A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS.

A central aspect in the study of the dynamics of sexually transmitted diseases is that of mixing. The study of the effects of social structure in disease dynamics has received considerable attention over the last few years as a result of the AIDS epidemic. In this paper, we formulate a generalization of the Blythe and Castillo-Chavez social/sexual framework for human interactions through the incorporation of age structure, and derive an explicit expression in terms of a preference function for the general solution to this formulation. We emphasize the role played by proportionate mixing, the only separable solution to this mixing framework, through the discussion of several specific cases, and we formulate an age-structured epidemic model for a single sexually active homosexual population, stratified by risk and age, with arbitrary risk- and age-dependent mixing as well as variable infectivity. In the special case of proportionate mixing in age and risk, an explicit expression for the basic reproductive number is computed.