Methods of Mathematical Modelling

Abstract

In order to explain the purpose of modelling, it is helpful to start by asking: what is a mathematical model? One answer was given by Rutherford Aris [4]: A model is a set of mathematical equations that … provide an adequate description of a physical system. Dissecting the words in his description, “a physical system” can be broadly interpreted as any real-world problem—natural or man-made, discrete or continuous and can be deterministic, chaotic, or random in behaviour. The context of the system could be physical, chemical, biological, social, economic or any other setting that provides observed data or phenomena that we would like to quantify. Being “adequate” sometimes suggests having a minimal level of quality, but in the context of modelling it describes equations that are good enough to provide sufficiently accurate predictions of the properties of interest in the system without being too difficult to evaluate. Trying to include every possible real-world effect could make for a complete description but one whose mathematical form would likely be intractable to solve. Likewise, over-simplified systems may become mathematically trivial and will not provide accurate descriptions of the original problem. In this spirit, Albert Einstein supposedly said, “Everything should be made as simple as possible, but not simpler” [107], though ironically this is actually an approximation of his precise statement [34].