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A nonstandard numerical scheme of predictor-corrector type for epidemic models. (English) Zbl 1198.65116
Summary: We construct and develop a competitive nonstandard finite difference numerical scheme of predictor-corrector type for the classical SIR epidemic model. This proposed scheme is designed with the aim of obtaining dynamical consistency between the discrete solution and the solution of the continuous model. The nonstandard finite difference scheme with Conservation Law (NSFDCL) developed here satisfies some important properties associated with the considered SIR epidemic model, such as positivity, boundedness, monotonicity, stability and conservation of frequency of the oscillations. Numerical comparisons between the NSFDCL numerical scheme developed here and Runge-Kutta type schemes show its effectiveness.

65L06Multistep, Runge-Kutta, and extrapolation methods
Full Text: DOI
[1] Mickens, Ronald E.: Nonstandard finite difference models of differential equations, (1994) · Zbl 0810.65083
[2] Mickens, R. E.: Applications of nonstandard finite difference schemes, (2000) · Zbl 0989.65101
[3] Mickens, R. E.: Numerical integration of population models satisfying conservation laws: NSFD methods, Journal of biological dynamics 1, No. 4, 427-436 (2007) · Zbl 1284.92116
[4] Mickens, R. E.: Dynamic consistency: a fundamental principle for constructing nonstandard finite difference schemes for differential equations, Journal of difference equations and applications 11, No. 7, 645-653 (2005) · Zbl 1073.65552 · doi:10.1080/10236190412331334527
[5] Anguelov, R.; Lubuma, J. M. -S.: Contributions to the mathematics of the nonstandard finite difference method and applications, Numerical methods for partial differential equations 17, No. 5, 518-543 (2001) · Zbl 0988.65055 · doi:10.1002/num.1025
[6] Anguelov, Roumen; Lubuma, J. M. S.: Nonstandard finite difference method by nonlocal approximation, Mathematics and computers in simulation 61, No. 3--6, 465-475 (2003) · Zbl 1015.65034 · doi:10.1016/S0378-4754(02)00106-4
[7] Dimitrov, Dobromir T.; Kojouharov, Hristo V.: Positive and elementary stable nonstandard numerical methods with applications to predator--prey models, Journal of computational and applied mathematics, No. 1--2, 98-108 (2006) · Zbl 1087.65068 · doi:10.1016/j.cam.2005.04.003
[8] Solis, Francisco J.; Chen-Charpentier, Benito: Nonstandard discrete approximations preserving stability properties of continuous mathematical models, Mathematical computer modelling 40, 481-490 (2004) · Zbl 1112.65070 · doi:10.1016/j.mcm.2004.02.028
[9] Chen-Charpentier, B. M.; Dimitrov, D. T.; Kojouharov, H. V.: Combined nonstandard numerical methods for odes with polynomial right-hand sides, Mathematics and computers in simulation 73, 105-113 (2006) · Zbl 1105.65084 · doi:10.1016/j.matcom.2006.06.008
[10] Villanueva, Rafael; Arenas, Abraham J.; González-Parra, Gilberto: A nonstandard dynamically consistent numerical scheme applied to obesity dynamics, Journal of applied mathematics 2008, 14 (2008) · Zbl 1157.92028 · doi:10.1155/2008/640154
[11] Arenas, Abraham J.; Moran\hat o, José Antonio; Cortés, Juan Carlos: Non-standard numerical method for a mathematical model of RSV epidemiological transmission, Computers mathematics with applications 56, 670-678 (2008) · Zbl 1155.92337 · doi:10.1016/j.camwa.2008.01.010
[12] Moghadas, S. M.; Alexander, M. E.; Corbett, B. D.; Gumel, A. B.: A positivity preserving Mickens-type discretization of an epidemic model, Journal of difference equations and applications 9, No. 11, 1037-1051 (2003) · Zbl 1033.92030 · doi:10.1080/1023619031000146913
[13] Piyawong, W.; Twizell, E. H.; Gumel, A. B.: An unconditionally convergent finite-difference scheme for the SIR model, Applied mathematics and computation 146, 611-625 (2003) · Zbl 1026.92041 · doi:10.1016/S0096-3003(02)00607-0
[14] Jansen, H.; Twizell, E. H.: An unconditionally convergent discretization of the SEIR model, Mathematics and computers in simulation 58, 147-158 (2002) · Zbl 0983.92025 · doi:10.1016/S0378-4754(01)00356-1
[15] Gumel, A. B.; Mickens, R. E.; Corbett, B. D.: A non-standard finite-difference scheme for a model of HIV transmission and control, Journal of computational methods in sciences and engineering 3, No. 1, 91-98 (2003) · Zbl 1036.65057
[16] Jódar, Lucas; Villanueva, Rafael J.; Arenas, Abraham J.; González, Gilberto C.: Nonstandard numerical methods for a mathematical model for influenza disease, Mathematics and computers in simulation 79, 622-633 (2008) · Zbl 1151.92018 · doi:10.1016/j.matcom.2008.04.008
[17] Lambert, J. D.: Computational methods in ordinary differential equations, (1973) · Zbl 0258.65069
[18] Duncan, C. J.; Duncan, S. R.; Scott, S.: Whooping cough epidemic in London, 1701--1812: infection dynamics seasonal forcing and the effects of malnutrition, Proceedings of the royal society of London, series B 263, 445-450 (1996)
[19] Mickens, R. E.: Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Wiley interscience 23, No. 3, 672-691 (2006) · Zbl 1114.65094 · doi:10.1002/num.20198
[20] Brauer, F.; Castillo-Chavez, C.: Mathematical models in population biology and epidemiology, (2001) · Zbl 0967.92015