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Application of Homotopy Analysis Method for Solving an SEIRS Epidemic Model
Mathematical Modelling and Applications
Volume 4, Issue 3, September 2019, Pages: 36-48
Received: Jun. 10, 2019; Accepted: Jul. 15, 2019; Published: Sep. 3, 2019
Authors
Inyama Simeon Chioma, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Ekeamadi Godsgift Ugonna, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Uwagboe Osazee Michael, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Omame Andrew, Department of Mathematics, Federal University of Technology, Owerri, Nigeria
Mbachu Hope Ifeyinwa, Department of Statistics, Imo State University, Owerri, Nigeria
Uwakwe Joy Ijeoma, Alvan Ikoku College of Education, Owerri, Nigeria
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Abstract
In this paper, we modified the model of [23] and then applied a new semi-analytic technique namely the Homotopy Analysis Method (HAM) in solving the SEIRS Epidemic Mathematical Model. The modified SEIRS model wasfirst formulated and adequately analyzed. We investigated the basic properties of the model by proving the positivity of the solutions and establishing the invariant region. We further obtained the steady states: disease-free equilibrium (DFE) and endemic equilibrium (EE), then we went further to determine the local stability of the DEF and EE using the basic reproduction number which was calculated. We also applied Lyaponuv method to prove the global stability of endemic equilibrium, The HAM was applied to obtain an accurate solution to the model in few iterations. Finally, a numerical solution (simulation) of the model was obtained using MAPLE 15 computation software.
Keywords
SEIRS Model, Homotopy Analysis Method (HAM), Local Stability, Disease-free Equilibrium, Endemic Equilibrium
Inyama Simeon Chioma, Ekeamadi Godsgift Ugonna, Uwagboe Osazee Michael, Omame Andrew, Mbachu Hope Ifeyinwa, Uwakwe Joy Ijeoma, Application of Homotopy Analysis Method for Solving an SEIRS Epidemic Model, Mathematical Modelling and Applications. Vol. 4, No. 3, 2019, pp. 36-48. doi: 10.11648/j.mma.20190403.11
References
[1]
Avordeh et al (2012) Mathematical Model for the control of malaria- Case study: Chorkor polyclinic, Accra Ghana, Global. Advanced Research journal of Medicine and Medical Sciences, 1 (5) 108-118.
[2]
Awrejcewicz, J. et al (1998) Asymptotic Approaches in Nonlinear Dynamics New Trends and Applications; Journal-Springer Series in Synergetics, Publisher Springer-Verlag berlin Heidelberg ISSN 0172-7389.
[3]
Falana, A. and Eighedion, E.E. (2014), Homotopy Analysis Method of one dimensional heat conduction in A Bar with temperature dependent on thermal conductivity. The International Journal of Engineering and Science (IJES) Vol. 3 Issue 5 pages pp-37-46 ISSN (e) : 2319-1813 ISSN (p): 2319-1805.
[4]
Fallahzadeh, A. and Shakibi, K. (2015) A method to solve Convection-Diffusion equation based on Homotopy Analysis Method; Journal of Interpolation and Approximation in Scientific Computing. No. (1) 1-8.
[5]
Gupta, V. G. and Gupta, S. (2012) Application of Homotopy Analysis Method(HAM) for solving nonlinear Cauchy Problem; Journal-Survey in Mathematics and its Applications, Vol. 7, 105-116 ISSN 1842-6298.
[6]
He, J. H. (1999) Variational iteration method- a kind of nonlinear analytical technique: some examples. International Journal of nonlinear Mechanics Vol. 34, No.4, pp 699-708.
[7]
Ibrahim, M. O and Egbetade, S. A. (2013) on the Homotopy Analysis Method for an SEIR Tuberculosis model, American Journal of Applied Mathematics and Statistics Vol1, No pp.71-75.
[8]
Jafarian, A. et al (2013) Homotopy Analysis Method for solving coupled Ramani equations. Journal-Rom. J. Phys. 58, 694-705.
[9]
Jifeng, C. et al (2015) Homotopy Analysis Method for nonlinear Periodic Oscillating equations with absolute value term Mathematical Problems in Engineering Vol.(7) ID 132651.
[10]
Johansson, P. and Leander, J. (2010) Mathematical Modelling of Malaria-Methods for Simulation of Epidemics; A Report from Chalmers University of Technology Gothenbury.
[11]
Koella, J. C.(1991) On the use of Mathematical Models of Malaria transmission, Acta Tropical (Elsevier) Vol. 49 pp1-25.
[12]
Liao, S. J. (2004) Homotopy Analysis Method for nonlinear Problems; Appl. Math Computer, Vol.174 pp. 449-513.
[13]
Liao, S. J. (2003) Beyond Perturbation. Introduction to the Homotopy Analysis Method, Boca Raton Chapman and Hall CRC press.
[14]
Lui, G. L. (1997) New research directions in singular Perturbation theory: artificial Parameter method and inverse-perturbation technique conf. of 7th Modern Mathematics and Mechanics, Shanghai, pp.47-53.
[15]
Momoh, A. A. et al (2012) Mathematical Modeling of Malaria Transmission in north senatorial zone of Taraba State Nigeria, IOSR Journal ofMathematics, Vol. 3 pp 7-13.
[16]
Momoh A. A, Ibrahim M. O. A. Tahir and Ibrahim (2015). Application of Homotopy Analysis Method for Solving SEIR models of Epidemics. Nonlinear Analysis and Differential Equation s Vol. 3, pp 53-68.
[17]
Nirmala, P. Subramanian S. P. et al (2015) SEIR Model of Seasonal Epidemic Diseases using HAM. Applications and Applied Mathematics: An International Journal (AAM) vol. 10, Issue pp. 1066-1081 ISSN: 1932-9466.
[18]
Srikumar, P. (2013) Assessment of Homotopy Analysis Method and Modified Homotopy perturbation Method for strongly Nonlinear Oscillator, International Journal of nonlinear Science vol. 16 No. 4 pp 291-300 (World Academic Press, England).
[19]
Umana, R. A, Omame, A. and Inyama S. C. Global Stability Analysis of the impact of media coverage on the control of infectious diseases. FUTO JNLS 2016 Volume -2, issue -2, Pp.173-194.
[20]
Van den Driessche and Watmough (2002) Reproduction numbers and Sub-threshold endemic equilibria for compartmental models of disease transmission; Journal mathematics Biosciences vol. 180 ISSUE 1-2.
[21]
Zadeh, K. S. (2010) An integro-patiall differential equations for modellingbiofluids flow in fractured biomaterials, Journal of Theoretical Biology, vol. 273 pp 72-79.
[22]
Zhuo, X. and Cui, J. (2011) Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate. National Natural Science Foundation of China (No. 11071011).
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