Empirical Methods for The Estimation of the Mixing Probabilities for Socially Structured Populations from A Single Survey Sample

The role of variability of sexual behavior in the transmission dynamics of HIV and AIDS has been illustrated, through the use of mathematical models, by several investigators. Models that capture heterogeneities due to rates of sexual partner change, changing behaviors, and demographic factors are invaluable in the study of the dynamics of sexually-transmitted diseases. Models that describe the processes of mixing between individuals and/or pair formation/dissolution have been formulated in great generality by Blythe, Busenberg, and Castillo-Chavez. Mathematical formulas describing all forms of mixing for one-and two-sex populations as structured deviations from random mixing have been obtained by Busenberg and Castillo-Chavez. In this paper we describe some practical methods for estimating the deviations from random mixing from a single survey sample. This method can be potentially very useful if one considers the difficulties—technical and political—involved in the gathering of behavioral and mixing data. We include a description of the role of the estimated mixing probabilities in models for the spread of HIV, a discussion of alternatives and possible extensions of the methods described in this article, and an outline of future directions of research. We note that despite the fact that the mixing probabilities {p,-;(T)} are time-dependent, we are able to make use of time-independent parameters—the matrix of constant quantities (Φ¡¡} which are related to the initial deviations from random mixing—in the estimation of the dynamic mixing probabilities {p,;(T)}.