor. Biol., 262 (2010), 330–338.

13. T. Khan, G. Zaman and M. I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis b, J. Biol. Dyn., 11 (2017), 172–189.

14. T. Zhang, K. Wang and X. Zhang, Modeling and analyzing the transmission dynamics of HBV epidemic in Xinjiang, China, PloS One, 10 (2015), e0138765.

15. S. Spencer, Stochastic epidemic models for emerging diseases, PhD thesis, University of Nottingham, 2008.

16. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 2010.

17. Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal.-Real, 14 (2013), 1434–1456.

18. Q. Liu, D. Jiang, N. Shi, et al., Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay, Discrete Cont. Dyn. - B, 22 (2017), 2479–2500.

19. Y. Cai and W. Wang, Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonlinear Anal.- Real, 30 (2016), 99–125.

20. X. Mao, G. Marion and E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stoch. Proc. Appl., 97 (2002), 95–110.

21. Y. Cai, Y. Kang, M. Banerjee, et al., Complex Dynamics of a host-parasite model with both horizontal and vertical transmissions in a spatial heterogeneous environment, Nonlinear Anal.- Real, 40 (2018), 444–465.

22. L. Allen, An introduction to stochastic epidemic models, Mathematical Epidemiology, Lect. Notes Math., 1945 (2008), 81–130.

23. K. Bao and Q. Zhang, Stationary distribution and extinction of a stochastic SIRS epidemic model with information intervention, Adv. Differ. Equ., 352 (2017), doi:10.1186/s13662-017-1406-9.

24. T. Khan, A. Khan and G. Zaman, The extinction and persistence of the stochastic hepatitis B epidemic model, Chaos, Solitons Fract., 108 (2018), 123–128.

25. R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stoch. Proc. Appl., 108 (2003), 93–107.

26. R. Rudnicki, K. Pichr and M. Tyrankamiska, Markov semigroups and their applications, Lect. Notes Phys., 597 (2002), 215–238.

27. K. Pichr and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249 (2000), 668–685.

28. R. Rudnicki, On asymptotic stability and sweeping for Markov operators, B. Pol. Acad. Sci. Math., 43 (1995), 245–262.

29. K. Pichr and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl., 215 (1997), 56–74.

30. R. Rudnicki, Asymptotic properties of the Fokker-Planck equation, Springer Berlin Heidelberg, (1995), 517–521.

31. P. Martin, Statistical physics: Statics, dynamics, and renormalization, World Scientific, 2000.

32. O. Diekmann and J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000.

33. R. Lipster, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217–228.

34. F. Baudoin and M. Hairer, A version of Hörmanders theorem for the fractional Brownian motion, Probab. Theory Rel., 139 (2007), 373–395.

35. G. Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II), Probab. Theory Rel., 90 (1991), 377–402.

36. D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546.

© 2019 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

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved