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**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.**
*(English)*
Zbl 0764.92017

An important aspect in the study of the dynamics of sexually transmitted diseases is that of heterogeneous mixing, i.e. heterogeneous contacts between individuals. As a result of the AIDS epidemic, the effects of social structure in disease dynamics have received considerable attention over the last few years. In this paper, a generalization of the Blythe and Castillo-Chavez framework for social/sexual human interactions is formulated through the incorporation of age structure. The mixing function is axiomatically characterized, and its role in disease dynamics is described by deriving an expression for the incidence (the number of new cases per unit time).

An explicit expression for the general solution of the mixing problem in terms of a preference function is derived. It is shown that proportionate mixing is the only separable solution to this mixing framework, and several specific cases of proportionate mixing are discussed. An age- structured epidemic model is formulated for a single sexually active homosexual population, stratified by risk and age, with arbitrary risk- and age-dependent mixing as well as variable infectivity, and an explicit expression for the basic reproduction number is computed in the case of proportionate mixing in age and risk.

An explicit expression for the general solution of the mixing problem in terms of a preference function is derived. It is shown that proportionate mixing is the only separable solution to this mixing framework, and several specific cases of proportionate mixing are discussed. An age- structured epidemic model is formulated for a single sexually active homosexual population, stratified by risk and age, with arbitrary risk- and age-dependent mixing as well as variable infectivity, and an explicit expression for the basic reproduction number is computed in the case of proportionate mixing in age and risk.

Reviewer: G.Hasibeder (Wien)

### MSC:

92D30 | Epidemiology |