After Turing: Mathematical modelling in the biomedical and social sciences - From animal coat patterns to brain tumours to saving marriages

Abstract

Turing's 1952 paper on reaction diffusion models for spatial pattern formation was important in the early development of the application of mathematical modelling in biology and medicine. We describe here three very different problems which have been studied in depth and which have proved informative and useful in understanding specific phenomena. We describe an early study of a reaction diffusion model which helped explain the diverse coat patterns observed on animal coats. We then describe a basic, but surprisingly informative and accurate model, currently used medically, for quantifying the growth of gliomablastoma brain tumours. It enhances imaging techniques beyond any brain scanning procedure currently available and is used to estimate patient life expectancy and explain some current patient brain tumour anomalies. Finally we describe a modelling example from the social sciences, which quantifies marital interaction which was used to predict divorce with surprising accuracy and has helped design a new scientific marital therapy which is currently used. © 2012 Springer-Verlag.