metastablex 0.1.1

Metastable dynamics detection for complex systems

🧠 MetastableX

A Unified Computational Framework for Metastability and Critical Transitions in Complex Systems

Bridging statistical physics, machine learning and epidemiology to detect systemic instability in real-world dynamics.

⚙️ PyTorch • 📊 Time-Series • 🧠 Complex Systems • 🏥 Public Health • 🌍 Open Science

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📌 Abstract

Complex systems—from biological organisms to healthcare infrastructures—operate near the edge of stability, where small perturbations can trigger large-scale transitions.

We introduce MetastableX, a computational framework for detecting:

• metastable regimes
• early warning signals
• critical transitions

using time-series analysis, statistical physics, and machine learning.

The framework combines:

• stochastic dynamical systems modeling
• information-theoretic metrics
• learned representations via neural networks

and is validated on real-world epidemiological data from SIH/SUS (Brazil).

We show that healthcare systems exhibit signatures consistent with marginal criticality, enabling early detection of systemic stress and potential collapse.

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🧠 1. Introduction

Many natural and social systems exhibit nonlinear dynamics characterized by abrupt transitions.

Examples include:

• cardiac arrhythmias
• ecosystem collapse
• financial crises
• hospital overload

Traditional statistical approaches fail to capture these transitions because they rely on:

• averages
• linear correlations

Instead, such systems require a structural and dynamical analysis framework.

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⚙️ 2. Theoretical Framework

2.1 Stochastic Dynamics

We model the system as:

$$ \frac{dx}{dt} = f(x) + \sigma \eta(t) $$

where:

• $x(t)$ is the system state
• $f(x)$ encodes deterministic dynamics
• $\eta(t)$ is stochastic noise

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2.2 Potential Landscape

The system can be represented through an effective potential:

$$ U(x) = -\frac{\sigma^2}{2}\log P(x) $$

Interpretation:

Regime	Landscape
Stable	deep wells
Metastable	shallow wells
Critical	flattened

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2.3 Criticality Condition

We hypothesize that complex systems operate near:

$$ \lambda \approx 0 $$

where $\lambda$ is the Lyapunov exponent, defining the boundary between:

• order ($\lambda < 0$)
• chaos ($\lambda > 0$)

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2.4 Information-Theoretic Principle

We define an informational optimality condition:

$$ \max (H + F) $$

where:

• $H$ = Shannon entropy
• $F$ = Fisher information

This condition corresponds to maximum structural adaptability.

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⚠️ 3. Early Warning Signals

Variance

$$ \sigma^2 = \frac{1}{N}\sum (x_i - \mu)^2 $$

Autocorrelation

$$ AC_1 = \frac{ \sum (x_t-\mu)(x_{t-1}-mu) }{ \sum (x_t-\mu)^2 } $$

Interpretation

Signal	Meaning
↑ Variance	instability
↑ Autocorrelation	critical slowing down

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🧬 4. Information Geometry

Entropy

$$ H = -\sum p_i \log p_i $$

Fisher Information

$$ F = \int \frac{(\nabla p)^2}{p} dx $$

Together, they define a statistical manifold where:

• entropy → exploration
• Fisher → curvature

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🤖 5. Machine Learning Formulation

MetastableX extends classical physics with a learned representation:

features = metastable_features(x)
y_pred = model(features)

The model learns:

• latent regime transitions
• nonlinear instability patterns
• hidden structural signals

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🏥 6. Application to Health Systems

Healthcare systems behave as complex adaptive systems.

We model hospitalization dynamics using:

• SIH/SUS data
• municipal time series
• population-normalized rates

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6.1 Data Source

SIH/SUS — Brazilian Hospital Information System

Each record corresponds to a hospitalization (AIH).

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6.2 Rate Definition

$$ Rate = \frac{\text{Hospitalizations}}{\text{Population}} \times 10^4 $$

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6.3 Structural Monitoring

For each municipality we compute:

• variance
• autocorrelation
• entropy
• metastable score

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📊 7. Results

Empirical analysis shows that:

• hospital systems exhibit metastable regimes
• transitions are preceded by early warning signals
• critical phases correspond to system overload risk

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🧠 8. Interpretation

We interpret healthcare systems as operating near:

marginal criticality

This enables:

• adaptability
• responsiveness
• but also vulnerability to collapse

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🌍 9. Implications

MetastableX enables:

• early detection of hospital stress
• structural monitoring of public health systems
• real-time systemic risk assessment

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⚙️ 10. Implementation

Installation

pip install metastablex

Usage

from metastablex import MetastableModel

model = MetastableModel()
model.fit(series)

score = model.predict(series)

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📊 11. Dashboard

The framework includes an interactive dashboard for:

• time-series visualization
• regime classification
• risk ranking

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🧱 12. Architecture

metastablex/
├── ml/
├── physics/
├── regimes/
├── model.py

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🔬 13. Scientific Contributions

This work contributes:

• a unified framework for metastability detection
• integration of physics + ML + epidemiology
• empirical validation on large-scale health data

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🚀 14. Future Work

• multiscale modeling
• spatial epidemiology
• digital twin of health systems
• real-time surveillance

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👨‍🔬 Author

Lucas Amaral Dourado

• Medical Student — Federal University of Tocantins
• Biomedical Engineering Student — UNINTER

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📜 License

MIT License

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MetastableX — Toward a physics of complex adaptive systems