A Leslie Matrix Approach to an Age-Structured Epidemic Joe Gani * Linda Stals * June 28, 2012 Abstract We consider a Leslie-type matrix approach to an SIR epidemic in dis- crete time. We give examples of the population of susceptibles, infectives and removals for different birthrates and two different infection rates. Fi- nally, when the infection rate depends on the number of infectives, we derive conditions for a steady state. 1 Introduction An interesting problem in epidemiology is the modelling of age-structured epi- demics. Several such studies in continuous time have been carried out in recent years, among them [2, 3, 5, 7]. For example, the authors of [2] derive a set of partial differential equations for the time-dependent age-specific densities of susceptible, infective and removed individuals in an SIR epidemic. Among other results, the authors obtain con- ditions for the existence of steady states. Their calculations are lengthy and result in a complicated system of equa- tions, from which, it is not easy to get an intuitive insight. One is led to consider if a simpler model based on a discrete time Leslie-type matrix [4], might serve equally well as a model. Models of this type have already been used by Zhou, Ma and Brauer [8] for the transmission of SARS in China, and by Powell, Slapnicar and van der Werf [6] for the spread of plant pathogens. The purpose of this paper is not to develop new mathematical methods, but rather to indicate that discrete time age-structured epidemic models can char- acterise SIR epidemics very adequately. We propose to examine this alternative, and find conditions under which steady-state solutions exist. * Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia 1