Threshold and Stability Results for an Epidemic Model with an Age-Structured Meeting Rate

Abstract

This paper examines threshold and stability results for simple mathematical models for the transmission of infectious diseases with permanent natural immunity. Examples of such diseases are measles, chicken-pox, hepatitis, and mumps. An important feature of this work is the introduction of an age structure into the population amongst whom the disease is spreading and, in particular, the realization of the fact that the contact rate itself may depend on age. Equilibrium and stability analyses are performed on these models. These results are in part directed towards establishing conditions sufficient for the existence of a nonzero equilibrium disease level to be possible. Conjectures about the existence of a nonzero solution to a set of partial integrodifferential equations are examined. These conditions determine the circumstances under which the disease will persist. Particular emphasis is devoted to the case where the meeting rate depends on age.