The distribution of birth intervals under a class of stochastic fertility models

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Abstract
Characterization ofthe distribution ofintervals between successive births affords both practical and theoretical advantages in the analysis of fertility data. Moreover, attempts to formulate a mathematical model describing the process of reproduction lead naturally to a consideration of the distribution of birth intervals. A class ofmodels has been formulated, applying to a woman who is not pregnant and fecundable at marriage and who after a variable length of time enters a pregnancy, which terminates in either a miscarriage, stillbirth, or live birth, with a given probability depending on the duration of the pregnancy. Each pregnancy is followed by a period of post-partum infecundability (physiological sterility), whose length depends on the outcome of the pregnancy. Eventually, the woman returns to the non-pregnant fecundable state. The durations of pregnancy and of post-partum infecundability are treated as variables whose distributions can depend on the next state (such as the outcome ofpregnancy) as well as on the state being currently occupied. Expressions for the mean and variance of the intervals to the first live birth and the intervals between subsequent live births, as based on this formulation, are given. With the aid of numerical examples, these characteristics of the birth intervals are examined, particularly with respect to the effects of including terms for foetal deaths in the model. The mean length and the variance of the intervals increase with the frequency offoetal death and the duration of the associated infecundable periods. Generally, the variance of birth intervals is greater than expected from estimates in which the occurrence of foetal deaths is ignored. For intervals after the first, similar mean values may result either from a lowered fecundability (probability of conceptio) or from an increase in the duration of post-partum physiological sterility. If this change resulted from a decreased fecundability the variance of the birth intervals would be much greater than if the change were due to prolongation of post-partum infecundability. Applications ofthese results are possible even in the absence of reliable data for the frequency of foetal deaths.