Mathematical Characterization of a Three-Layer Ontological Framework for Dengue Early Warning Using Joint Episodes

This research presents evidence that the three ontological layers of the Tau Sistémico paradigm—Intensification (Layer 1), Permeability (Layer 2), and Structural Reorganization (Layer 3)—can be rigorously characterized and operationalized mathematically using recurrence quantification analysis, Joint episode detection, and calibrated confidence scoring. The framework is demonstrated through a dengue early-warning application developed within the RECD + Ontología del Presente research program. This application resulted from eight sequential phases of methodological development, culminating in controlled synthetic validation. The primary limitation is the scarcity of observed Joint episodes, particularly in Iquitos. To maintain ontological fidelity while allowing for calibration, we created a synthetic generator that produces controlled realizations of Layer 1 + Layer 2 configurations under different regimes. This approach enables systematic calibration of the Confidence Score and reveals regime-specific behavior that historical episodes alone cannot support. The work is deposited to support reproducibility and to solicit constructive scholarly feedback. Comments can be submitted via the Zenodo platform or through the associated repository. Subsequent revisions will incorporate results from real-world operational validation with extended Puerto Rico datasets. --- Abstract-- Effective early-warning systems for dengue are hindered by the lack of mechanistically grounded analytical units that link local dynamical processes to outbreak-scale reorganization. We introduce a three-layer ontological framework, where Layer 1 represents local intensification (high recurrence-based M and trapping time), Layer 2 signifies sustained permeability (low antisynchrony), and Layer 3 corresponds to detectable structural reorganization. The fundamental unit of analysis is the **Joint episode**—a coherent interval of Layer 2 containing Layer 1 intensification—which serves as the basis for a single, actionable alert per episode accompanied by an interpretable Confidence Score. Only 11 Joint episodes were identified for San Juan and three for Iquitos. Consequently, we developed a controlled synthetic generator that produces realistic Layer 1 + Layer 2 configurations under three regimes: San Juan-like (high-intensity, frequent Layer 2), Iquitos-like (low-intensity, rare sustained Layer 2), and noisy balanced. Weightings that emphasized duration and Joint Score performed best under high-prevalence conditions (Alto F1 up to 0.55). In the Iquitos-like regime, no episodes reached the "Alto" category, and the mean confidence remained low, indicating a qualitatively distinct dynamical regime rather than mere data scarcity. This work establishes that the three ontological layers of the Tau Sistémico paradigm can be rigorously characterized and operationalized mathematically: Layer 1 through recurrence metrics (M, trapping time), Layer 2 through sustained low antisynchrony, and their joint realization through the Joint episode and calibrated Confidence Score. This pipeline enhances both conceptual coherence and operational utility by replacing per-timestep classification with episode-level alerts and categorical prioritization. --- Mathematical Characterization of the Three Layers-- The results demonstrate that the three ontological layers can be rigorously mathematically characterized using recurrence quantification analysis. Layer 1 intensification is represented by the recurrence rate M and trapping time. Layer 2 permeability is characterized by the duration of sustained intervals of low antisynchrony. The Joint episode is defined by the maximal temporal window within which Layer 1 excursions occur while Layer 2 is in a state. The Joint Score combines their intensities, while the four-component Confidence Score estimates the latent probability of transitioning to Layer 3. This mathematical operationalization is a significant contribution, as it provides precise and reproducible operational definitions for abstract ontological categories within the Systemic Tau paradigm, grounded in established recurrence metrics. --- Data and Code Availability-- All code for Phases 1–8, the synthetic episode generator, weight-grid evaluation routines, and supporting scripts are available in the project repository: https://github.com/johelpadilla/ontologia-presente (subdirectory: preprints/recd_dengue_joint_episodes_2026). Synthetic episode datasets (containing 2,200 episodes), calibration tables, reliability plots, the external validation protocol, and planning documents are also included. The real incidence data used in this study are sourced from the public DengAI competition. Related prior work (same research program): - https://doi.org/10.5281/zenodo.19654897 - https://doi.org/10.5281/zenodo.20723966