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. We develop a general mathematical framework for modelling senescence in which individuals are viewed as comprising a fixed number of interacting intra-organismal sub-systems, each characterised by stochastic failure and repair rates. In many organisms, `failure begets failure' because sub-systems are typically interdependent. We use diffusion approximations to derive an expression for population-level mortality curves in this setting, and demonstrate that greater interdependence leads to faster rates of senescence. 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 of population-level measurements from individual-level trajectories, causing late-life mortality plateaus. 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. We provide necessary and asymptotically sufficient conditions for Gompertz-Makeham mortality in our framework, with falsifiable predictions in terms of a scaling law for mortality near complete failure. We also show that the simplest null model in our framework leads to exact Gompertz-Makeham curves. Our work establishes a general framework for modelling demographic senescence and selective disappearance from biological first principles, and provides concrete predictions for when Gompertz-Makeham mortality emerges in such models.