Stochastic Modelling of Early-Life Stress and Long-Term Psychological Well-Being: Thresholds in a Shot-Noise Model of Allostatic Load

Early-life stress is reliably associated with poorer psychological well-being decades later, but the mechanism by which an exposure that ends in childhood continues to act in adulthood is not settled. This paper formalises the question as a problem in stochastic processes. Allostatic load is modelled as a shot-noise process driven by a Poisson stream of stressors with exponentially distributed magnitudes and a homeostatic recovery rate. Two results follow. First, in the constant-parameter model the influence of a childhood stressor on adult load decays exponentially and is numerically extinct within a decade, so the state variable cannot carry the observed persistence; persistence must instead enter through a parameter. Second, when childhood exposure is allowed to lower the recovery rate, the stationary load distribution is exactly Gamma with shape equal to the ratio of the stressor arrival rate to the recovery rate, and two exposure thresholds appear in closed form. At the first, the modal load leaves zero and the typical state changes from recovered to elevated. At the second, located by a digamma equation, the rate of threshold crossings is maximised; beyond it episodes become fewer but longer, so disturbance shifts from episodic to chronic. The exceedance probability, episode rate and mean episode duration all have closed forms, each confirmed against exact simulation. The model further predicts that exposures of equal count but earlier timing carry greater adult risk, a prediction count-based epidemiology cannot test.