Mathematical Modeling and Simulation Study for the Control and Transmission Dynamics of Measles
American Journal of Applied Mathematics
Volume 5, Issue 4, August 2017, Pages: 99-107
Received: Apr. 15, 2017;
Accepted: May 2, 2017;
Published: Jul. 6, 2017
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Selam Nigusie Mitku, School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia
Purnachandra Rao Koya, School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia
In this paper, a five compartmental model has been considered and investigated the transmission dynamics of measles disease in the human populations. The only one infected compartment in the standard model has been split into two: Infected catarrh, and infected eruption. Measles is a deadly disease that is very common and contagious in the world. However, if enough care is taken one can survive easily against Measles disease. The Measles disease has no specific treatment but vaccination is available. It has been shown that the model has a positive solution and is bounded. The basic reproduction number is derived using the next generation matrix method. The disease free equilibrium point is found and endemic equilibrium state is identified. It is shown that the disease free equilibrium point is locally and globally asymptotically stable if the reproduction number takes a value less than one unit and unstable if it is more than one unit. Numerical simulation study is conducted using ode 45 of MATLAB. The results and interpretations are elaborated and included in the text. Description of the model, Mathematical analysis, stability analysis, and simulation studies are conducted and the results are included. The standard model and the proposed models have been compared and the observations are presented in a tabular form.
Selam Nigusie Mitku,
Purnachandra Rao Koya,
Mathematical Modeling and Simulation Study for the Control and Transmission Dynamics of Measles, American Journal of Applied Mathematics.
Vol. 5, No. 4,
2017, pp. 99-107.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A. Mazer Sankale. Guide de medicine en Afrique ET Ocean Indien, EDICEF, Paris 1988.
Daley D. J. and Gani J. 2005. Epidemic Modeling and Introduction, New York: Cambridge University Press.
N. F. Britton 2003. Essential Mathematical Biology, parts of Chapter 3: Infectious Diseases.
Dancho Desaleng, Purnachandra Rao Koya. The Role of Polluted Air and Population Density in the Spread of Mycobacterium Tuberculosis Disease. Journal of Multidisciplinary Engineering Science and Technology JMEST. Vol. 2, Issue 5, May 2015, http://www.jmest.org/wp-content/uploads/JMESTN42350782.pdf
A. A. Momoh, M. O. Ibrahim, I. J. Uwanta, S. B. Manga. Mathematical modeling for control of measles epidemiology. International Journal of Pure and Applied Mathematics Volume 87, No. 5, 2013, 707-718.
Stephen Edward, Kitengeso Raymond E., Kiria Gabriel T., Felician Nestory, Mwema Godfrey G., Mafarasa Arbogast P. A Mathematical Model for Control and Elimination of the Transmission Dynamics of Measles. Applied and Computational Mathematics. Vol. 4, No. 6, 2015, pp. 396-408. doi: 10.11648/j.acm.20150406.12
Demsis Dejene, Purnachandra Rao Koya. Population Dynamics of Dogs Subjected To Rabies Disease. Journal of Mathematics IOSR-JM, e-ISSN: 2278-5728; p-ISSN: 2319-765X. Volume 12, Issue 3, Version IV, May – June 2016, Pp 110-120 www.iosrjournals.org
Tadele Degefa Bedada, Mihretu Nigatu Lemma and Purnachandra Rao Koya. Mathematical Modeling and simulation study of Influenza disease, Journal of Multidisciplinary Engineering Science and Technology JMEST. Vol. 2, Issue 11, November 2015, Pp 3263 – 69. ISSN: 3159-0040. http://www.jmest.org/wp-content/uploads/JMESTN42351208.pdf
N. Chitins, J. Hyman and J. Cushing. Determining important parameters in the spread of malaria through the sensitivity analysis of a malaria model, Bull. Math. Biology, 70, 2008, 1272 - 1296.
Arino J., Brauer F., Van den Driessche P., Watmough J., and Wu J. 2008. A model for influenza with vaccination and antiviral treatment, Journal of Theoretical Biology, 253, 118 -130.
Anes Tawhir 2012. Modeling and Control of Measles Transmission in Ghana, Master of Philosophy thesis, Kwame Nkrumah University of Science and Technology.
M. O. Fred, J. K. Sigey, J. A. Okello, J. M. Okwyo and G. J. Kang'ethe 2014. Mathematical Modeling on the Control of Measles by Vaccination: Case Study of KISII Country, Kenya. The SIJ Transactions on Computer Science Engineering and Its Applications CSEA, Vol 2, pp. 61-69.
Arino J., Brauer F., van den Driessche P., Watmough J., and Wu J. 2006. Simple models for containment of a pandemic, Journal of the Royal Society Interface, 3, 453 -457.
N. R. Derrick, S. L. Grossman, Differential Equation with Applications, Addison Wesley Publishing Company, Inc. Philippines 1976.