Skip to main content

Advertisement

SpringerLink
Log in

Numerical study for epidemic model of hepatitis-B virus

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

Hepatitis-B is a highly infectious disease and causes many worldwide deaths. In this article, a mathematical model of hepatitis-B virus is described. The optimal existence and optimality conditions for the existence of solutions of the concerned model are established. The numerical behavior of the continuous system, susceptible–exposed–acute–carrier–hospitalized–recovered, of hepatitis-B virus is also discussed. Stability of the equilibrium points is studied using Routh–Hurwitz stability criteria. To study this model numerically, a nonstandard finite difference (NSFD) scheme is designed which has the most important features, for instance, unconditional stability, positivity and convergence of the continuous system. Basic reproduction number \(R_0\) that imparts a decisive role in disease prediction as well as stability analysis is described. It is observed that disease vanishes when the value of \(R_0\) is less then unity and it persists for \(R_0>1\). Stability analysis is carried out, and two bench mark results for the system to be locally asymptotically stable are established. Comparison of the NSFD scheme with two renowned techniques, RK-4 and Forward Euler’s method, is shown in this study. Numerical simulations show that proposed NSFD scheme is convergent to the steady states at all step sizes. It shows that the proposed scheme is independent of the step size while the other two existing schemes do not observe this property. Also, the other methods do not preserve some of the main conditions of the continuous system at certain time steps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. L. Zou, W. Zhang, S. Ruan, Modeling the transmission dynamics and control of hepatitis-B virus in China. J. Theor. Biol. 262(2), 330–338 (2010)

    Article  MathSciNet  Google Scholar 

  2. World Health Organization. http://www.who.int/news-room/fact-sheets/detail/hepatitis-B (2019)

  3. S.A. Ali, R.M. Donahue, H. Qureshi, S.H. Vermund, Hepatitis-B and hepatitis-C in Pakistan: prevalence and risk factors. Int. J. Infect. Dis. 13(1), 9–19 (2009)

    Article  Google Scholar 

  4. M.A. Khan, S. Islam, J.C. Valverde, S.A. Khan, Control strategies of hepatitis B with three control variables. J. Biol. Syst. 26, 1–21 (2018)

    Article  MathSciNet  Google Scholar 

  5. M.A. Khan, S. Islam, G. Zaman, Media coverage campaign in hepatitis B transmission model. Appl. Math. Comput. 331, 378–393 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Y. Nakata, T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl. 363(1), 230–237 (2010)

    Article  MathSciNet  Google Scholar 

  7. H.W. Hethcote, The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  8. R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations (World Scientific, Singapore, 1994)

    MATH  Google Scholar 

  9. W. Qin, L. Wang, X. Ding, A non-standard finite difference method for a hepatitis-B virus infection model with spatial diffusion. J. Differ. Equ. Appl. 20(12), 1641–1651 (2014)

    Article  MathSciNet  Google Scholar 

  10. J. Cresson, F. Pierret, Non standard finite difference scheme preserving dynamical properties. J. Comput. Appl. Math. 303, 15–30 (2016)

    Article  MathSciNet  Google Scholar 

  11. M. Suleman, S. Riaz, Unconditionally stable numerical scheme to study the dynamics of hepatitis-B disease. Punjab Univ. J. Math. 49, 99–118 (2017)

    MathSciNet  MATH  Google Scholar 

  12. N. Ahmed, N. Shahid, Z. Iqbal, M. Jawaz, M. Rafiq, S.S. Tahira, M.O. Ahmad, Numerical modeling of SEIQV epidemic model with saturated incidence rate. J. Appl. Environ. Biol. Sci 8(4), 67–82 (2018)

    Google Scholar 

  13. N. Ahmed, S.S. Tahira, M. Rafiq, M.A. Rehman, M. Ali, M.O. Ahmad, Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model. Open Math. 17(1), 313–330 (2019)

    Article  MathSciNet  Google Scholar 

  14. J.E. Macías-Díaz, N. Ahmed, M. Rafiq, Analysis and nonstandard numerical design of a discrete three-dimensional hepatitis-B epidemic model. Mathematics 7(12), 1157 (2019)

    Article  Google Scholar 

  15. A.Z. Zaher, A.M.A. Moawad, K.S. Mekheimer, M.M. Bhatti, Residual time of sinusoidal metachronal ciliary flow of non-Newtonian fluid through ciliated walls: fertilization and implantation. Biomech. Model Mechanobiol. (2021). https://doi.org/10.1007/s10237-020-01405-5

    Article  Google Scholar 

  16. L. Zhang, M.M. Bhatti, M. Marin, K.S. Mekheimer, Entropy analysis on the blood flow through anisotropically tapered arteries filled with magnetic zinc-oxide (ZnO) nanoparticles. Entropy 22, 1070 (2020). https://doi.org/10.3390/e22101070

    Article  ADS  Google Scholar 

  17. L. Zou, W. Zhang, S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China. J. Theor. Biol. 21(262), 330–338 (2010). https://doi.org/10.1016/j.jtbi.2009.09.035

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Li, Z. Ma, Qualitative analyses of SIS epidemic model with vaccination and varying total population size. Math. Comput. Model. 1(35), 1235–1243 (2002). https://doi.org/10.1016/S0895-7177(02)00082-1

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Ullah, M.A. Khan, J.F. Gómez-Aguilar, Mathematical formulation of hepatitis-B virus with optimal control analysis. Optim. Control Appl. Methods 40(3), 529–544 (2019)

    Article  MathSciNet  Google Scholar 

  20. P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1–2), 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  21. W.J. Edmunds, G.F. Medley, D.J. Nokes, The transmission dynamics and control of hepatitis-B virus in The Gambia. Stat. Med. 15(20), 2215–2233 (1996)

    Article  Google Scholar 

  22. C.W. Shepard, E.P. Simard, L. Finelli, A.E. Fiore, B.P. Bell, Hepatitis-B virus infection: epidemiology and vaccination. Epidemiologic Rev. 28(1), 112–125 (2006)

    Article  Google Scholar 

  23. J. Pang, J.A. Cui, X. Zhou, Dynamical behavior of a hepatitis-B virus transmission model with vaccination. J. Theor. Biol. 265(4), 572–578 (2010)

    Article  MathSciNet  Google Scholar 

  24. World Health Organization. Pakistan. http://www.who.int/countries/pak/en/. Accessed May 15 (2016)

  25. G.F. Medley, N.A. Lindop, W.J. Edmunds, D.J. Nokes, Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control. Nat. Med. 7(5), 619–624 (2001)

    Article  Google Scholar 

  26. World Health Organization. Hepatitis-B. http://www.who.int/mediacentre/factsheets/fs204/en/. Accessed July 1 (2016)

Download references

Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project R-2021-50.

Author information

Authors and Affiliations

  1. Department of Mathematics, Lahore College for Women University, Lahore, Pakistan

    Tahira Sumbal Shaikh & Nimra Fayyaz

  2. Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan

    Nauman Ahmed & Naveed Shahid

  3. Department of Mathematics, University of Management and Technology, Lahore, Pakistan

    Naveed Shahid

  4. Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, Pakistan

    Muhammad Rafiq

  5. Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al Majma’ah, 11952, Saudi Arabia

    Ilyas Khan

  6. Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Wadi-al-dawser, Saudi Arabia

    Kottakkaran Sooppy Nisar

Authors
  1. Tahira Sumbal Shaikh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nimra Fayyaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nauman Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Naveed Shahid
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Muhammad Rafiq
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ilyas Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Kottakkaran Sooppy Nisar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ilyas Khan.

Ethics declarations

Conflict of Interest

The authors declare no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shaikh, T.S., Fayyaz, N., Ahmed, N. et al. Numerical study for epidemic model of hepatitis-B virus. Eur. Phys. J. Plus 136, 367 (2021). https://doi.org/10.1140/epjp/s13360-021-01248-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-021-01248-8

Search

Navigation