The Formal Demography of Peak Population.

When will the human population peak? In this article, we build on classical results by Ansley Coale, who showed that when fertility declines steadily, births reach their maximum before fertility reaches replacement level, and the decline in total population size does not occur until several decades after fertility has reached that level. We extend Coale's results by modeling longevity increases, net immigration, and a slowdown in fertility decline that resembles current projections. With these extensions, our models predict a typical lag between replacement-level fertility and population decline of about 35 to 40 years, consistent with projections by the United Nations and about 15 years longer than the lag predicted by Coale. Our analysis helps reveal underlying factors in the timing of peak population.

Vaccinating the oldest against COVID-19 saves both the most lives and most years of life.

Many competing criteria are under consideration for prioritizing COVID-19 vaccination. Two criteria based on age are demographic: lives saved and years of future life saved. Vaccinating the very old against COVID-19 saves the most lives, but, since older age is accompanied by falling life expectancy, it is widely supposed that these two goals are in conflict. We show this to be mistaken. The age patterns of COVID-19 mortality are such that vaccinating the oldest first saves the most lives and, surprisingly, also maximizes years of remaining life expectancy. We demonstrate this relationship empirically in the United States, Germany, and South Korea and with mathematical analysis of life tables. Our age-risk results, under usual conditions, also apply to health risks.

A cohort model of fertility postponement.

We introduce a new formal model in which demographic behavior such as fertility is postponed by differing amounts depending only on cohort membership. The cohort-based model shows the effects of cohort shifts on period fertility measures and provides an accompanying tempo adjustment to determine the period fertility that would have occurred without postponement. Cohort-based postponement spans multiple periods and produces "fertility momentum," with implications for future fertility rates. We illustrate several methods for model estimation and apply the model to fertility in several countries. We also compare the fit of period-based and cohort-based shift models to the recent Dutch fertility surface, showing how cohort- and period-based postponement can occur simultaneously.

How slowing senescence translates into longer life expectancy.

Mortality decline has historically been largely a result of reductions in the level of mortality at all ages. A number of leading researchers on ageing, however, suggest that the next revolution of longevity increase will be the result of slowing down the rate of ageing. In this paper, we show mathematically how varying the pace of senescence influences life expectancy. We provide a formula that holds for any baseline hazard function. Our result is analogous to Keyfitz's 'entropy' relationship for changing the level of mortality. Interestingly, the influence of the shape of the baseline schedule on the effect of senescence changes is the complement of that found for level changes. We also provide a generalized formulation that mixes level and slope effects. We illustrate the applicability of these models using recent mortality decline in Japan and the problem of period to cohort translation.