Addiction Recovery: Relapse-Risk Landscape

Recovery Landscape Controls

Craving / Trigger Intensity 40%

Coping-Skill Strength 50%

Social Support Level 40%

Recovery Status

Recovery (Abstinent)

Days in Recovery

Relapse Risk (est.)

—%

Relapse Events

Landscape Position

-1.00

How it works

Recovery is modelled as a particle undergoing noisy (Langevin) motion on a bistable potential V(x) with two wells — Recovery at x ≈ -1 and Relapse/Use at x ≈ +1 — separated by a barrier at x = 0. This double-well / stochastic-escape framework is the standard way computational psychiatry models relapse as a transition between attractor states under stress and craving noise.

V(x)  = k·(x² - 1)² + s·x
dV/dx = 4k·x·(x² - 1) + s

Langevin update (Euler–Maruyama):
  x(t+dt) = x(t) - dV/dx·dt + sqrt(2D·dt)·ξ,   ξ ~ N(0,1)

k = barrier height   (↑ with Coping-Skill Strength)
s = tilt toward the Recovery well (↑ with Social Support)
D = noise intensity   (↑ with Craving / Trigger Intensity)

The instantaneous relapse-risk estimate uses the Kramers escape-rate approximation for thermally (here, stress-) driven barrier crossing:

r ≈ (ω_well·ω_barrier)/(2π) · exp(-ΔV/D)
ΔV = V(0) - V(well)   (barrier height above the current well)

Coping-Skill Strength — deepens the potential wells and raises the barrier k, making both states more stable and harder to leave.
Social Support Level — tilts the landscape (s) so the Recovery well sits lower in energy, biasing the system to stay there.
Craving / Trigger Intensity — sets the noise level D driving random excursions toward the barrier.
Trigger Craving Event — an instantaneous kick toward x > 0 plus a brief noise spike, modelling exposure to a real relapse trigger.

This is a simplified illustrative model, not a clinical predictor — but the underlying bistable-potential / Kramers-rate mechanism is the same one used in real stress-vulnerability models of addiction relapse.

Recovery well (x < 0) Relapse well (x > 0) Current state

⚡ Craving spike!

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