Dynamical Features and Vaccination Strategies in an SEIR Epidemic Model

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Dynamical Features and Vaccination Strategies in an SEIR Epidemic Model

Author Affiliations

¹ Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, PAKISTAN
² Department of Mathematics, University of Malakand, Chakdara, Dir, Khyber Pakhtunkhwa, PAKISTAN

Res. J. Recent Sci., Volume 2, Issue (10), Pages 48-56, October,2 (2013)

Abstract

An epidemic model with a vaccination program is investigated in this paper. The vaccine induced reproduction number R₀(k) is determined and the impact of vaccination in reducing R₀(k) is discussed. The local and global stabilities of both the disease-free and endemic equilibrium are derived. A control problem is formulated to control the disease by using an optimal control theory. Numerical simulations and optimal analysis of the model show that proper use of control measures can significantly decrease the number of infected humans.

References

Kar T.K., Batabyal A., Stability analysis and optmal control of an SIR epidemic model with vaccination, Biosystems 104, 127-135 (2011)
Ullah R., Zaman G., Islam S., Global dynamics of avianhuman influenza with horizontal transmission in human population, Life Sci. J. 9, 5747-5753 (2012)
Zaman G., Kang Y.H. and Jung I.H., Stability analysis and optimal vaccination of an SIR epidemic models, BioSystems 93, 240-249 (2008)
Riedel S., Edward Jenner and the history of smallpox and vaccination, Proceedings, 18, 21–25 (2005)
Shulgin B., Stone L., Agur Z., Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol. 60, 1123-1148 (1998)
Kribs-Zaleta C.M., Velasco-Hernandez J.X., A simple vaccination model with multiple endemic states, Math. Biosci., 164, 183-201 (2000)
Farrington C.P., On vaccine efficacy and reproduction numbers, Math. Biosci., 185, 89-109 (2003)
Lasalle J.P., The Stability of Dynamical Systems, SIAM, Philadelphia PA (1976)
Ullah R., Zaman G., Islam S., Prevention of influenza pandemic by multiple control strategies, J. Appl. Math., vol 2012, Article ID 294275, 14 pages, doi:10.1155/2012/294275 (2012)
Zaman G., Khan M.A., Islam S., Jung I.H., Chohan M.I., Modeling dynamical interactions between leptospirosis infected vector and human population, Appl. Math. Sci. 26, 1287-1302 (2012)
Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F., The Mathematical Theory of Optimal Processes, Wiley, Hoboken, NJ, USA, (1962)

[ref1] Kar T.K., Batabyal A., Stability analysis and optmal control of an SIR epidemic model with vaccination, Biosystems 104, 127-135 (2011)

[ref2] Ullah R., Zaman G., Islam S., Global dynamics of avianhuman influenza with horizontal transmission in human population, Life Sci. J. 9, 5747-5753 (2012)

[ref3] Zaman G., Kang Y.H. and Jung I.H., Stability analysis and optimal vaccination of an SIR epidemic models, BioSystems 93, 240-249 (2008)

[ref4] Riedel S., Edward Jenner and the history of smallpox and vaccination, Proceedings, 18, 21–25 (2005)

[ref5] Shulgin B., Stone L., Agur Z., Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol. 60, 1123-1148 (1998)

[ref6] Kribs-Zaleta C.M., Velasco-Hernandez J.X., A simple vaccination model with multiple endemic states, Math. Biosci., 164, 183-201 (2000)

[ref7] Farrington C.P., On vaccine efficacy and reproduction numbers, Math. Biosci., 185, 89-109 (2003)

[ref8] Lasalle J.P., The Stability of Dynamical Systems, SIAM, Philadelphia PA (1976)

[ref9] Ullah R., Zaman G., Islam S., Prevention of influenza pandemic by multiple control strategies, J. Appl. Math., vol 2012, Article ID 294275, 14 pages, doi:10.1155/2012/294275 (2012)

[ref10] Zaman G., Khan M.A., Islam S., Jung I.H., Chohan M.I., Modeling dynamical interactions between leptospirosis infected vector and human population, Appl. Math. Sci. 26, 1287-1302 (2012)

[ref11] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F., The Mathematical Theory of Optimal Processes, Wiley, Hoboken, NJ, USA, (1962)