Introduction

Mixing patterns in multi-group populations are now recognized to have an important role in the population dynamics of disease (Hethcote and Yorke 1984, Sattenspiel 1987b, Anderson et al 1990). Initially in response to the resurgence of gonorrhea and later with the rapid growth of the AIDS epidemic, selective mixing has become a major focus for epidemiological modelers. Various methods for summarizing the structure of selective mixing have been proposed (Gupta and Anderson 1989, Blythe et al. 1991, Koopman et al. 1991, Morris 1991). Simulation studies show that these effects can be both strong and variable (Hyman and Stanley 1988, Haraldsdottir et al. 1992, Morris 1995), and that they can bias the estimates of other epidemiological parameters if they are not taken into account (Koopman et al. 1991). Analytic expressions for the effect of mixing on the reproductive rate (or number) of a disease and the definition of core groups are beginning to be developed (Diekmann et al 1990, Jacquez et al 1993).

One of the major issues in modeling the mixing patterns of a multi-group population concerns the solution of multiple matching constraints in nonequilibrium populations. Constraints are imposed by the symmetry inherent in contact processes, i.e., if I meet you, then you have to meet me. This is a generalized version of the ‘two-sex problem’ familiar to demographers. In its classical form this problem arises in life table modeling when births are projected on the basis of two-sex populations. The birth process implies a matching process between the age-structured populations of males and females, and these constraints become complicated when vital dynamics are considered (Pollard 1948, Schoen 1982).