Joint Reproduction Number and Spatial Connectivity Structure Estimation via Graph Sparsity-Promoting Penalized Functional

TL;DR

This work tackles the challenge of real-time epidemiological surveillance under poor quality counts by jointly estimating the time-varying reproduction number $\boldsymbol{\mathsf{R}}$ across territories and the spatial connectivity encoded in a Laplacian $\mathsf{L}$. It extends a scaled Poisson spatiotemporal framework to multivariate data and introduces a variational objective that fuses a $D_{\mathsf{KL}}$ data fidelity with temporal sparsity and graph-based spatial regularization, while concurrently learning the connectivity structure. The optimization uses an alternating scheme with Chambolle–Pock primal–dual updates for $\boldsymbol{\mathsf{R}}$ and a quadratic program for $\mathsf{L}$, yielding a joint estimator denoted $\widehat{\boldsymbol{\mathsf{R}}}^{\mathsf{Joint}}$, $\widehat{\mathsf{L}}^{\mathsf{Joint}}$. Numerical experiments on synthetic data show substantial improvements in reproduction number accuracy and near-perfect recovery of the connectivity; application to 39-country COVID-19 data reveals meaningful clusters and coherent dynamics, with code publicly available. This approach provides robust, interpretable spatiotemporal indicators useful for epidemiological surveillance and decision making.

Abstract

During an epidemic outbreak, decision makers crucially need accurate and robust tools to monitor the pathogen propagation. The effective reproduction number, defined as the expected number of secondary infections stemming from one contaminated individual, is a state-of-the-art indicator quantifying the epidemic intensity. Numerous estimators have been developed to precisely track the reproduction number temporal evolution. Yet, COVID-19 pandemic surveillance raised unprecedented challenges due to the poor quality of worldwide reported infection counts. When monitoring the epidemic in different territories simultaneously, leveraging the spatial structure of data significantly enhances both the accuracy and robustness of reproduction number estimates. However, this requires a good estimate of the spatial structure. To tackle this major limitation, the present work proposes a joint estimator of the reproduction number and connectivity structure. The procedure is assessed through intensive numerical simulations on carefully designed synthetic data and illustrated on real COVID-19 spatiotemporal infection counts.

Figures (3)

• Figure 1: COVID-19 infection counts reported in France, Italy, and the United-Kingdom from Sept. $1$, $2020$ to Oct. $1$, $2021$. ft:jhu
• Figure 2: Spatially structured synthetic data.Top left: Graph with $C = 9$ vertices split into $I = 3$ clusters; top right:$I = 3$ realistic reproduction numbers constructed following du2023compareddu2024synthetic; bottom: synthetic multivariate scaled Poisson counts in each of the $I=3$ clusters.
• Figure 3: Left: Map of the countries from the real data experiments. Colors indicate clusters derived from the spatial structure infered by $\mathsf{Joint}$. Hashed countries are isolated ones. Right: Reproduction numbers estimated from the $\mathsf{Joint}$ method organized by clusters.