The standard age - structured disease models involve recovery rates which may depend on chronological age. Models have been studied by H. Thieme, C. Castillo - Chavez, and others with recovery rates depending on disease age and exponentially distributed natural life spans. Here, we formulate SIR models with natural death rates depending in an arbitrary way on chronological age and recovery rates depending in an arbitrary way on disease age. We focus on the relation between the basic reproductive number, the mean life span, and the age at infection. Two limiting cases which can be analyzed are models with mean infective period much shorter than the mean life span and SI models with no recovery. For intermediate cases there are open questions about stability of the endmic equilibrium.