Topological Percolation in Urban Dengue Transmission: A Multi-Scale Analysis of Spatial Connectivity

TL;DR

This study introduces a model-free, topological framework to analyze the spatial structure of urban dengue transmission in Recife from 2015 to 2024 by applying zero-dimensional persistent homology to Vietoris–Rips filtrations. By parameterizing the filtration with distance-percentiles, the authors identify distinct geometric percolation regimes and quantify transitions via empirical scales such as $\varepsilon_{\text{crit}}$, $\varepsilon_{\text{collapse}}$, and the global percolation index $P$, revealing that incidence alone cannot determine spatial organization. A striking 2020 rupture showcases how external perturbations to mobility alter spatial topology, despite high case counts. The results demonstrate that percolation-based topological observables provide interpretable, scalable measures of epidemic structure, with potential utility for surveillance and cross-city comparisons across vector-borne diseases.

Abstract

We investigate the spatial organization of dengue cases in the city of Recife, Brazil, from 2015 to 2024, using tools from statistical physics and topological data analysis. Reported cases are modeled as point clouds in a metric space, and their spatial connectivity is studied through Vietoris-Rips filtrations and zero-dimensional persistent homology, which captures the emergence and collapse of connected components across spatial scales. By parametrizing the filtration using percentiles of the empirical distance distribution, we identify critical percolation thresholds associated with abrupt growth of the largest connected component. These thresholds define distinct geometric regimes, ranging from fragmented spatial patterns to highly concentrated, percolated structures. Remarkably, years with similar incidence levels exhibit qualitatively different percolation behavior, demonstrating that case counts alone do not determine the spatial organization of transmission. Our analysis further reveals pronounced temporal heterogeneity in the percolation properties of dengue spread, including a structural rupture in 2020 characterized by delayed or absent spatial percolation. These findings highlight percolation-based topological observables as physically interpretable and sensitive descriptors of urban epidemic structure, offering a complementary perspective to traditional spatial and epidemiological analyses.

Figures (4)

• Figure 1: Global fusion curves $N(\varepsilon)$ for the zero-dimensional persistent homology of dengue cases in Recife (2015--2024), restricted to $\varepsilon \leq 1$ km. Years with intense outbreaks exhibit rapid component coalescence at short spatial scales, while milder years show gradual and diffuse percolation behavior.
• Figure 2: Spatial distribution of connected components of dengue cases in Recife at the critical percolation scale $\varepsilon^*$. Left: 2015 ($\varepsilon^* = 140$ m), characterized by early emergence of large, compact clusters. Right: 2020 ($\varepsilon^* = 210$ m), exhibiting fragmented and spatially dispersed components. Colors indicate distinct connected components.
• Figure 3: Relationship between the critical percolation scale $\varepsilon^*$ and annual dengue incidence in Recife (2015--2024). Lower values of $\varepsilon^*$ indicate faster spatial percolation and are strongly associated with higher incidence levels. The dashed curve represents a LOWESS smoothing shown only as a guide to the eye.
• Figure 4: Relationship between the global percolation index $P$ and annual dengue incidence in Recife (2015--2024). Smaller values of $P$, corresponding to more rapid spatial percolation, are associated with increased epidemic intensity. The dashed curve represents a LOWESS smoothing shown only as a guide to the eye.

Theorems & Definitions (2)

• Definition 1: Geometric percolation
• Definition 2