Numerical Solution and Analysis for Acute and Chronic Hepatitis B

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Muhammad Farman, Muhammad Farman, Zafar Iqbal, Zafar Iqbal, Aqeel Ahmad, Ali Raza, Ali Raza, Ehsan Ul Haq, Ehsan Ul Haq

Abstract

In this article, we present the transmission dynamic of the acute and chronic hepatitis B epidemic problem to control the spread of hepatitis B in a community. In order to do this, first we present sensitivity analysis of the basic reproduction number R0. We develop a unconditionally convergent nonstandard finite difference scheme by applying Mickens approach φ(h) = h + O(h^2) instead of h to control the spread of this infection, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis B individuals and maximize the number of susceptible and recovered individuals. The stability analysis of the scheme has been developed by theorems which shows the both stable locally and globally. Comparison is also made with standard nonstandard finite difference scheme. Finally numerical simulations are also established to investigate the influence of the system parameter on the spread of the disease.

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References

  1. J. Biazar, Solution of the epidemic model by Adomian decomposition method, Appl. Math. Comput. 173 (2006), 1101-1106.
  2. S. Busenberg, P. Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 28 (1990), 65-82.
  3. A.M.A. El-Sayed, S.Z. Rida and A.A.M. Arafa, On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, Int. J. Nonlinear Sci. 7 (2009), 485-495.
  4. O.D. Makinde, Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy, Appl. Math. Comput. 184 (2007), 842-848.
  5. A.A.M. Arafa, S.Z. Rida and M. Khalil, Fractional modeling dynamics of HIV and 4 T-cells during primary infection, Nonlinear Biomed. Phys. 6 (2012), 1-7.
  6. C.M. Kribs-Zaleta, Structured models for heterosexual disease transmission, Math. Biosci. 160 (1999), 83-108.
  7. B. Buonomo and D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate, Ricerche Mat. 57 (2008), 261-281.
  8. WHO, Hepatitis B Fact Sheet No. 204, The World Health Organisation, Geneva, Switzerland, 2013, http://www.who.int/mediacentre/factsheets/fs204/en/.
  9. Canadian Centre for Occupational Health and Safety, Hepatitis B, http : //www.ccohs.ca/oshanswers/diseases/hepatitisb.html.
  10. A. V. Kamyad, R. Akbari, A. K Heydari and A. Heydari Mathematical Modeling of Transmission Dynamics and Optimal Control of Vaccination and Treatment for Hepatitis B Virus, Comput. Math. Methods Med. 2014 (2014), Article ID 475451.
  11. C.M. Stanca, R.M. Ruy, W.N. Patrick and S.P. Alan, Modeling the mechanisms of acute hepatitis B virus infection, J. Theor. Biol. 247 (2007), 23-35.
  12. A. Perelson, Modelling viral and immune system dynamics. Nature Rev. Immunol. 2 (2002), 28-36.
  13. A. Perelson and R. Ribeiro, Hepatitis B virus kinetics and mathematical modeling. Sem. Liv. Dis. 24 (2004), 11-15.
  14. M. Nowak, S. Bonhoeffer,A. Hill, R. Boehme, H. Thomas, H. McDade, Viral dynamics in hepatitis B infection. Proc. Natl Acad. Sci. USA 93 (1996), 4398-4402.
  15. S. Lewin, R. Ribeiro, T. Walters, G. Lau, S. Bowden, S. Locarnini and A. Perelson, Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed. Hepatology 34 (2001), 1012-1020.
  16. P. Colombatto, L. Civitano, R. Bizzarri, F. Oliveri, S. Choudhury, R. Gieschke, F. Bonino and M.R. Brunetto, A multiphase model of the dynamics of HBV infection in HBeAg-negative patients during pegylated interferon-a2a, lamivudine and combination therapy. Antiviral Therapy 11 (2006), 197-212.
  17. J. Wang, J. Pang and X. Liu,Modelling diseases with relapse and nonlinear incidence of infection: A multi-group epidemic model, J. Biol. Dyn. 8 (2014), 99-116.
  18. J. Wang, R. Zhang and T. Kuniya, The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes, J. Biol. Dyn. 9 (2015), 73-101.
  19. G. Zaman, Y.H. Kang and I.H. Jung, Stability and optimal vaccination of an SIR Epidemic Model, BioSystems 93 (2008), 240249.
  20. G. Zaman, Y.H. Kang and I.H. Jung, Optimal treatment of an SIR epidemic model with time delay, BioSystems 98 (2009), 43-50.
  21. D. Lavanchy, Hepatitis B virus epidemiology, disease burden, treatment, and current and emerging prevention and control measures, J. Viral Hepat. 11 (2004), 97-107.
  22. A.S. Lok, E.J Heathcote and J.H. Hoofnagle, Management of hepatitis B, 2000 Summary of a workshop, Gastroenterology 120 (2001), 18281853.
  23. B.J.McMahon, Epidemiology and natural history of hepatitis B, Semin. Liver Dis. 25 (2005), 38.
  24. M.K. Libbus and L.M. phillips, Public health management of perinatal hepatitis B virus, Public Health Nurs. 26 (2009), 353-361.
  25. J.E. Maynard, M.A. Kane and S.C. Hadler, Global control of hepatitis B through vaccination role of hepatitis B vaccine in the expanded programme on immunization, Rev. Infect. 2 (1989), S574-S578.
  26. S. Thornley, C. Bullen and M. Roberts,Hepatitis B in a high prevalence NewZealand population: A mathematical model applied to infection control policy, J. Theor. Biol. 254 (2008), 599-603.
  27. C.W. Shepard, E.P. Simard, L. Finelli, A.E. Fiore and B.P. Bell, Hepatitis B virus infection epidemiology and vaccination, Epidemiol. Rev. 28 (2006), 112-125.
  28. R. Williams, Global challenges in liver disease, Hepatology 44 (2006), 521-526.
  29. 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.
  30. R. E. Mickens, Exact solutions to a finite difference model of a nonlinear reactions advection equation: Implications for numerical analysis, Numer. Methods Partial Differ. Equ. 5 (1989), 313-325.
  31. R. E. Mickens, Applications of Nonstandard finite difference Schemes, World Scientific, Singapore (2000).
  32. R. Anguelov and J.M.S Lubuma, Nonstandard finite difference method by nonlocal approximations, Math. Comput. Simul. 61 (2003), 465-475.
  33. R.E. Mickens, Nonstandard finite difference Models of differential equations, World Scientific, Singapore (1994).
  34. R. Anguelov and J.M.S. Lubuma, Contributions to the mathematics of the nonstandard finite differencemethodandapplications, Numer. Methods Partial Differ. Equ. 17 (2001), 518-543.
  35. J.M.S. Lubuma and K.C. Patidar, Non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions, Appl. Math. Comput. 187(2) (2007), 1147-1160.
  36. L.W. Roeger, Exact difference schemes, in A. B. Gumel Mathematics of Continuous and Discrete Dynamical Systems, Contemp. Math., Vol. 618, Amer. Math. Soc., Providence, RI, (2014), 147-161.