Application of Symmetry and Symmetry Analyses to Systems of First-Order Equations Arising from Mathematical Modelling in Epidemiology

We examine the integrability, in terms of the Painlevé singularity analysis and of the Lie symmetry analysis, of systems of nonlinear first-order ordinary differential equations that 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 methods of Dynamical Systems. The importance to obtain complete understanding of 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. The concept of decomposed systems is introduced as illustrating some rather attractive mathematical properties of certain classes of systems of equations arising in Mathematical Modelling. Such systems allow the mathematician some opportunity to enjoy Mathematics even in the context of quite strict applications.