A Mathematical Model for Control and Elimination of the Transmission Dynamics of Measles
Applied and Computational Mathematics
Volume 4, Issue 6, December 2015, Pages: 396-408
Received: Aug. 10, 2015;
Accepted: Sep. 7, 2015;
Published: Sep. 29, 2015
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Stephen Edward, Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma (UDOM), Dodoma, Tanzania
Kitengeso Raymond E., Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma (UDOM), Dodoma, Tanzania
Kiria Gabriel T., Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma (UDOM), Dodoma, Tanzania
Felician Nestory, Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma (UDOM), Dodoma, Tanzania
Mwema Godfrey G., Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma (UDOM), Dodoma, Tanzania
Mafarasa Arbogast P., Department of Mathematics, College of Natural and Mathematical Sciences, University of Dodoma (UDOM), Dodoma, Tanzania
Despite the availability of measles vaccine since 1963, the infectious disease is still endemic in many parts of the world including developed nations. Elimination of measles requires maintaining the effective reproduction number less than unity, Re <1 as well as achieving low levels of susceptibility. Infectious diseases are great field for mathematical modeling, and for connecting mathematical models to primary or secondary data. In this project, we concentrated on the mathematical model for control and elimination of transmission dynamics of measles. We have obtained disease free equilibrium (DFE) point, effective reproduction number and basic reproduction number for the model. Simulations of different variables of the model have been performed and sensitivity analysis of different embedded parameters has been done. MATLAB has been used in simulations of the ordinary differential equations (ODEs) as well as the reproduction numbers.
Kitengeso Raymond E.,
Kiria Gabriel T.,
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.
A. Mazer, Sankale, Guide de medicine en Afrique et Ocean Indien, EDICEF, Paris (1988).
M. B. A. Oldstone, Viruses, Plagues, and History, Oxford University Press, New York, USA (1998), 73-89.
Department of Health Statistics and Informatics. WHO mortality Database .Geneva: World Health Organization, 2011.
WHO vaccine-preventable diseases: monitoring system. Geneva: World Health Organization, 2010.
Burton A, Gacic-Dobo M, Karimov R, Kowalski R. A computational logic-based representation of the WHO and UNICEF estimates of national immunization coverage. http://www.doc.ic.ac.uk/~rak/papers/wuenic.pdf (accessed April 14, 2012).
Harpaz R. Completeness of measles case reporting: review of estimates for the United States. J Infect Dis 2004; 189 (suppl 1): S185–90.
Catherine Comiskey B A (Mod). (1988) A mathematical Model for Measles Epidemics in Ireland. Msc Thesis. National Institute of Higher Education.
Ministry of Health, Manatu Hauora. (1998) Predicting and Preventing Measles Epidemics in New Zealand: Application of Mathematical Model.
M. G. Roberts and M. I. Tobias (2000). Predicting and Preventing measles epidemics in New Zealand: Application of Mathematical Model. Cambridge University Press, Vol 124,pp. 279-287.
C. R. Mac Intyre, N. J. Gay, H. F. Gidding, B. P. Hull, G. L. Gilbert and P. B. McIntyre (2002). A Mathematical model to the impact of the measles control campaign on the potential for measles transmission in Australia. International Journal of Infectious Diseases, Vol 6,pp. 277-282.
J. Mossong and C. P. Muller. (2003) Modeling measles re-emergence as a result of waning of immunity in vaccinated populations. Science direct, Vaccine 21, pp. 4597-4603.
O. M. Tessa. (2006) Mathematical Model for control of measles by vaccination. Abdou Moumouni University, Niamey, Niger, pp. 31-36.
E. A. Bakare, Y. A. Adekunle and K. O. Kadiri. (2012) Modelling and Simulation of the Dynamics of the Transmission of Measles. International Jounal of Computer Trends and Technology, Vol 3, pp. 174-178.
S. O. Siabouh and I. A. Adetunde. (2013) Mathematical Model for the study of measles in Cape Coast Metropolis. International Journal of Mordern Biology and Medicine, Vol 4(2), pp. 110-133.
A. A. Momoh, M. O. Ibrahim, I. J. Uwanta, S. B. Manga. (2013) Mathematical Model for control of Measles Epidemiology. International Journal of Pure and Applied Mathematics, Vol 67, pp. 707-718.
A. A. Momoh, M. O. Ibrahim, I. J. Uwanta, S. B. Manga. (2013) Modelling the effects of vaccination on the transmission dynamics of measles. International Journal of Pure and Applied Mathematics, Vol 88, pp. 381-390.
M. O. Fred, J. K. Sigey, J. A. Okello, J. M. Okwyo and G. J. Kang’ ethe. (2014) Mathematical Modelling 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.
J. M. Ochoche and R. I. Gweryina. (2014) A Mathematical Model of Measles with Vaccination and Two Phases of Infectiousness. IOSR Jounal of Mathematics, Vol 10, pp. 95-105.
G. Bolarian. (2014) On the Dynamical Analysis of a new Model for Measles Infection. International Journal of Mathematics Trends and Technology, Vol 7, pp. 144-155.
Verguet S, et al. Controlling measles using supplemental immunization activities: A mathematical model to inform optimal policy. Vaccine (2015), http://dx.doi.org/10.1016/j.vaccine.2014.11.050.
Van Den Driessche, P and Watmough, J (2002). Reproduction numbers and Sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, Vol 180, pp. 29-48.
Schenzle D. (1984). An age-structured model of pre- and post-vaccination measles transmission. Math Med Biol Vol 1(2), pp.169–191.
R. F. Grais, M. J. Ferrari, C. Dubray, O. N. Bjørnstad, B. T. Grenfell, A. Djibo, F. Fermon, P. J. Guerin.(2006) Estimating transmission intensity for a measles epidemic in Niamey, Niger: lessons for intervention. Royal society of tropical medicine and hygiene,Vol100,pp. 867 – 873.
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts. (2009) The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface, (doi:10.1098/rsif.2009.0386).
Anes Tawhir. (2012). Modelling and Control of Measles Transmission in Ghana. Master of Philosophy thesis. Kwame Nkrumah University of Science and Technology.
Stephen, E., Dmitry K and Silas M.(2014). Modeling and Stability Analysis for a Varicella Zoster Virus Model with Vaccination. Applied and Computational Mathematics. Vol. 3, No. 4, pp. 150-162.doi: 10.11648/j.acm.20140304.16.
Stephen, E., Dmitry K and Silas M.(2014). Modeling the Impact of Immunization on the epidemiology of Varicella Zoster Virus. Mathematical theory and Modeling.Vol.4, No. 8, pp.46-56.
Stephen, E. and N. Nyerere. (2015). A Mathematical Model for the Dynamics of Cholera with Control Measures. Applied and Computational Mathematics. Vol.4, No.2, pp.53-63. doi: 10.11648/j.acm.20150402.14.
N. Nyerere, L. S. Luboobi and Y. Nkansah-Gyekye. Modeling the Effect of Screening and Treatment on the Transmission of Tuberculosis Infections. Mathematical theory and Modeling. Vol.4, No. 7, 2014, pp.51-62.
N. Nyerere, L. S. Luboobi and Y. Nkansah-Gyekye. Bifurcation and Stability analysis of the dynamics of Tuberculosis model incorporating, vaccination, Screening and treatment, Communications in Mathematical biology and Neuroscience, Vol 2014(2014), Article ID 3.
Chitnis, N., Hyman, J. M., and Cusching, J. M. (2008). Determining important Parameters in the spread of malaria through the sensitivity analysis of a mathematical Model. Bulletin of Mathematical Biology 70(5):1272–12.