A simple construction of nonstandard finite-difference schemes for small nonlinear systems applied to SIR models

摘要

We present a simple mathematical construction for nonstandard finite-difference (NSFD) schemes for small systems of nonlinear differential equations using standard differential equation approximation techniques such as introducing artificial viscosity and a predictor-corrector scheme. The construction creates an explicit scheme. We begin by considering the NSFD scheme of Mickens and that of Erdogan and Ozis for first-order equations. The latter's method formulates a method to calculate some of the denominator functions and nonlocal approximations that characterize Mickens' scheme. We extend the analysis of Erdogan and Ozis by presenting a construction for second-order nonlinear autonomous functions that naturally generates the numerator and denominator functions for Mickens' equivalent schemes. We extend this result to systems of up to three differential equations and demonstrate the simply constructed scheme's computational advantages to handle transitions between linear and nonlinear problems as well as with the inclusion or exclusion of nonhomogeneous constant terms. We include applications using the SIR model for whooping cough with a constant and nonconstant, for example, seasonally affected, transmission rate.