American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2013, 1(4), 71-75
DOI: 10.12691/ajams-1-4-4
Open AccessArticle

On the Homotopy Analysis Method for an Seir Tuberculosis Model

M.O. Ibrahim1 and S.A. Egbetade2,

1Department of Mathematics, University of Ilorin, Ilorin, Nigeria

2Department of Mathematics & Statistics, The Polytechnic, Ibadan, Nigeria

Pub. Date: September 21, 2013

Cite this paper:
M.O. Ibrahim and S.A. Egbetade. On the Homotopy Analysis Method for an Seir Tuberculosis Model. American Journal of Applied Mathematics and Statistics. 2013; 1(4):71-75. doi: 10.12691/ajams-1-4-4


In this paper, we provide a very accurate, non-perturbative, semi-analytical solution to a system of nonlinear first-order differential equations modeling the transmission of tuberculosis (TB) in a homogeneous population. Our analysis is based on Homotopy Analysis Method (HAM). Maple 15 software is used to carry out the computations. Our results show the validity and potential of HAM for computing the solution of nonlinear equations.

uberculosis homotopy analysis method series solution nonlinear equations mathematical model

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[1]  Blower, S.M., McLean, A.R., Porco, T.C., Small, P.M., Hopewell, P.C., Sanchez, M.A. and Moss, A.R., “ The intrinsic transmission dynamics of tuberculosis epidemics,” Nat. Med., 1(8). 815-821. 1995.
[2]  Song, B., Castillo-Chavez, C. and Aparicio, J-P., “Tuberculosis with fast and slow dynamics: the role of close and casual contact,” Math. Biosci., 180. 187-205. 2002.
[3]  Egbetade, S.A. and Ibrahim, M.O., “Global stability results for a tuberculosis epidemic model,” Res. J. Maths & Stat., 4(1). 14-20. 2012.
[4]  Colijn, C., Cohen, T. and Murray, M. (2006) Mathematical models of tuberculosis: accomplishments and future challenges. Proc. Natl. Acad. USA 103(18), 1-28.
[5]  World Health Organisation . Tuberculosis factsheet. 2010.
[6]  Ibrahim, M.O., Ejieji, C.N. and Egbetade, S.A., “A mathematical model for the epidemiology of tuberculosis with estimate of the basic reproduction number,” IOSR J. of Maths., 5(5). 46-52. 2013.
[7]  World Health Organisation. Tuberculosis control. 2009. http://who/globalreport2009/pdf.
[8]  World Health Organisation. WHO global TB
[9]  Egbetade, S.A., Ibrahim, M.O. and Ejieji, C.N., “On existence of a vaccination model of tuberculosis disease pandemic,” Int. J. Engrg. and Sc., 2(7). 41-44. 2013.
[10]  Waaler, H., Geser, A. and Anderson, S., “The use of mathematical models in the study of the epidemiology of tuberculosis,” Am. J. Public Health, 52. 1002-1013.1062.
[11]  Castillo-Chavez, C. and Feng, Z., “To treat or not to treat: the case of tuberculosis,” J. Math. Biol., 35(6). 629-656. 1997.
[12]  Aparicio, J.P., Capurro, A.F. and Castillo-Chavez, C., “Transmission and dynamics of tuberculosis on generalized households,” J. Theor. Biol., 206. 327-341. 2000.
[13]  Dye, C., “Global epidemiology of tuberculosis,” Lancet, 367(9514). 938-940.2006.
[14]  Cohen, T., Colijn, C., Finklea, B. and Murray, M. “Exogeneous re-infection and the dynamics of tuberculosis epidemics: local effects in a network model of transmission,” J.R. Soc. Interface, 4(14). 523-531.2007.
[15]  Roeger, L.W., Feng, Z. and Castillo-Chavez, C. “Modeling TB and HIV co-infections,” Mathematical Biosciences and Engineering, 6(4). 815-837. 2009.
[16]  White, P.J. and Garnett, G.P., “Mathematical modeling of the epidemiology of tuberculosis,” In: Michael, E., Speer, R.C. eds. Advances in Experimental Medicines and Biology Vol. 673, Modeling Parasite Transformation and Control. NY:Springer+Business Media, LLC Landes Biosciences, 127-140.
[17]  Oxlade, O., Sterling, T.R. and Schwartzman, K., “Developing a tuberculosis transmission model that accounts for changes in population health,” Medical Decision Making, 31. 53-68. 2011.
[18]  Brauer, F., “Basic ideas of mathematical epidemiology. Mathematical approaches for energy and re-emerging infectious diseases: Models, Methods and Theory,” (eds. Castillo-Chavez, C., Blower, S., van der Driessche, P., Kirschner, D. and Yakubu, A.A.). Berlin, Springer-Verlag.
[19]  Liao, S.J., The proposed homotopy analysis method for the solutions of nonlinear problems. Ph.D. Thesis. Shanghai Jiao Tong University, Shanghai, China.1992.
[20]  Alexander, J.L. and York, J.A., “The homotopy continuation method: numerically implementable topological procedures,” Trans Am. Math. Soc., 242. 271-284. 1978.
[21]  Liao, S.J., “A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics,” Int. J. Nonlinear Mech., 32(5). 815-822. 1997.
[22]  Liao, S.J. and Chwang, A.T., “Application of homotopy analysis method in nonlinear oscillations,” Trans. ASME J. Appl. Mech., 65. 914-922. 1998.
[23]  Liao, S.J., “An explicit totally analytic approximate solution for Blasius viscous flow problems,” Int. J. Nonlinear Mechanics, 34. 759-778. 1999.
[24]  Liao, S.J., “Analytic approximation of the drag coefficient for the viscous flow past a sphere,” Int. J. Nonlinear Mech., 37, 1-18.2002.
[25]  Liao, S.J.2003 Beyond Perturbation: Introduction to the homotopy analysis method. Chapman and Hall, CRC Press, Boca Raton.
[26]  Liao, S.J., “On the analytic solution of magnetohydrodynamic flow of non-Newtonian fluids over a stretching sheets,” J. Fluid Mech., 488, 189-212. 2003.
[27]  Liao, S.J. and Magyari, E., “Exponentially decaying boundary layers and limiting cases of families of algebraically decaying ones,” ZAMP, 57(5). 777-792. 2006.
[28]  Hayat, T., Khan, M. and Asghar, S., “Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid,” Acta. Mech., 168. 213-232. 2004.
[29]  Zhu, S.P., “An exact explicit solution for the evaluation of American put options,” Quantitative Finance, 6. 229-242. 2006.
[30]  Zhu, S.P., “A closed form analytical solution for the evaluation of convertible bonds with constant dividend yield,” Anzian J., 47, 477-494. 2006.
[31]  Song, H. and Tao, L., “Homotopy analysis of 1D unsteady nonlinear ground water flow through porous media,” J. Coastal Res., 50. 292-295. 2007.
[32]  Abbasbandy, S., “Application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Phys. Lett. A, 361. 478-483. 2007.
[33]  Awawdeh, F., Adawi, A. and Mustafa, Z., “Solutions of the SIR models of epidemics using HAM,” Chaos, Solitons and Fractals, 42. 3047-3052. 2009.
[34]  Matinfar, M. and Saeidy, M., “Application of homotopy analysis method to fourth order parabolic partial differential equations,” Applications and Applied Mathematics, 5(9).70-80. 2010.
[35]  Li, Y., Nohara, B.T. and Liao, S.J., “Series solutions of coupled Van der Pol equation by means of homotopy analysis method,” Journal of Mathematical Physics, 51. 063517. 2010.
[36]  Hassan, H.N. and El-Tawil, M.A., “A new technique of using homotopy analysis method for solving high-order nonlinear differential equations,” Mathematical methods in Applied Science, 34. 728-742. 2011.
[37]  Arafa, A.A.M., Rida, S.Z. and Khalil, M., “Solutions of fractional order model of childhood diseases with constant variation strategy,” Math. Sc. Lett., 1(1). 17-23. 2012.
[38]  Vahdati, S., Tavassoli, K.M. and Ghasemi, M., “Application of homotopy analysis method to SIR epidemic model,” Research Journal of Recent Sciences, 2(1). 91-96. 2013.
[39]  Lyapunov, A.M., “General problems on stability of motion,” Taylor and Francis, London.
[40]  Awrejcewicz, J., Andrianov, I.V. and Manevitch, L.I., “Asymptotic approaches in nonlinear dynamics,” Springer-Verlag, Berlin.
[41]  Adomian, G., “A review of the decomposition method and some recent results for nonlinear equations,” Comp. Math. Appl., 21. 101-127. 1991.
[42]  He, J.H., “Homotopy perturbation techniques,” Comput. Methods Appl. Mech. Engrg., 178. 257-262. 1999.
[43]  Sajid, M. and Hayat, T., “Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations,” Nonlinear Anal.: Real World Appl., 9. 2296-2301. 2008.
[44]  Liang, S.X. and Jeffery, D.J., “Comparison of homotopy analysis method and homotopy perturbation method through an evaluation equation,” Comm. Nonlinear Sci. Numer. Simul., 14. 4057-4064. 2009.
[45]  Egbetade, S.A. and Ibrahim, M.O., “Stability analysis of equilibrium states of an SEIR tuberculosis model,” Journal of the Nigerian Association of Mathematical Physics, 20. 119-124. 2012.
[46]  Maplesoft 15, Waterloo Maple Inc. Ontario, Canada. 2011.