Mathematics · A dynamical model of allostatic load
Why some stress leaves a mark that outlasts it.
Allostatic load is usually one number. This model splits it into four: what happened (exposure), how much regulation it demands now (demand), accumulated wear (load), and how much demand the body can absorb without damage (capacity). Because load erodes capacity, the system can tip into a depleted state that persists after the stressor is gone.
A theoretical model with illustrative parameters. Nothing here is fitted to data, and it says nothing about any individual.
Demand against capacity
Load accumulates fast only while demand is above capacity.
Load
The two basins
Capacity across, load up, at the demand reached at the end of the run. The blue path is this run. Filled circles are stable states and the open circle is the tipping point. The dashed line through it is the boundary between the two basins: a run that ends on the wrong side does not come back when the stressor stops. Thin grey lines are where load (solid) or capacity (dotted) stop changing.
Which stressor caused how much load
Shapley values: each stressor's average extra load across every order in which the stressors could have been added. They share out the interactions fairly and add up exactly to the total. A negative value is a buffer.
The model
Exposure with memory. Each stressor fades at its own rate: an acute infection over days, bereavement or financial strain over years.
Demand. The sum of each stressor's remembered exposure, weighted by its potency, plus pairwise terms. Positive pairs are synergies (financial strain with discrimination); negative ones are buffers (social support with caregiving). Earlier exposure can also sensitise the response to a later one.
Load grows slowly with any demand, much faster when demand exceeds capacity (the damage term is quadratic in the excess), and is repaired over time.
Capacity relaxes toward a baseline that falls with age, is worn down by load, and is built up by moderate, recoverable demand (hormesis).
dL/dt = αD + β[D − C]₊² − ρL dC/dt = κ(C₀ − C) − δL + η h(D)
How it fits the landscape
Capacity sets how deep a person's basin is on the landscape. Chronic stress pushes them toward the edge, and once over it, removing the stressor is not enough. The model also predicts the warning signs: recovery from a small challenge slows sharply as the edge approaches. In these simulations, the slowest recovery rate falls from 1.0 to 0.33 per year just before tipping, while far from the edge it does not change at all.
Two details were needed for this to hold. Capacity cannot fall below zero, and ordinary everyday demand has to sit within the range where both states exist. With no everyday demand at all, load always drains and the system recovers.
Fitting it to people
Load and capacity are not measured directly. They are read through biomarkers on several timescales: cortisol and heart-rate variability (fast), CRP and HbA1c (medium), epigenetic pace of ageing and grip strength (slow). In simulated people, the repair and erosion rates are recovered almost exactly from someone pushed through the tipping point, but are biased by about 30% from short episodes they recover from. Challenge tests and natural experiments that produce real overload are what pin them down.
This is open problem 4. If you work on stress physiology, life-course epidemiology or state-space models, get in touch.