Abstract
In classical mumps models, individuals are generally assumed to be uniformly mixed (homogeneous), ignoring population heterogeneity (preference, activity, etc.). Age is the key to catching mixed patterns in developing mathematical models for mumps. A continuous heterogeneous age-structured model for mumps with vaccines has been developed in this paper. The stability of age-structured models is a difficult question. An explicit formula of R (0) was defined for the various mixing modes (isolation, proportional and heterogeneous mixing) with or without the vaccine. The results show that the endemic steady state is unique and locally stable if R (0) > 1 without any additional conditions. A number of numerical examples are given to support the theory.