Stochastic failure accumulation as a foundation for exponential mortality and selective disappearance

Most organisms become increasingly likely to die as they grow older, a phenomenon known as demographic senescence. Despite this statement saying nothing about the particular shape of a mortality curve, the Gompertz-Makeham law remains remarkably accurate in a broad range of species. We develop a general mathematical framework in which individuals are modelled as comprising a fixed number of interacting intra-organismal sub-systems, each characterised by stochastic failure and repair rates, such that the number of failed sub-systems follows a birth-death process. In many organisms, failure begets failure because sub-systems are typically interdependent. We use diffusion approximations to demonstrate that this interdependence generically produces Gompertz-Makeham mortality curves. Since individuals who die can no longer age, observed cohorts become increasingly composed of lucky individuals that avoided death by (stochastically) taking paths in failure space associated with lower mortality despite no intrinsic differences in their quality. Selective disappearance of unlucky individuals generates deviations from Gompertz-Makeham predictions at advanced ages, producing a late-life mortality plateau. We show that while these deviations must always exist, they may often be too small to detect, either because the failure accumulation process is stereotyped or because detection requires unreasonably large cohort sizes. Our work establishes Gompertz-Makeham curves arising from stochastic failure accumulation as a null expectation in organisms with many interdependent sub-systems.