The probable numbers of kin in a multi-state population: a branching process approach

Recent progress in mathematical kinship modelling has allowed one to predict the probable numbers of kin for a typical population member. In the models, kin may be structured by age and sex, both in static or time-variant demographies. Knowing the probable numbers of kin in different stages – such as parity, health status, or geographic location – however, remains an open challenge in Kinship Demography. Knowing how population structure delimits kin to distinct stages is an advance – for instance, the probability of having one sister at home and one sister away has different social implications from the probability of having two sisters. We present a novel analytical framework, grounded in branching process theory, that provides kin-number distributions jointly structured by age and stage. Using recursive compositions of probability generating functions (PGFs), we derive the joint age, stage, and age × stage kin-number distributions. All marginal distributions over either dimension naturally emerge. Simple extensions of the PGF approach additionally yield: the joint distribution of an individual’s own stage and their kin’s stage; the probable numbers of kin deaths, both in total and by generation number; and the probabilities of being kinless and/or orphaned. We demonstrate the framework through novel results in an application using UK parity-specific fertility and mortality data. Highlights A new method calculates probability generating functions for the number of kin structured by age and stage The model allows predicting the probable numbers of kin organised by age and stage Recursive nesting of probability generating functions in branching processes is used An application is presented highlighting the novel results