Approximate Solution of Typhoid Fever Model by Variational Iteration Method

Adebisi A. F., Peter O. J., Ayoola T. A., Oguntolu F. A., Ishola C. Y.

Abstract


In this paper, a deterministic mathematical model involving the transmission dynamics of typhoid fever is presented and studied. Basic idea of the disease transmission using compartmental modeling is discussed. The aim of this paper is to apply Variational Iteration Method (VIM) to solve typhoid fever model for a given constant population. This mathematical model is described by nonlinear first order ordinary differential equations. First, we find the solution of the model by using Variation Iteration Method (VIM). The validity of the VIM in solving the model is established by classical fourth-order Runge-Kutta method (RK4) implemented in Maple 18. In order to show the efficiency of the method we compare the solutions obtained by VIM and RK4. We illustrated the profiles of the solutions of each of the compartments, from which we speculate that the VIM and RK4 solutions agreed well.


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Abbasbandy, E & Shivanian, E. (2009). Application of the variational iteration method for system of nonlinear Volterra’sintegro-differential equations, Mathematical and Comp. Applic., 14(2), 147-158.

Abdou, M . A & Soliman, A . A (2005). New application of Variational iteration method, Physica D211 (1-2):1-8

Abdou, M . A & Soliman, A . A (2005). Variational iteration method for solving Burger’s and coupled Burger’s equation,J.Comp.Appl.Math 181(2):245-251

Akinboro, F. S., Alao, S., & Akinpelu, F. O. (2014). Numerical solution of SIR model using differential transformation method and variational iteration method. General Mathematics Notes, 22(2), 82-92.

Batiha, B., Noorani, M. S. M. & Hashim, I. (2007). Application of variational iteration method to a general Riccati equation, International Mathematical Forum, 2(56), 2758 – 2770.

Gasem, M.H., Dolmans, W. M., Keuter, M. M., Djokomoeljanto, R. R. (2001). Poor food hygiene and housing as risk factors for typhoid fever in Semarang, Indonesia. Trop Med Int Health. 6(6): p. 484–90. PMID: 11422963

Hosoglu, S.,Aldemir, M., Akalin, S., Geyik, M. F. (2006). Risk factors for typhoid fever among adult patients in Diyarbakir, Turkey. EpidemiolInfect, 134(3): p. 612–6. PMID: 16288686

Kalajdzievska, D. “Modeling the Effects of Carriers on the Transmission Dynamics of Infectious Diseasesâ€. Math Biosci Eng, 8(3), 711–722

Kariuki, C. (2011). Characterization of Multidrug-Resistant Typhoid Outbreaks in Kenya. J.C.Micbol, 42(4), 1477-1482.

Lauria, D. T., Maskery, T., Poulos, C.,Whittington, D. (2009). An optimization model for reducing typhoid cases in developing countries without increasingpublic spending, Vaccine, JVAC-8805 http://dx.doi.org/10.1016/j.vaccine.2008.12.032

Lu, J. (2007). Variational iteration method for solving a nonlinear systemof second-order boundary value problems, Computer and Mathematics with Applications, 54, 1133 – 1138 .

Moatlhodl, K.& Gosaamang, K. R. (2017). Mathematical Analysis of Typhoid Infection with Treatment. Journal of Mathematical Sciences: Advances and Applications, 40, 75-91.

Muhammad, A. K., Muhammad, P. I., Saeed, K.. Ilyas, S. Sharidan, G. & Taza, M.(2015). Mathematical Analysis of Typhoid Model with Saturated Incidence Rate. Advanced Studies in Biology, 7( 2), 65 - 78.

Mushayabasa, S. (2011). Impact of vaccines on controlling typhoid. Journal of modern mathematics and Statistics, 5(2), 54-59.

Nthiiri, J. K.., Lawi, G. O., Akinyi, C. O., Oganga, D. O., Muriuki, W .C, Musyoka M. J, Otieno P. O. & Koech, L. (2016). Mathematical modelling of typhoid fever disease incorporating protection against infection. British Journal of mathematics and computer science, 14(1), 1-10.

Peter , O. J., Ibrahim , M. O., Akinduko, O. B., & Rabiu, M. (2017). Mathematical model for the control oftyphoid fever. IOSRJournal of Mathematics (IOSR-JM), 13(4), 60–66.

Peter, O. J, Afolabi, O. A, Oguntolu F. A, Ishola C. Y, Victor, A. A (2018).Solution of a Deterministic Mathematical Model of Typhoid Fever by Variational Iteration Method.Science World Journal Vol 13(No 2) 2018

Peter, O. J & Ibrahim, M. O. (2017). Application of Differential Transform Method in Solving a Typhoid Fever Model. International Journal of Mathematical analysis and Optimization. 1(1), 250-260.

Peter, O. J, Akinduko, O. B, Ishola C. Y, Afolabi, O. A. Series Solution of Typhoid Fever Model Using Differential Transform Method. Malaysian Journal of Computing, 3(1): 67-80, (2018).

Shanahan, F. M. (1998).Molecular analysis of and identification of antibiotic resistance- genes in clinical isolates of Salmonella typhi from India, J. C. Micbio. 36, 1595-1600.

Virginia, E. Pitzer, Cayley, C. Bowles, Stephen Baker, Gagandeep Kang, V., Jeremy, J., Farrar, T., Bryan, T., Grenfell, G. (2014). Predicting the Impact of Vaccination on the Transmission Dynamics of Typhoid in South Asia: .A Mathematical Modeling Study. PLoSNeglTrop Dis 8(1), 26-42. doi:10.1371/journal.pntd.0002642

Watson, C. H., & Edmunds, W. (2015). Review of typhoid fever transmission dynamic models and economic evaluations of vaccination, Vaccine 33, 42-54.http://dx.doi.org/10.1016/j.vaccine.2015.04.013

Vollaard A.M., et al., Risk factors for typhoid and paratyphoid fever in Jakarta, Indonesia. JAMA, 2004.291(21): p. 2607–15. PMID: 15173152

Wazwaz, A-M. (2007). the variational iteration method for solving linear and nonlinear systems of PDEs, Computer and Mathematics with Applications, 54, 895 – 902 .

Yuliyanto, D. & mungkasi, S. (2017). Variational iteration method for solving population dynamics model of two species, Journal of physics: Conference series, 2(21), 32 – 40.


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