OPTIMAL CONTROL ANALYSIS OF THE DYNAMICAL SPREAD OF MEASLES

Adewale; Olopade; Ajao; Adeniran

doi:10.29121/granthaalayah.v4.i5.2016.2692

Authors

S. O. Adewale Department of Pure and Applied Mathematics, LadokeAkintola University of Technology (LAUTECH), Ogbomoso, NIGERIA
I. A. Olopade Department of Mathematics and Computer Science, Elizade University, Ilara-Mokin, NIGERIA
S. O. Ajao Department of Pure and Applied Mathematics, LadokeAkintola University of Technology (LAUTECH), Ogbomoso, NIGERIA
G. A. Adeniran Department of Pure and Applied Mathematics, LadokeAkintola University of Technology (LAUTECH), Ogbomoso, NIGERIA

DOI:

https://doi.org/10.29121/granthaalayah.v4.i5.2016.2692

Keywords:

Measles, Reproduction Number, Optimal Control, Vaccination, Isolation, Epidemic

Abstract [English]

In this paper, a five (5) compartmental model is presented to study the transmission dynamics of Measles in a population at any point in time. The model is rigorously analyzed to gain insight into the dynamical features of Measles and also, optimal control theory is applied to give an optimality system which we used to minimize the number of infected individuals and propose the most suitable control strategy for the spread of measles. It is shown that the model has a diseases free equilibrium which is globally asymptotically stable (GAS). Also, there exists a unique endemic equilibrium point which is locally stable whenever the associated threshold quantity exceeds (one) unity. We also show that there exists a solution for the optimality system. From the result, it was observed that vaccine control strategy is more efficient in reducing the number of infected individuals as compared to other control strategies.

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References

Adeoye I.A, Dairo M.D, Adekunle L.V, Adedokun H.O, and Makanjuola J. (2010): Investigation of a measles outbreak in a Rural Nigerian community – The Aladura experience. African Journal of Microbiology Research Vol. 4(5), pp. 360-366.

Adewale S.O, Podder C.N, and Gumel A.B (2009): Mathematical Analysis of a TB Transmission Model with DOTS. Canadian Applied Mathematical Quarterly Volume 17, number 1, Spring 2009.

Adewale S.O, Olanrewaju P.O, Taiwo S.S, Anake T.A and Famewo,M.M (2012) Mathematical Analysis of the effect of Immunization on the dynamical spread of Measles. International Electronic Journal of Pure & Applied Maths.

Adewale S.O, Mohammed I.T and Olopade I.A (2014) Mathematical Analysis of Effect of Area on the Dynamical Spread of Measles. International Organization of Scientific Research. DOI: https://doi.org/10.9790/3021-04324357

Agusto F.B, Marcus N, Okosun K.O (2012): Application of Optimal Control To The Epidemiology of Malaria. Electronic Jour. Of Diff. Eqns, Vol 81, Pp 1-22

Annual Report of the National Disease Surveillance Centre,2000. Dublin, Health Protection Surveillance Centre; 2001. ISSN:1649-0436. Available from: http://www.hpsc.ie/hpsc/AboutHPSC/AnnualReports/File,520,en.pdf.

Ca´ceres VM, Strebel PM, Sutter RW (2000): Factors determining prevalence of maternal antibody to measles virus throughout infancy: a review. Clin Infect Dis 2000;31: 110–19. DOI: https://doi.org/10.1086/313926

Carabin H, Edmunds WJ, Kou U, van den Hof S & Nguyen VH (2002). The average cost of measles cases and adverse events following vaccination in industrialised countries. DOI: https://doi.org/10.1186/1471-2458-2-22

BMC Public Health 2:22 (http://www.biomedcentral.com/1471-2458/2/22).

Driessche P. van den and Watmough J.(2002): Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math.Biosci. 180 (2002), 29{48}. DOI: https://doi.org/10.1016/S0025-5564(02)00108-6

Fleming W, and RishelR(1975): Deterministic and Stochastic Optimal Control. Springer Verlag, New York DOI: https://doi.org/10.1007/978-1-4612-6380-7

Grais RF, Dubray C, Gerstl S (2007): Unacceptably high mortality related to measles epidemics in Niger, Nigeria, and Chad. PLoS Med 2007;4:e16. DOI: https://doi.org/10.1371/journal.pmed.0040016

Grais R.F, Ferrari M.J, Dubray C, Bjornstad O.N, Grenfell B.T, Djibo A, Fermon, F, Guerin P.J (2006): Estimating Transmission intensity for a measles epidemic in Niamey, Niger: Lessons for intervention. Transactions of Royal Society of Tropical Medicine and Hygiene (2006)100,867-873.

Hattaf. K, and Yousfi. N (2011): Dynamics of HIV infection Model with Therapy and cure rate. International Journal of Tomography and Statistics, 16(11), 74-80.

Hattaf. K, and Yousfi. N (2012): Two Optimal Treatments of HIV Infection Model. World Journal of Modelling and Simulation, Vol. 8(2012), No.1, Pp. 27-35

Hethcote H.W. and Waltman P. (1973):Optimal vaccination schedules in a deterministic epidemic model.Math. Biosci. 18 (1973), 365-382. DOI: https://doi.org/10.1016/0025-5564(73)90011-4

Hethcote H. W. andThieme H. R.(1985): Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci. 75 (1985), 205{227}. DOI: https://doi.org/10.1016/0025-5564(85)90038-0

Hethcote H.W. (1989):Optimal ages for vaccination for measles. Math. Biosci. 89 (1989), 29 -52.

Hethcote H.W., (2000): The mathematics of infectious diseases. Society for Industrial and Applied Mathematics Siam Review, Vol. 42, No. 4, pp. 599–653. DOI: https://doi.org/10.1137/S0036144500371907

Joshi, H. R. (2002): Optimal Control of an HIV Immunology Model, Optim. Control Appl. Math, 23(2002), 199-213. DOI: https://doi.org/10.1002/oca.710

Lakshmikantham V, Leela S, and Martynyuk A. A. (1989). Stability analysis of nonlinear systems.Marcel Dekker, Inc., New York and Basel.

Lara J Wolfson, Rebecca F Grais,Francisco J Luquero, Maureen E Birmingham and Peter M Strebel (2009): Estimates of measles case fatality ratios: a comprehensive review of community-based studies. International Journal of Epidemiology 2009;38:192–205doi:10.1093/ije/dyn224. DOI: https://doi.org/10.1093/ije/dyn224

Lukes,D.L (1982): Differential Equation: Classical to Controlled. Academic Press, New york

Ousmane MOUSSA TESSA (2006): Mathematical model for control of measles by vaccination.MSAS Symposium 2006.

Pontryagin, L.S; Boltyanski, V.G; Gamkrelidze, R.V; Mishchenko, E.F (1962): The Mathematical Theory of optimal processes, Wiley, New York, 1962.

Ramsay M, Gay N, Miller E, White J, Morgan-Capner P & Brown D (1994). The epidemiology of measles in England and Wales: Rationale for the 1994 national vaccination.

Roberts M.G and Tobias M.I (2000): Predicting and Preventing Measles epidemics in New Zealand: application of a mathematical model. Epidemiol. Infect. (124) 279-287.

Spotlight on Measles 2010: Measles outbreak in Ireland 2009-2010. DOI: https://doi.org/10.5339/qmj.2010.2.12

BMC Public Health 2:22 (http://www.biomedcentral.com/1471-2458/2/22)

Strebel P, Cochi S, Grabowsky M (2003): The unfinished measles immunization agenda. J Infect Dis 2003;187(Suppl 1):S1–S7. DOI: https://doi.org/10.1086/368226

World Health Organization, (2004): Measles vaccines: WHO position paper. Wkly. Epidemiol. Rec. 79, 130—142.

World Health Organization (2009): Measles Fact Sheet No 286. Geneva Switzerland: www,who,int/mediacentre/factsheets/fs286/en.

World Health Organization(2011): WHO Vaccine Preventable Diseases: Monitoring System2011 Global Summary, available at:http://www.who.int/immunization_monitoring/en/globalsummary/timeseries/tsincidencemea.htm.