Application of symmetry and singularity analyses to systems of first-order equations arising from mathematical modelling in epidemiology

Abstract:
We examine the integrability, in terms of the Painlev\'e singularity analysis and of the Lie symmetry analysis, of systems of nonlinear first-order ordinary di®erential equations which arise in the particular area of Mathematical Modelling known as Rational Epidemiology. These analyses are presented as being complementary to the standard analysis using the method of Dynamical Systems. The importance of obtaining as complete an understanding of the evolution of epidemics – one need think only of the potential for devastation by HIV-AIDS in African countries or the more recent threat posed by SARS – demands that all possible approaches of analysis be used. Particular attention is given to decomposed systems as illustrating some rather attractive mathematical properties.