Discrete Models for Biomedical Phenomena: Nonstandard Finite Difference Scheme and Integer-Valued Population
(Ronald Mickens, Clark Atlanta University)

Biomedical systems can be modeled mathematically by differential equations (DE).  However, the typical case is that general closed form solutions are not known.  A way to deal with this situation is to construct discrete representations of the DE's and use them to obtain numerical solutions. However, major difficulties are the nonuniqueness of the discrete construction process and the existence of numerical instabilities (NI), i.e., the existence of solutions to the discrete equations not corresponding to any solution of the original DE's.  We demonstrate that nonstandard finite difference (NSFD) schemes can be constructed such that the elementary NI's do not occur.  The guiding principle used is that of "dynamical consistency."  A related topic, of special importance for interacting populations models, is the matter of small populations. We present several "toy" models to illustrate that it is possible to construct integer valued schemes, i.e., both the input and output populations take only integer values.  An overview will be given of various issues related to the applicability of NSFD and integer-valued populations methods for the modeling of biomedical systems.