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Technical Overview: Computational Model of Craving Dynamics

 

This document outlines the architecture and parameters of the mathematical model used to simulate the temporal dynamics of substance craving in recovery.

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1. Model Architecture: A Multi-Component Exponential Decay Framework
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The core model is a deterministic-stochastic hybrid, built upon a decaying exponential baseline that is modulated by three key dynamic components:

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  • A. Exponential Decay Baseline: The foundational craving level without acute triggers. It follows a standard exponential decay function:
    Baseline(t) = Initial_Craving * e^(-λt)
    where λ (lambda) is the decay rate constant and t is time in months.

  • B. Incubation of Craving: A Gaussian function that models the clinically observed phenomenon where cue-reactivity can increase after the initial withdrawal period.
    Incubation(t) = A_inc * exp(-(t - T_peak)² / (2 * σ²))
    where A_inc is the incubation strength, T_peak is the time of peak incubation (in days), and σ (sigma) controls the width of the incubation window.

  • C. Stochastic Spiking Processes: Two independent Poisson-like processes generate short-duration craving spikes, simulating real-world triggers:

    • Cue-Induced Spikes: Triggered by environmental cues (e.g., people, places, things).

    • Stress-Induced Spikes: Triggered by negative affective states or life stressors.
      The magnitude of each spike is a random proportion of the current baseline craving level.

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The final craving at any time t is the sum of the incubated baseline and the superimposed spikes:
Total_Craving(t) = Baseline(t) * (1 + Incubation(t)) + Cue_Spikes(t) + Stress_Spikes(t)

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2. Parameterization and Data-Informed Assumptions
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Substance-specific parameters were calibrated by synthesizing findings from longitudinal studies on addiction recovery, neurobiological data on withdrawal timelines, and clinical observations of relapse risk. Key parameters for each substance include:

  • Base Decay Rate (λ): Determines the natural decline of craving. Substances with higher physical dependence (e.g., Opioids, Meth) have slower decay rates (~0.018-0.022), while others like Cannabis decay faster (~0.040).

  • Cue & Stress Parameters: Define the frequency and intensity of daily spikes. Stimulants like Meth and Cocaine have higher cue reactivity, while Opioids have higher stress reactivity.

  • Incubation of Craving Peak (T_peak): Based on preclinical and clinical data. For example, Opioids and Meth show a protracted incubation period (peaks at 120-150 days), whereas Alcohol and Cannabis peak earlier (45-60 days).

  • Safe Zone Months: An estimate of the time until the craving baseline stabilizes near zero, derived from clinical recovery literature (e.g., 12 months for Cannabis, 30+ months for Opioids).

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3. Modeling the Impact of Interventions (Treatment Efficacy)
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The model incorporates therapeutic interventions as modulators of the core parameters. A composite Treatment Effect score is calculated from user-input sliders (0 to 1, representing adherence/efficacy), with heavier weighting for evidence-based practices like Cognitive Behavioral Therapy (CBT) and Monitoring.

This score directly influences the model by:

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  • Accelerating Decay: λ_effective = λ_base * Decay_Factor * (1 + Treatment_Effect)

  • Reducing Spike Frequency: Freq_effective = Freq_base * (1 - 0.7 * Monitoring * Spike_Reduction)

  • Reducing Spike Magnitude: Mag_effective = Mag_base * (1 - 0.6 * Monitoring)

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4. Stochastic Recovery Trajectories: The "Recovery Type"
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To reflect population heterogeneity, the model probabilistically assigns one of three recovery trajectories based on the composite Treatment Effect score:

  • Rapid Responder (~50% for Alcohol): High decay factor and significant spike reduction.

  • Typical Responder (~45% for Alcohol): Moderate improvements.

  • Persister (~5% for Alcohol): Slow decay and minimal spike reduction, representing a more treatment-resistant profile.

The probabilities for these types are substance-specific, reflecting known clinical outcomes (e.g., a higher persister rate for Opioids and Meth).

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5. Food Craving Overlay: A Biological Comparator
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To contextualize substance craving against a normative biological drive, a food craving model is included. It features:

  • A constant Baseline Hunger level.

  • Deterministic, time-locked spikes at typical meal times (breakfast, lunch, dinner).

  • Stochastic, low-magnitude spikes representing food cues.
    This provides a visual benchmark, illustrating how substance craving spikes are often higher in amplitude and less predictable than natural hunger signals.

 
Summary
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This model provides a simplified yet quantitatively grounded simulation of craving dynamics. As one very prominent addiction scientist put it - "Craving is probably the most studied and least understood component of addiction." The Model has a myriad of limitations. We have attempted to provide an accurate, generalized graphical representation  that integrates established mathematical functions with clinically informed parameters to illustrate the probabilistic nature of the phenomenon of craving in early recovery and the measurable impact of various treatment modalities over time.

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