School of Mathematical and Statistical Sciences,

University of Natal, Durban 4041, SOUTH AFRICA

E-mail: leachp@nu.ac.za, leach@math.aegean.gr

**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.