The formal demography of kinship III: Kinship dynamics with time-varying demographic rates

Background Kinship models generally assume time-invariant demographic rates, and compute the kinship structures implied by those rates. It is important to compute the consequences of time variation in demographic rates for kinship stuctures. Objectives Our goal is to develop a matrix model for the dynamics of kinship networks subject to arbitrary temporal variation in survival, fertility, and population structure. Methods We develop a system of equations for the dynamics of the age structure of each type of kin of a Focal individual. The matrices describing survival and fertility vary with time. The initial conditions in the time-invariant model are replaced with a set of boundary conditions for initial time and initial age. Results The time-varying model maintains the network structure of the time-invariant model. In addition to the results of the time-invariant model, it provides kinship structures by period, cohort, and age. It applies equally to historical sequences of past rates and to projections of future rates. As an illustration, we analyze the kinship structure of Sweden from 1891 to 2120. Contribution The time-varying kinship model makes it possible to analyze the consequences of changing demographic rates, in the past or the future. It is easily computable, requires no simulations, and is readily extended to include additional, more distant relatives in the kinship network. The method can also be used to show the growth of families, lineages, and dynasties in populations across time and place and between social groups.