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The transmission dynamic of different hepatitis B-infected individuals with the effect of hospitalization. (English) Zbl 1447.92439

Summary: We propose an epidemic model for the transmission of hepatitis B virus along with the classification of different infection phases and hospitalized class. We formulate the model and discuss its basic mathematical properties, e.g. existence, positivity, and biological feasibility. Exploiting the next generation matrix approach, we find the basic reproductive number of the model. We perform sensitivity analysis to illustrate the effect of various parameters on the transmission of the disease. We investigate stability of the equilibria of the model in terms of the basic reproduction number. Conditions for the stability of the proposed model are obtained using various approaches. Finally, we perform the numerical simulations to discuss sensitivity analysis and to support our analytical work.

MSC:

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
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[1] R.M. Anderson and R.M. May, Infectious Disease of Humans, Dynamics and Control, Oxford University Press, Oxford, UK, 1991. [Google Scholar]
[2] M.H. Chang, Hepatitis virus infection, Semin. Fetal. Neonatal Med. 12(3) (2007), pp. 160-167. doi: 10.1016/j.siny.2007.01.013[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[3] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), pp. 69-96. doi: 10.3934/dcdsb.2006.6.69[Crossref], [Web of Science ®], [Google Scholar] · Zbl 1088.92049
[4] T. Khan and G. Zaman, Classification of different Hepatitis B infected individuals with saturated incidence rate, Springer Plus 5 (2016), p. 16. doi: 10.1186/s40064-015-1619-x[Crossref], [PubMed], [Google Scholar]
[5] T. Khan, G. Zaman, M.I Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B. J. Bio. Dynam. 1(11) (2016), pp. 172-189[Google Scholar] · Zbl 1447.92438
[6] D. Lavanchy and B. Hepatitis, Disease burden treatment and current and emerging prevention and control measures, J. Viral. Hepat. 11 (2004), pp. 97-107. doi: 10.1046/j.1365-2893.2003.00487.x[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[7] A.S. Lok, E.J. Heathcote, and J.J. Hoofnagle, Management of hepatitis B, 2000 - summary of a workshop, Gastroenterology 120 (2001), pp. 1828-1853. doi: 10.1053/gast.2001.24839[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[8] J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol. 269(1) (2011), pp. 266-272. doi: 10.1016/j.jtbi.2010.10.028[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1307.92348
[9] B.J. McMahon, Epidemiology and natural history of hepatitis B, Semin. Liver Dis. 25(Suppl 1) (2005), pp. 3-8. doi: 10.1055/s-2005-915644[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[10] C.W. Shepard, L Finelli, and B.P. Bell, hepatitis B virus infection: Epidemiology and vaccination, Epidemiol. Rev. 28 (2006), pp. 112-125. doi: 10.1093/epirev/mxj009[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[11] S. Thornley, C. Bullen, and M. Roberts, Hepatitis B in a high prevalence New Zealand Population a mathematical model applied to infection control policy, J. Theor. Biol. 254 (2008), pp. 599-603. doi: 10.1016/j.jtbi.2008.06.022[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1400.92543
[12] P. van den Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29-48. doi: 10.1016/S0025-5564(02)00108-6[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1015.92036
[13] P. Van Den Driessche, J. Watmough, Mathematical Epidemiology, Springer-Verlag, Victoria, BC, 2008. [Crossref], [Google Scholar] · Zbl 1159.92034
[14] G.F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, P. Magal and S. Ruan, eds., Lecture Notes in Math. 1936, Springer-Verlag, Berlin, New York, 2008, pp. 1-49. [Google Scholar]
[15] WHO hepatitis B, Fact sheet No. 204, 2014; software available at http://www.who.int/meadiacenter/factsheet/fs204/en/index.html[Google Scholar]
[16] G. Zaman, Y.H. Kang, and I.H. Jung, Stability and optimal vaccination of an SIR epidemic model, BioSystems 93 (2008), pp. 240-249. doi: 10.1016/j.biosystems.2008.05.004[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[17] G. Zaman, Y.H. Kang, and I.H. Jung, Optimal treatment of an SIR epidemic model with time delay, BioSystems 98 (2009), pp. 43-50. doi: 10.1016/j.biosystems.2009.05.006[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[18] S.J. Zhao, Z.Y. Xu, and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int. J. Epidemol. 29(4) (2000), pp. 744-752. doi: 10.1093/ije/29.4.744[Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[19] L. Zou, W. Zhang, and S. Ruan, Modeling the transmission dynamics and control of Hepatitis B virus in Chain, J. Theor. Biol. 262 (2010), pp. 330-338. doi: 10.1016/j.jtbi.2009.09.035[Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1403.92316
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