Mathematical Model of Hepatitis B Virus With Effect of Vaccination and Treatments

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Saif H. Elkhadir, Ali E. M. Saeed, Abdelfatah Abasher


In this paper, a mathematical model of hepatitis B virus with vaccination and treatments is studied, Stability analysis discussed and the disease-free equilibrium and endemic equilibrium points obtained, the basic reproductive number R0 determined and became the threshold for equilibrium points stability. The study showed when R0 < 1 the disease-free equilibrium point was stable, whereas R0 > 1 the virus is endemic and the endemic equilibrium point is stable. The sensitivity analysis for the parameters that could reduce the spread of hepatitis B virus is studied. Finally the numerical simulation are established by using SageMath software package to show the effect of vaccination and treatments. We found that vaccination and also treatments give an effect on value of R0. Increasing the value of the vaccine in the immunized compartment or in the suspected compartment may decrease the value of R0 which mean reduce the spread of the disease.

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  1. Hepatitis B Foundation,, accessed 20 August 2022.
  2. J. Zhang, S. Zhang, Application and Optimal Control for an HBV Model with Vaccination and Treatment, Discrete Dyn. Nat. Soc. 2018 (2018), 2076983.
  3. M. Aniji, N. Kavitha, S. Balamuralitharan, Analytical solution of SEICR model for Hepatitis B virus using HPM, AIP Conf. Proc. 2112 (2019), 020024.
  4. L.K. Beay, Kasbawati, S. Toaha, Effects of Human and Mosquito Migrations on the Dynamical Behavior of the Spread of Malaria, AIP Conf. Proc. 1825 (2017), 020006.
  5. E.D. Wiraningsih, F. Agusto, L. Aryati et al. Stability Analysis of Rabies Model With Vaccination Effect and Culling in Dogs, Appl. Math. Sci. 9 (2015), 3805-3817.
  6. J. Cao, Y. Wang, A. Alofi, et al. Global Stability of an Epidemic Model With Carrier State in Heterogeneous Networks, IMA J. Appl. Math. 80 (2014), 1025–1048.
  7. N. Scott, M. Hellard, E.S. McBryde, Modeling Hepatitis C Virus Transmission Among People Who Inject Drugs: Assumptions, Limitations and Future Challenges, Virulence. 7 (2015), 201–208.
  8. J.M. Haussig, S. Nielsen, M. Gassowski, et al. A large proportion of people who inject drugs are susceptible to hepatitis B: Results from a bio-behavioural study in eight German cities, Int. J. Infect. Dis. 66 (2018), 5–13.
  9. J. Wong, M. Payne, S. Hollenberg, A Double-Dose Hepatitis B Vaccination Schedule in Travelers Presenting for Late Consultation, J. Travel Med. 21 (2014), 260–265.
  10. A.V. Kamyad, R. Akbari, A.A. Heydari, et al. Mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis B virus, Comput. Math. Methods Med. 2014 (2014), 475451.
  11. M.A.E. Osman, I.K. Adu, Simple mathematical model for malaria transmission, J. Adv. Math. Comput. Sci. 25 (2017), JAMCS.37843.
  12. P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.
  13. D.R. Merkin, Introduction to the theory of stability, Springer New York, 1997.