The aim of this work is to establish a matnematical model for measles epidemics and to predict levels of vaccination coverage require! in Ireland in order to eradicate tne disease The emphasis througnout has oeen to derive the parameters of the model using data collected in Ireland. To achieve this a nonlinear differential equation model first oroposed by Anderson R M. and May R.M. has been adopted and adjusted to meet our application. In Chapter 1 we introduce the concept of mathematically modelling the dynamics of an infectious disease and we also propose a simple constant parameter model We then move on in Chapter 2 to discuss what is known as ”the force of the infection". This is then calculated for Ireland by testing over 100 blood samples for measles antibodies. In Chapter 3 we estimated the Irish interepidemic period using Hopf's bifurcation theorem. In Chapter 4 we move on to the more detailed model with age dependence. We also estimate the age dependent survival rate jj(a) for the Irish population. Finally, in Chapters 5 and 6, we look at immunisation and the results predicted by the model. In Chapter 5 we derive c(a), the Irish age dependent vaccination rate This is accomplished by computerising over 4,000 immunisations. We also predict how the reproductive rate, R , of the disease will change with vaccination. In Chapter 6 we numerically analyse the model with the Irish age dependent parameters and we predict the levels of vaccination required in order to eradicate measles in Ireland.