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A spatiotemporal negative binomial model with dynamic dispersion: An application to Tuberculosis infections

This paper introduces a novel spatiotemporal negative binomial INGARCH model with dynamic dispersion and flexible spatial correlation structures to accurately analyze and predict Tuberculosis infection patterns across Sao Paulo, Brazil, demonstrating superior performance over standard baselines in capturing spatial heterogeneity and temporal volatility for public health decision-making.

Original authors: Rodrigo B. Silva, Luiza S. C. Piancastelli, Wagner Barreto-Souza

Published 2026-09-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rodrigo B. Silva, Luiza S. C. Piancastelli, Wagner Barreto-Souza

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of public health, tracking infectious diseases is a constant race against time and uncertainty. Health officials rely on counting cases to understand how a disease moves, but these counts are rarely simple. They fluctuate wildly from month to month and vary drastically from one town to the next. A disease might surge in a crowded city center while remaining quiet in a rural village, only to spike again later. To make sense of this chaos, statisticians use models—mathematical frameworks that try to predict what will happen next based on what has happened before. For decades, these models often assumed that the "noise" or unpredictability in the data stayed the same over time. However, real-world outbreaks are rarely that steady. They breathe, swell, and shrink. The challenge for scientists is to build a model that can capture this living, breathing volatility, especially when the disease spreads across a map where neighbors influence one another.

This is the precise problem tackled by a team of researchers studying tuberculosis in the state of São Paulo, Brazil. Tuberculosis remains a stubborn global health threat, and in Brazil, its spread is shaped by deep inequalities in income, sanitation, and access to healthcare. The researchers focused on 61 small administrative regions, known as microregions, within São Paulo, analyzing monthly reports of new infections spanning twenty-four years, from 2001 to 2024. They found that the disease did not behave like a steady stream. Instead, the number of cases in any given month was highly unpredictable, with some areas experiencing sudden, intense bursts of infection while others remained relatively calm. Crucially, they observed that this unpredictability itself changed over time. In some periods, the data was tightly clustered around an average; in others, the numbers swung wildly. Furthermore, the infection in one town was often linked to what was happening in its neighbors, creating a complex web of regional influence.

To untangle this, the researchers developed a new statistical tool designed specifically for this kind of messy, real-world data. They created a model that allows both the average number of cases and the level of unpredictability to change dynamically as time passes and as the disease moves across the map. Unlike older models that forced the data to fit a rigid pattern, this new approach lets the "volatility" breathe. It acknowledges that a disease outbreak is not a static event but a shifting landscape where the rules of spread can tighten or loosen depending on local conditions. The team tested two different ways of defining how these regions interact. One method treated regions as neighbors only if they shared a physical border, like pieces of a puzzle. The other, more innovative approach, used a smooth mathematical curve to measure how influence fades as the distance between two towns increases, regardless of whether they share a border. This second method allowed the model to learn directly from the data how far the influence of an outbreak typically reaches.

When they applied this new framework to the twenty-four years of tuberculosis records, the results were striking. The model proved far superior to standard methods used by health agencies. While older models often produced predictions that were too narrow, failing to account for the wild swings in case numbers, the new model successfully captured the true range of uncertainty. It managed to predict the spread of the disease with a level of accuracy that kept its confidence intervals correct about 95 percent of the time, even in the most volatile urban centers and the quietest rural microregions. The analysis revealed that the disease's baseline presence is heavily concentrated in dense metropolitan hubs, but the way it spreads is driven by short-range interactions. The data suggested that the influence of an outbreak in one town on its neighbors drops off quickly with distance, a pattern that the new, distance-based model captured with greater nuance than the simple border-sharing method.

The study also highlighted the limitations of previous approaches. Models that assumed the unpredictability of the data remained constant over time failed to reflect reality, often underestimating the risk during sudden spikes in infection. Similarly, models that treated all regions as having the same underlying rules for how the disease spreads struggled to handle the vast differences between a bustling city and a small town. By allowing the statistical properties of the disease to vary from place to place and from month to month, the researchers provided a much clearer picture of the epidemic's behavior. Their work suggests that to effectively manage tuberculosis, health officials need tools that can adapt to the changing nature of the threat, recognizing that the volatility of an outbreak is just as important to track as the number of cases itself.

The implications of this work extend beyond just tuberculosis. The methodology offers a robust way to monitor any disease that spreads through a population in a patchwork of locations, where the intensity of the spread changes over time. By accurately modeling how volatility shifts and how influence decays across a landscape, public health authorities can better anticipate localized outbreaks. This capability is vital for making informed decisions about where to send resources, how to time interventions, and how to protect vulnerable communities before a crisis deepens. The researchers have shown that by treating disease data as a dynamic, evolving system rather than a static set of numbers, we can build a more reliable foundation for protecting public health.

Technical Summary: A Spatiotemporal Negative Binomial Model with Dynamic Dispersion

Problem Statement
Tuberculosis (TB) surveillance data, particularly in Brazil, exhibit complex spatiotemporal dynamics characterized by pronounced spatial heterogeneity, temporal volatility, and significant overdispersion. Standard statistical frameworks often fail to capture these features simultaneously. While integer-valued time series models (INGARCH) and spatial models are well-established individually, their integration for count data remains underdeveloped. Existing multivariate INGARCH models typically assume static dispersion parameters and rely on rigid, pre-specified spatial weight matrices (e.g., binary adjacency), which may not adequately reflect the continuous decay of disease transmission across geographic distances. Furthermore, standard Poisson or fixed-dispersion models often fail to provide reliable uncertainty quantification during periods of localized outbreaks or volatility.

Methodology
The authors propose a Negative Binomial spatial INGARCH model with dynamic dispersion (NB tvd-SPINGARCH). This framework extends the univariate time-varying dispersion INGARCH (tvd-INGARCH) model of Barreto-Souza et al. (2026) to a multivariate spatial setting.

Model Structure: The model assumes that, conditional on historical information $\mathcal{F}_{t-1}$ $F_{t - 1}$ , the count vector $\mathbf{Y}_t$ $Y_{t}$ follows independent Negative Binomial distributions for each microregion $i$ $i$ . Both the conditional mean ( $\mu_{it}$ $μ_{i t}$ ) and the dispersion parameter ( $\phi_{it}$ $ϕ_{i t}$ ) are modeled as dynamic processes:
- Mean Process: $\lambda_{it} = \log(\mu_{it})$ follows a log-linear recursion incorporating autoregressive terms (past counts of region $i$ ) and spatial interaction terms (weighted past counts of neighboring regions $j$ ).
- Dispersion Process: $\nu_{it} = \log(\phi_{it})$ follows a similar log-linear recursion, allowing the dispersion to evolve over time and space, thereby capturing changing volatility.
Spatial Dependence: The model evaluates two distinct spatial weighting schemes ( $w_{ij}$ $w_{ij}$ ) to quantify inter-regional spillovers:
1. Discrete Adjacency: A binary structure where weights are non-zero only if regions share a border (normalized by the number of neighbors).
2. Continuous Matérn Correlation: A novel formulation where weights are derived from the Matérn correlation function, parameterized by smoothness ( $\alpha$ ) and scale ( $\beta$ ). This allows spatial dependence to decay continuously with distance rather than being confined to administrative boundaries.
Estimation: Parameter estimation is performed via Conditional Maximum Likelihood (CML). Due to the computational complexity of jointly estimating region-specific parameters and the global Matérn parameters ( $\alpha, \beta$ $α, β$ ), the authors employ a two-step profile-likelihood iterative scheme:
1. Fixing regional parameters to optimize spatial parameters ( $\alpha, \beta$ ).
2. Fixing spatial parameters to optimize regional parameters ( $\vartheta$ ) independently for each location.
  This process iterates until convergence.
Uncertainty Quantification: Standard errors and confidence intervals are derived using a parametric bootstrap procedure, simulating trajectories from the fitted model to estimate sampling variability.

Key Contributions

Methodological Extension: The paper generalizes the time-varying dispersion INGARCH framework to a multivariate spatial network, enabling the joint estimation of dynamic means and dynamic variances across multiple geographic units.
Flexible Spatial Weighting: It introduces the Matérn correlation function into spatiotemporal INGARCH modeling as a continuous alternative to rigid adjacency matrices. This allows for the estimation of spatial range and smoothness directly from the data.
Empirical Application: The framework is applied to 24 years of monthly TB notification data from 61 microregions in São Paulo, Brazil, demonstrating the necessity of dynamic dispersion for accurate modeling of infectious disease volatility.

Results

Model Fit: The NB tvd-SPINGARCH models (both Adjacency and Matérn variants) substantially outperformed standard Poisson SPINGARCH and the global-parameter glmSTARMA framework (Maletz et al., 2026) in terms of log-likelihood, AIC, and BIC.
Uncertainty Quantification: The proposed models achieved empirical coverage close to the nominal 95% level across diverse microregions (from dense urban centers like São Paulo to rural areas like Dracena). In contrast, the equidispersed Poisson model produced artificially narrow intervals with very low coverage, and the glmSTARMA model showed degraded performance in heterogeneous urban centers due to its global parameterization.
Spatial Structure: While information criteria (AIC/BIC) were non-decisive between the Adjacency and Matérn formulations, simulation studies evaluating the reproduction of spatial dependencies favored the Matérn formulation. The Matérn model more accurately captured the decay of spatial correlations over distance and the structure of lag-1 cross-correlations compared to the discrete adjacency approach.
Data Insights: The analysis confirmed marked spatial heterogeneity in baseline incidence and identified that overdispersion is dynamic, driven by localized outbreaks. The results suggest that TB transmission is predominantly short-ranged, with spatial interaction decaying over distance.

Significance
The paper claims that the proposed methodology provides a robust statistical tool for public health surveillance. By accurately modeling spatiotemporal volatility and dynamic dispersion, the NB tvd-SPINGARCH framework supports more reliable prediction intervals and uncertainty quantification compared to fixed-dispersion baselines. This capability is essential for anticipating localized outbreaks, evaluating regional interventions, and optimizing resource allocation. The authors posit that the methodology offers a versatile approach for analyzing high-dimensional, volatile count processes in epidemiology and environmental monitoring beyond the specific context of Tuberculosis.

Technical Summary: A Spatiotemporal Negative Binomial Model with Dynamic Dispersion

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