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A Dynamic Latent Space Model for Healthcare Mobility Networks: the Italian National Health Service case

This paper introduces a Bayesian dynamic latent space model with a hurdle negative binomial likelihood to analyze and quantify the evolving structural asymmetries, pandemic disruptions, and territorial disparities in patient mobility flows across 109 Italian Local Health Authorities from 2018 to 2024.

Original authors: Cecilia Manente, Marco Alfò, Silvia D'Angelo

Published 2026-06-01

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CC BY 4.0

Original authors: Cecilia Manente, Marco Alfò, Silvia D'Angelo

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

Imagine Italy's healthcare system as a massive, bustling train network. Instead of trains, the passengers are patients needing hip replacements. Instead of stations, the stops are local health districts (called ASLs).

Usually, people get treated at the station closest to their home. But sometimes, they hop on a train to a different station—perhaps because that station has a better reputation, more specialized doctors, or simply because the local one is full. This paper is about mapping out exactly where these "passengers" are going, why they are going there, and how the map has changed over the last few years.

Here is the breakdown of the research in simple terms:

The Problem: A Map That Was Too Blurry

For a long time, researchers looked at this train network by grouping entire regions together (like "The North" vs. "The South"). It was like looking at a map where Italy was just three big blobs. This hid the real story. Inside those big blobs, some small towns were sending almost everyone away, while others were keeping everyone close.

The authors wanted to zoom in. They looked at 109 specific local districts instead of just the big regions. They tracked over 800,000 hip replacement surgeries between 2018 and 2024.

The Challenge: The "Ghost" Flows and The "Crowded" Trains

Two things made this data tricky to analyze:

The Ghosts (Zero Flows): Most pairs of districts don't exchange any patients. If you tried to draw a line between every possible pair of towns, 75% of the lines would be empty. Standard math tools get confused by so many empty spots.
The Crowded Trains (Overdispersion): When patients do travel, they don't travel in small, equal groups. A few famous hospitals get thousands of patients, while others get just a handful. The data is "spiky," not smooth.

The Solution: A "Magic Invisible Map"

To solve this, the authors built a new kind of statistical model. Think of it as a magic invisible map where every local health district is a dot.

The Invisible Distance: On this map, dots that are close together represent districts that naturally exchange patients (maybe they are neighbors or have similar hospital quality). Dots far apart rarely exchange patients.
The Moving Map: This isn't a static map; it's an animation. The dots move slightly every year, showing how the relationships between districts change over time.
The "Hurdle" Mechanism: The model has a special trick for the "ghosts." It asks two questions:
1. Will a patient travel at all? (The hurdle).
2. If they do, how many will go? (The count).
  This allows the model to handle the empty spots and the crowded trains separately.
The "Sender" and "Receiver" Roles: The model realizes that some districts are naturally "generators" (they send many people out) and some are "magnets" (they pull many people in). It separates these roles so it doesn't confuse a big city (which has lots of people just because it's big) with a truly attractive hospital.

What They Found

1. The North-South Divide is Real
The invisible map confirmed what people suspected: The South is mostly "sending" patients, and the North is mostly "receiving" them. Southern districts often look like they are on the edge of the map, while Northern districts are clustered in the center, acting as major hubs.

2. The Islands Have Their Own Rules
Sicily and Sardinia formed their own distinct clusters. They don't just look like "Southern Italy"; they have unique patterns. Sardinian patients seem to look toward the Northeast of Italy, while Sicilian patients look toward the Northwest and the South. It's like they have their own private train lines that don't match the mainland routes.

3. The Pandemic Shook the Map
When the pandemic hit in 2020, the map got messy. Elective surgeries (like hip replacements) were paused. When they started again in 2021 and 2022, the districts didn't all recover at the same speed. The "dots" on the map spread out further, showing that the system became more uneven. Some areas bounced back quickly; others lagged behind.

4. Size Matters (But Not How You Think)
If you just look at raw numbers, big cities like Milan and Rome look like the biggest magnets. But the authors' model corrected for population size. When they did that, the "real" magnets turned out to be some smaller, specialized hubs in Sardinia and Sicily that pull in patients from their own islands and beyond, despite having fewer people living there.

Why This Matters

This study didn't just count patients; it revealed the hidden geography of healthcare. It showed that the way patients move isn't random—it follows a structured, evolving pattern.

By using this "magic map," health officials can see:

Which districts are struggling to keep their own patients.
Which districts are becoming overloaded with outsiders.
How the system is recovering after big shocks (like the pandemic).

The paper concludes that this mathematical tool is a flexible way to watch the health system's "heartbeat" over time, helping to spot inequalities and structural problems that a simple count of patients would miss.

Technical Summary: A Dynamic Latent Space Model for Healthcare Mobility Networks

Problem Statement
Healthcare mobility—the phenomenon where patients seek treatment outside their territory of residence—creates significant inequality and financial imbalances in decentralized health systems. In Italy, persistent North–South asymmetries in patient flows among Local Health Authorities (ASLs) have reinforced disparities within the National Health Service (SSN). While existing monitoring frameworks typically aggregate data at the regional level, this obscures substantial within-region heterogeneity. Furthermore, traditional quantitative approaches, such as gravity models, treat origin–destination pairs as independent, failing to capture structural network features like reciprocity, sender/receiver heterogeneity, and higher-order clustering. Additionally, mobility data at the ASL level exhibit specific statistical challenges: high sparsity (excess zeros), strong overdispersion, and temporal dynamics that static models cannot adequately address.

Methodology
The authors propose a Bayesian dynamic latent space model (LSM) for directed, weighted networks, specifically tailored to healthcare mobility data. The framework integrates three critical statistical components:

Hurdle Negative Binomial Likelihood: To address the dual nature of the data, the model decomposes the data-generating process into two parts:
- A binary component (logistic hurdle) determining the probability of a positive flow ( $\pi_{ij}^{(t)}$ ), distinguishing structural zeros from sampling zeros.
- A zero-truncated negative binomial component modeling the magnitude of positive flows, accounting for the strong overdispersion observed in the data.
Dynamic Latent Space: Nodes (ASLs) are embedded in a low-dimensional Euclidean space (fixed at $d=2$ for interpretability). The probability of interaction decreases as the latent distance between nodes increases. Temporal dependence is modeled via a Gaussian random walk, where latent positions evolve smoothly over time ( $x_i^{(t)} | x_i^{(t-1)} \sim N(x_i^{(t-1)}, \sigma_X^2 I_2)$ ), capturing the evolving geometry of the healthcare system.
Multiplicative Sender and Receiver Effects: To explicitly model directional asymmetry, the mean of the positive flows incorporates node-specific parameters: $\gamma_i$ (sender effect, propensity to send patients) and $\theta_j$ (receiver effect, attractiveness). These are estimated separately from the latent positions, allowing the latent space to represent the symmetric, aggregate structure of patient exchange.
Covariates and Exposure: The model incorporates geographical distance as a deterrent in the hurdle component and includes an exposure term ( $E_{ij}^{(t)}$ ) based on marginal flow volumes to control for differences in territorial size and hospital capacity.

Estimation is performed using a Metropolis–Hastings within Gibbs MCMC algorithm. The posterior inference involves updating scalar parameters via random-walk steps and updating latent coordinates individually (node-by-node, year-by-year) with Procrustes transformations to ensure identifiability regarding rotation and translation.

Key Results
The model was applied to administrative discharge records for hip replacement procedures involving 109 Italian ASLs from 2018 to 2024.

Model Selection: A specification combining the multiplicative sender/receiver effects, geographical distance in the hurdle component, and exposure adjustment (Mult Geo Offset) provided the best overall fit (lowest DIC, highest cosine similarity), successfully reproducing the relational structure of the network.
Latent Geometry: The estimated latent space revealed persistent clustering by macro-area. Notably, the island regions (Sicily and Sardinia) formed distinct clusters, reflecting unique logistical and referral patterns. The dispersion of latent positions increased monotonically from 2018 to 2024, with a notable acceleration between 2019 and 2021, coinciding with the disruption of elective surgery during the COVID-19 pandemic.
Directional Heterogeneity: The inclusion of the exposure term significantly altered the interpretation of receiver effects. Without exposure correction, large northern facilities dominated due to absolute volume. With exposure correction, the model identified smaller hub ASLs in Sardinia and Sicily as having high structural attractiveness and outward propensity, independent of their size. Conversely, the Campania cluster consistently showed high outward propensity (passive mobility), while Lombardy ASLs demonstrated strong local retention (low sender effects).
Overdispersion Dynamics: The dispersion parameter ( $r$ ) showed a temporary increase (reduction in overdispersion) during 2020–2021, indicating that flows became more concentrated along established corridors during the pandemic. Post-2021, $r$ declined, suggesting a return to higher heterogeneity in flow intensities as activity resumed unevenly across regions.
Simulation Performance: Simulation studies confirmed the MCMC algorithm's ability to recover latent geometry with high accuracy (Procrustes correlations > 0.96) and sender/receiver effects (correlations > 0.98) across varying levels of sparsity and overdispersion.

Significance and Claims
The paper claims to provide a flexible, unified framework for the statistical analysis of dynamic healthcare mobility networks that simultaneously addresses excess zeros, overdispersion, and network dependence. Its primary contributions are:

Methodological Integration: It extends latent space models to directed, weighted networks with hurdle negative binomial likelihoods and explicit sender/receiver effects, separating directional heterogeneity from the symmetric latent structure.
Granular Insight: By analyzing data at the ASL level rather than the regional level, the model reveals nuanced relational structures, including local hubs and peripheries, which are obscured in aggregated analyses.
Policy Relevance: The framework offers a tool for monitoring the evolving geometry of healthcare systems, quantifying the impact of external shocks (like the pandemic), and uncovering structural asymmetries in patient flows. This supports the evaluation of territorial healthcare provision and the effectiveness of decentralization policies.

The authors emphasize that while the model captures the underlying geometry and structural asymmetries effectively, the primary inferential objective is the recovery of the latent network structure rather than the exact prediction of small flow counts. The methodology is presented as directly applicable to any context involving directed weighted flows among territorial units observed over time.