Dynamical Behavior of HBV in a Population

Abstract

The present study investigates a mathematical model for HBV carried out in a district of Kerman. The statistical sample comprises all men and women living in that district. Two different mathematical models are introduced for HBV related to this population. Data analysis was carried out with MATLAB programming. The results indicate that there is a meaningful relationship between the vaccination and epidemic disease. 

Keywords

• Mathematical models,epidemiology,Runge-Kutta method,differential equation systems

References

1. [1] M. Farkas, ”Dynamical Models in Biology”, Academic Press, (2001).
2. [2] Z. Ma, J. Li, ”Dynamical Modeling and Analysis of Epidemics”, World Scientific Publishing Co. Pte. Ltd., (2009).
3. [3] M. R. Molaei, T. Waezizadeh, ”A Mathematical Model for HBV”, J. Basic. Appl. Sci. Res., 2(9), 9407-9412(2012).
4. [4] J. Li, Z. Ma, F. Zhang, ”Stability Analysis for an Epidemic Model with Stage Structure”, Nonlinear Analysis RealWorld Application, (2008), 1672-1679.
5. [5] L. Cai, X. Li, ”Analysis of a SEIV Epidemic Model with a Nonlinear Incidence Rate”, Applied MathematicalModelling, 33,(2009), 2919–2926.
6. [6] L. Cai, X. Li, ”A Note on Global Stability of an SEI Epidemic Model with a Cute and Chronic Stages”, AppliedMathematics and Computation 196, (2008), 923–930.
7. [7] F. Brauer, P. Van den, J. Wu, ”Mathematical Epidemiology”, Springer-Verlag Berlin, (2008).
8. [8] E. Hairer, G. Wanner, ”Solving Ordinary Differential Equations II: Stiff and Differential-Algebric Problems”,Berlin, New York: Springer-Verlag, (1996).

9. [9] X. Dong, C. Wang, G. Xiong, ”Analysis and Simulations of Dynamic Models of Hepatitis B Virus”, Journal ofMathematics Research, 2, (2010), 12–18.
10. [10] H. Keyvani, M. Sohrabi, F. Zamani, H. Poustchi, H. Ashrafi, F. Saeedian, M. Mooadi, N. Motamed, H. Ajdarkosh,M. Khonsari, G. Hemmasi, M. Ameli, A. Kabir, M. Khodadost, ”A Population Based Study on Hepatitis B Virusin Northern Iran, Amol”, Hepat Mon, 14, (2014), 1–8.
11. [11] S. Alavian, B. Hajarizadeh, M. Ahmadzad-Asl, A. Kabir, K. Bagheri-Lankarani, ”Hepatitis B Virus Infection inIran: A Systematic Review”, Hepatitis Monthly, 8, (2008), 281–294.
12. [12] B. Behbahani, A. Mafi-Nejad, A. Tabei, S. Z. Lankarani, K. B. Torab, A. Moaddeb, ”Anti-HBC & HBV-DNADetection in Blood Donors Negative for Hepatitis B Virus Surface Antigen in Reducing Risk of TransfusionAssociated HBV Infection”, Indian J Med Res, 123, (2006), 37–42.
13. [13] S. Merat, H. Rezvan, M. Nouraie, A. Jamali, S. Assari, H. Abolghasemi, et al. ”The Prevalence of Hepatitis BSurface Antigen and Antihepatitis B Core Antibody in Iran: A Population-based Study” Arch Iran Med. 12(3),(2009), 225–231.
14. [14] F. Fathimoghaddam, M. R. Hedayati-Moghaddam, H. R. Bidkhori, S. Ahmadi, H. R. Sima, ”The Prevalence ofHepatitis B Antigen-positivity in the General Population of Mashhad, Iran”, Hepat Mon, 11(5), (2011), 346–350.
15. [15] G. Zaman, Y. H. Kang, I. H. Jung, ”Stability Analysis and Optimal Vaccination of an SIR Epidemic Model”,Biosystems, 93(3), (2008), 240–249.
16. [16] J. Pang, J. A. Cui, X. Zhou, ”Dynamical Behavior of a Hepatitis B Virus Transmission Model with Vaccination”,Journal of Theoretical Biology, 265(4), (2010), 572–578.
17. [17] S. Zhang, Y. Zhou, ”The Analysis and Application of an HBV Model”, Applied Mathematics Modelling, 36(3),(2012), 1302–1312.
18. [18] N. C. Grassly, C. Fraser, ”Mathematical Models of Infectious Disease Transmission”, Nature Reviews Microbiology,6, (2008), 477-487.
19. [19] A. Vahidian Kamyad, R. Akbari, A. A. Heydari, A. Heydari, ”Mathematical Modelling of Transmission Dynamicsand Optimal Control of Vaccination and Treatment for Hepatitis B Virus”, Computational and MathematicalMethods in Medicine, (2014).
20. [20] L. Min, Y. Su, Y. Kuang, ”Mathematical Analysis of a Basic Virus Infection Model with Application to HBVInfection”, Rocky Mountain Journal of Mathematics, 38(5), (2008), 1573–1585.
21. [21] WHO, Hepatitis B Fact Sheet No. 204, The World Health Organisation, Geneva, Switzerland, 2013,http://www.who.int/ mediacentre/factsheets/fs204/en/.
22. [22] L. Wang, R. Xu, ”Mathematical Analysis of an Improved Hepatitis B Virus Model”, International Journal ofBiomathematics, 5(1) (2012).
23. [23] P. Pasquini, B. Cvjetanovic, ”Mathematical Models of Hepatitis B Infection”, Annali dell’Istituto Superiore diSanit, 24(2), (1988), 245–250.
24. [24] C. Seeger, W. Mason, ”Hepatitis B Virus Biology”, Microbiology and Molecular Biology Reviews, 64, (2000),51–68.
25. [25] J. L. Hou, Z. H. Liu, F. Gu, ”Epidemiology Prevention of Hepatitis B Virus Infection”, International Journal ofMedical Sciences, 2, (2005), 50–57.
26. [26] X. Q. Zhao, ”Dynamical Systems in Population Biology”, Springer-Verlag New York, (2003).
27. [27] R. Akbari, A. Vahidian, A. A. Heydari, A. Heydari, ”The Analysis of a Disease-free Equilibrium of Hepatitis Bmodel”, Sahand communiation in mathematical analysis, 3(2), (2016), 1–11.
28. [28] K. Wang, A. Fan, A. Torres, ”Global Properties of an Improved Hepatitis B Virus Model”, Nonlinear analysis:Real World applications, 11(4), (2010), 3131–3138.
29. [29] A. R. Mclean, B. S. Blumbery, ”Modelling the Impact of Mass Vaccination Against Hepatitis B. I. Model Formulationand Parameter Estimation”, Proceedings of the royal society B, 256, (1994), 7–15.
30. [30] A. M. Elaiw, M. A. Alghamdi, S. Aly, ”Hepatitis B Virus Dynamics: Modeling, Analysis, and Optimal TreatmentScheduling”, Discrete Dynamics in Nature and Society, 2013, (2013), 1–10.
31. [31] S. R. Lewin, R. M. Ribeiro, T. Walters et al., ”Analysis of Hepatitis B Viral Load Decline under Potenttherapy:Complex Decay Profiles Observed, Hepatology, 34(5), (2001), 1012–1020.
32. [32] M. Y. Li, J. S. Muldowney, ”Global Stability for the SEIR Model in Epidemiology”, Mathematical Bioscience,125(2), (1995), 64–155.
33. [33] W. M. Schaffer, T. V. Bronnikova, ”Parametric Dependence in Model Epidemics”, Journal of Biological Dynamics,1(2), (2007), 183–195.
34. [34] E. Vynnycky, R. G. White, An Introduction to Infectious Disease Modelling”, Oxford: Oxford University Press,(2010).
35. [35] W. O. Kermack, A. G. McKendrick, ”Contributions to the Mathematical Theory of Epidemics. II. The Problem ofEndemicity”, Proceedings of the Royal Society A, 138(834),(1932), 55–83.