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Piecewise finite series solutions of seasonal diseases models using multistage Adomian method. (English) Zbl 1221.65201

Summary: Our aim is to apply the multistage Adomian decomposition method (MADM) to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models with periodic behavior. Here the concept of the MADM is introduced and then it is employed to obtain a piecewise finite series solution. The MADM is used here as a hybrid analytical-numerical technique for approximating the solutions of the epidemic models. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge-Kutta method solutions. Numerical comparisons show that the MADM is accurate, easy to apply and the calculated solutions preserve the periodic behavior of the continuous models. Moreover, the method has the advantage of giving a functional form of the solution for any time interval. Furthermore, it is shown that if the truncation order and the time step size are not properly chosen large computational work is required and inaccurate solutions may be obtained.

MSC:

65L99 Numerical methods for ordinary differential equations
92D30 Epidemiology
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] Dowell, Scott F., Seasonal variation in host susceptibility and cycles of certain infectious diseases, Emerg Infect Diseases, 7, 3, 369-374 (2005)
[2] Grossman, Z., Oscillatory phenomena in a model of infectious diseases, Theory Pop Biol, 18, 1980, 204-243 (2006) · Zbl 0457.92020
[3] Ma, J.; Ma, Z., Epidemics threshold conditions for seasonally forced SEIR models, Math Biosci Eng, 3, 1, 161-172 (2006) · Zbl 1089.92048
[4] Moneim, I. A.; Greenhalgh, D., Use of a periodic vaccination strategy to control the spread of epidemics with seasonally varying contact rate, Math Biosci Eng, 13, 591-611 (2005) · Zbl 1079.92058
[5] Schwartz, Ira B., Small amplitude, long period outbreaks in seasonally driven epidemics, J Math Biol, 30, 473-491 (1992) · Zbl 0745.92026
[6] Weber, A.; Weber, M.; Milligan, P., Modeling epidemics caused by respiratory syncytial virus (RSV), Math Biosci, 172, 95-113 (2001) · Zbl 0988.92025
[7] White, L. J.; Mandl, J. N.; Gomes, M. G.; Bodley-Tickell, A. T.; Cane, P. A.; Perez-Brena, P., Understanding the transmission dynamics of respiratory syncytial virus using multiple time series and nested models, Math Biosci, 209, 222-239 (2007) · Zbl 1120.92039
[8] Jódar, Lucas; Villanueva, Rafael J.; Arenas, Abraham J.; González, Gilberto C., Nonstandard numerical methods for a mathematical model for influenza disease, Math Comput Simul, 79, 3, 622-633 (2008) · Zbl 1151.92018
[9] Piyawong, W.; Twizell, E. H.; Gumel, A. B., An unconditionally convergent finite-difference scheme for the SIR model, Appl Math Comput, 146, 611-625 (2003) · Zbl 1026.92041
[10] Roberts, M. G.; Grenfell, B. T., The population dynamics of nematode infections of ruminants: the effect of seasonally in the free-living stages, Math Med Biol, 9, 1, 29-41 (1992) · Zbl 0767.92025
[11] Arenas, Abraham J.; Moranˆo, José Antonio; Cortés, Juan Carlos, Non-standard numerical method for a mathematical model of RSV epidemiological transmission, Comp Math Appl, 56, 670-678 (2008) · Zbl 1155.92337
[12] Adomianm, G., A review of the decomposition method in applied mathematics, Math Anal Appl, 135, 2, 501-544 (1988) · Zbl 0671.34053
[13] Adomian, G., Solving frontier problems of physics: the decomposition method (1994), Kluwer Academic Publishers.: Kluwer Academic Publishers. Boston · Zbl 0802.65122
[14] Achouri, Talha; Omrani, Khaled, Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian decomposition method, Commun Nonlinear Sci Num Simul, 14, 5, 2025-2033 (2009) · Zbl 1221.65270
[15] Hosseini, M. M.; Jafari, M., A note on the use of Adomian decomposition method for high-order and system of nonlinear differential equations, Commun Nonlinear Sci Num Simul, 14, 5, 1952-1957 (2009) · Zbl 1221.65162
[16] Ghosh, S.; Roy, A.; Roy, D., An adaptation of Adomian decomposition for numericanalytic integration of strongly nonlinear and chaotic oscillators, Comput Meth Appl Mech Eng, 196, 4-6, 1133-1153 (2007) · Zbl 1120.70303
[17] Ruan, J.; Lu, Z., A modified algorithm for Adomian decomposition method with applications to Lotka-Volterra systems Math, Comput Model, 46, 9-10, 1214-1224 (2007) · Zbl 1133.65046
[18] Chowdhury, M. S.H.; Hashim, I.; Mawa, S., Solution of preypredator problem by numericanalytic technique, Commun Nonlinear Sci Num Sim, 14, 4, 1008-1012 (2009) · Zbl 1221.65199
[19] Repaci, A., Nonlinear dynamical systems: on the accuracy of Adomian’s decomposition method, Appl Math Lett, 3, 4, 35-39 (1990) · Zbl 0719.93041
[20] Olek, Shmuel, An accurate solution to the multispecies lotka-volterra equations, SIAM Rev, 36, 3, 480-488 (1994) · Zbl 0802.92018
[21] Guellal, S.; Grimalt, P.; Cherruault, Y., Numerical study of lorenz’s equation by the Adomian method, Comput Math Appl, 33, 3, 25-29 (1997) · Zbl 0869.65044
[22] Vadasz, Peter; Olek, Shmuel, Convergence and accuracy of Adomians decomposition method for the solution of lorenz equations, Int J Heat Mass Transfer, 43, 10, 1715-1734 (2000) · Zbl 1015.76075
[23] Shawagfeh, N.; Kaya, D., Comparing numerical methods for the solutions of systems of ordinary differential equations, Appl Math Lett, 17, 3, 323-328 (2004) · Zbl 1061.65062
[24] Venkatarangan, S. N.; Rajalakshmi, K., A modification of Adomian’s solution for nonlinear oscillatory systems, Comput Math Appl, 29, 6, 67-73 (1995) · Zbl 0818.34006
[25] Hashim, I.; M Noorani, M. S.; Ahmad, R.; Bakar, S. A.; Ismail, E. S.; Zakaria, A. M., Accuracy of the Adomian decomposition method applied to the lorenz system, Chaos, Solitons and Fractals, 28, 5, 1149-1158 (2006) · Zbl 1096.65066
[26] Noorani, M. S.M.; Hashim, I.; Ahmad, R.; Bakar, S. A.; Ismail, E. S.; Zakaria, A. M., Comparing numerical methods for the solutions of the Chen system, Chaos, Solitons and Fractals, 32, 4, 1296-1304 (2007) · Zbl 1131.65101
[27] Abdulaziz, O.; Noor, N. F.M.; Hashim, I.; Noorani, M. S.M., Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos, Solitons and Fractals, 36, 5, 1405-1411 (2008)
[28] Al-Sawalha, M. M.; Noorani, M. S.M.; Hashim, I., Numerical experiments on the hyperchaotic Chen system by the Adomian decomposition method, Int J Comput Methods, 5, 3, 403-412 (2008) · Zbl 1257.70003
[29] Chen, B.; Company, R.; Jódar, L.; Roselló, M. D., Constructing accurate polynomial approximations for nonlinear differential initial value problems, Appl Math Comput, 193, 2, 523-534 (2007) · Zbl 1193.65111
[30] Kincaid, D.; Cheney, W., Numerical analysis (2002), Brooks/Cole Publishing Company: Brooks/Cole Publishing Company Pacific Grove, CA
[31] Jódar, L.; Villanueva, R. J.; Arenas, A., Modeling the spread of seasonal epidemiological diseases: theory and applications, Math Comput Model, 48, 548-557 (2008) · Zbl 1145.92336
[32] Arenas, Abraham J.; González, Gilberto; Jódar, Lucas, Existence of periodic solutions in a model of respiratory syncytial virus RSV, J Math Anal Appl, 344, 969-980 (2008) · Zbl 1137.92318
[33] Jódar, L.; Santonja, F.; González-Parra, G., Modeling dynamics of infant obesity in the region of Valencia, Spain, Comput Math Appl, 56, 3, 679-689 (2008) · Zbl 1155.92329
[34] Abbaoui, K.; Cherruault, Y., Convergence of Adomian’s method applied to differential equations, Comput Math Appl, 28, 5, 103-109 (1994) · Zbl 0809.65073
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