Asymptotically consistent non-standard finite-difference methods for solving mathematical models arising in population biology

Ever since the pioneering work of Kermack and McKendrick in the 1930s, numerous compartmental mathematical models have been used to help gain insights into the transmission and control mechanisms of many humandiseases. These models are often of the form of systems of non-linear 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 non-standard framework of Mickens, 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).