The Kinship Formula: inferring the numbers of all kin from any structured population projection model

Abstract

Demography influences kinship, however, as of yet, little is known of the effects of transition rates of a general, structured population, generation-overlapping, projection model on the number of kin, their structure, distribution, and how they co-vary. Thanks to a novel approach incorporating generation number into the population structure, we provide here a full description of the number of such "collaterals", in a one-sex, constant environment framework. This yields a formula for the expected number of kin (for any kin) in any class, of an individual in any class, that is much simpler than any approach published so far, easy and fast to compute and that allows future research focusing on the theory of kinship demography. This approach also leads to the first formulas for their variance-covariance and probability generating functions that yield their entire distribution. These results will prove important for various fields studying structured population dynamics. They will allow for instance, for population and behavioral ecologists and geneticists to better understand the kinship structure of the male or female lineage of a particular population and will provide a stepping stone for a closed-form formula for the number of collaterals in two-sex populations. For epidemiologists, it will allow to analyse the distribution of "kin", that is, of individuals infected by the same source, and to compare these distributions for strains with different transition rates. For family economists, it will allow, for instance, to understand the distribution of inherited capital among members of extended families. We further provide a formula for the unstructured number of kin and show that is symmetrical with regards to the kin relationships. The novel approach also allows to decompose the variance-covariance of the number of all kin in all classes of focal individual ego in a given class, according to projective stochasticity (called individual stochasticity in population ecology) and to genealogical stochasticity which characterizes uncertainty about the ancestors of ego.