A new age-structured diffusive model for the mathematical modelling of epidemics is suggested. The model can be considered as a generalization of two models suggested earlier for similar purposes. The Lie symmetry classification of the model is derived. It is shown that themodel admits an infinite-dimensional Lie algebra of invariance. Using the Lie symmetries, exact solutions, in particular those of the travelling wave types and in terms of special functions, are constructed. Examples of application of exact solutions with the correctly-specified parameters for calculation of the total number of infected individuals during an epidemic are presented.