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Asymptotically consistent nonstandard finite-difference methods for solving mathematical models arising in population biology. (English) Zbl 1086.65080

Mickens, Ronald E. (ed.), Advances in the applications of nonstandard finite difference schemes. Hackensack, NJ: World Scientific (ISBN 981-256-404-7/hbk). 385-421 (2005).
Summary: Ever since the pioneering work of W. O. Kermack and A. G. McKendrick [Proc. R. Soc. Lond., Ser. A 138, 55–83 (1932; Zbl 0005.30501)], numerous compartmental mathematical models have been used to help gain insights into the transmission and control mechanisms of many human diseases. These models are often of the form of systems of nonlinear differential equations, whose closed-form solutions are not easily obtainable (if at all), necessitating the use of numerical methods for their approximate solutions. Easy-to-use standard explicit finite-difference methods, such as the forward Euler and explicit Runge-Kutta methods, have often been used to solve these models. Unfortunately, these methods may suffer spurious behaviours, which are not the features of the continuous model being approximated, when certain values of the associated discretization and model parameters are used in the simulations.
The aim of this chapter is to investigate a class of finite-difference methods, designed via the nonstandard framework of R. E. Mickens [Nonstandard finite difference models of differential equations. Singapore: World Scientific (1994; Zbl 0810.65083)], for solving systems of differential equations arising in population biology. It will be shown that this class of methods can often give numerical results that are asymptotically consistent with those of the corresponding continuous model. This fact is illustrated using a number of case studies arising from population biology (human epidemiology and ecology).
For the entire collection see [Zbl 1079.65005].

MSC:

65L12 Finite difference and finite volume methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L20 Stability and convergence of numerical methods for ordinary differential equations
92D25 Population dynamics (general)
92D40 Ecology
92C60 Medical epidemiology
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