Eigenvector-Based Sensitivity Analysis of Contact Patterns in Epidemic Modeling

TL;DR

This study introduces an eigenvector-based sensitivity analysis framework that quantifies the influence of age-specific contact patterns on epidemic outcomes by applying perturbation analysis to the next-generation matrix, allowing us to pinpoint the age-group interactions most critical to transmission dynamics.

Abstract

Understanding how age-specific social contact patterns and susceptibility influence infectious disease transmission is crucial for accurate epidemic modeling. This study presents an eigenvector-based sensitivity analysis framework to quantify the impact of age-structured interactions on disease spread. By applying perturbation analysis to the Next Generation Matrix, we reformulate the basic reproduction number, $\mathcal{R}_0$, as a generalized eigenproblem, enabling the identification of key age group interactions that drive transmission. Using real-world COVID-19 contact data from Hungary, we demonstrate the framework's ability to highlight critical transmission pathways. We compare these findings with results obtained earlier using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC), validating the effectiveness of our approach. Additionally, we extend the analysis to contact structures in the UK and British Columbia, Canada, providing broader epidemiological insights. This work enhances our understanding of demographic interactions in epidemic propagation and offers a robust methodological foundation for improving infectious disease modeling and informing public health interventions.

Figures (7)

• Figure 1: Overview of the framework. This workflow illustrates the pipeline for analyzing the impact of contact patterns on the basic reproduction number, $\mathcal{R}_0$. It integrates scaled contact matrices from various social settings and employs the Next Generation Matrix methodology to compute $\mathcal{R}_0$ as the dominant eigenvalue of an epidemic model. The associated eigenvector is then used to derive $n_p$ sensitivity measures, demonstrating how variations in contact inputs affect $\mathcal{R}_0$. Automatic differentiation, implemented via PyTorch, is used to compute the gradients that underpin the sensitivity measures.
• Figure 2: Contact matrices illustrating social interactions in Hungary across different settings: Home, School, Work, and Other locations. Each heatmap shows the frequency of contacts between age groups (0--4 to 75+ years), where the horizontal axis represents survey participants (respondents), and the vertical axis corresponds to the individuals they reported having contact with (contactees). The color intensity reflects contact frequency, with darker shades indicating higher values. Contact rates are the average number of contacts per person per day. The matrices satisfy population-level reciprocity (Eqs. \ref{['total']}, \ref{['sym_contact']}). For visual clarity, alternating age group labels are shown along the axes. A consistent logarithmic color scale is applied to all heatmaps to allow direct comparison. The Full contact matrix is computed as a linear combination of the Home, School, Work, and Other matrices, as described in Eq. (\ref{['sym_contact']}). Due to the reciprocity condition, only its upper triangular section, containing $n_p$ unique values, is used for sensitivity analysis.
• Figure 3: The left panel shows the age-specific social contact matrix from Mossong et al. mossong2008, reflecting contact frequencies between age groups. The middle panel displays pairwise sensitivity scores, indicating how changes in age-group interactions affect $\mathcal{R}_0$. The right panel shows the group-level sensitivities, summarizing the total contribution of each age group to transmission potential. All results shown pertain to an influenza SEIR model (Pitman et al.).
• Figure 4: Sensitivity values are computed assuming a 50% reduction in susceptibility for the 0--19 age group. (Left) Sensitivity of the basic reproduction number $\mathcal{R}_0$ to contact perturbations. (Right) Sensitivity of mortality outcomes. Darker green shades indicate higher sensitivity values, meaning greater influence on the corresponding model output; lighter shades indicate lower sensitivity.
• Figure 5: Group-Level Sensitivity Analysis. (Left) Group-level sensitivity values for $\mathcal{R}_0$, reflecting the total influence of each age group on transmission dynamics. (Right) Group-level sensitivities for cumulative mortality. Bar heights represent overall impact, with darker red shades indicating higher sensitivity.