Optimal Control Problem of Vacination for The Spread of Measles Diseases Model
Main Article Content
Abstract
Measles is a disease in humans that is very contagious. Before the vaccine was known, the incidence of measles was very high, even the measles mortality rate reached 2.6 million every year. With the introduction of vaccines, the mortality rate in 2000-2016 can be reduced to 20.4 million deaths. Therefore, vaccination programs are very useful in reducing the incidence of measles. Unfortunately, we cannot know the optimal conditions for administering vaccines. The study of optimal control analysis of vaccination is needed in optimizing the prevention of the spread of measles. In this paper, a mathematical model which is a third-order differential equation system is constructed based on characteristic information on measles. The existence and locally stability of the equilibrium point are analyzed here. In addition, optimal control of the vaccination program also occurred. The results of our analysis suggest that the incidence of measles can decrease as the effectiveness of vaccination increases. But the effectiveness of vaccination is directly proportional to the costs incurred. If the cost incurred for the vaccination program more significant, the incidence of measles will decrease.
Article Details
Section
Applied Mathematics
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References
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[4] W. Orenstein, A. Hinman, B. Nkowane, J. Olive, and A. Reingold, œMeasles and Rubella Global Strategic Plan 20122020 midterm review, Vaccine, vol. 36, pp. A1A34, 2018.
[5] N. S. Lisa Indrian Dini, Pandu Riono, œKementerian kesehatan republik indonesia, Jurnal Kesehatan Reproduksi, vol. 7, no. April, pp. 119133, 2016.
[6] R. G. Halim, œCampak pada Anak, KALBE MEDICAL PORTAL, vol. 43, pp. 186189, 2016.
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[7] E. O. Oghre and I. I. Ako, œA mathematical model for measles disease, Far East Journal of Mathematical Sciences, vol. 54, no. 1, pp. 4763, 2011.
[8] O. Peter, O. Afolabi, A. Victor, C. Akpan, and F. Oguntolu, œMathematical model for the control of measles, Journal of Applied Sciences and Environmental Management, vol. 22, no. 4, p. 571, 2018.
[9] A. Momoh, M. Ibrahim, I. Uwanta, and S. Manga, œMATHEMATICAL MODEL FOR CONTROL OF MEASLES EPIDEMIOLOGY, International Journal of Pure and Applied Mathematics, vol. 87, no. 5, pp. 707718, 2013.
[10] Z. Bai and D. Liu, œModeling seasonal measles transmission in China, Communications in Nonlinear Science and Numerical Simulation, vol. 25, no. 1-3, pp. 1926, 2015.
[11] D. Suandi, œAnalisis Dinamik Model Penyebaran Penyakit, Kubik, vol. 2, no. 2, pp. 110, 2017.
[12] K. Dietz and J. Heesterbeek, œThe Basic Reproduction Ratio for Sexually Transmitted Diseases: I. Theoretical nldots, Mathematical Biosciences, 1991.
[13] L. Matthews, M. E. J. Woolhouse, and N. Hunter, œThe basic reproduction number for scrapie The basic reproduction number for scrapie The basic reproduction number for scrapie The basic reproduction number for scrapie, Proc. R. Soc. Lond. B, vol. 266, no. 1423, pp. 10851090, 1999.
[14] L. C. U. o. B. Evans, œAn Introduction to Mathematical Optimal Control Theory, Environment and Planning C Government and Policy, vol. 4, no. 2, pp. 128, 2006.
[15] L. Pontryagin, œOptimal regulation processes, American Mathematical Society Translations, pp. 125145, 1961.