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Contact Matrices in Compartmental Disease Transmission Models

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Mathematics of Public Health

Part of the book series: Fields Institute Communications ((FIC,volume 88))

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Compartmental disease transmission models often stratify populations by factors such as risk, age, and geography. In such models, average rates of contact between each combination of population strata can be summarized as contact matrices. These matrices are key determinants of epidemic dynamics and intervention impact. In this chapter, we review the definition and application of contact matrices in compartmental transmission models. We explore different types of contacts and sources of data to support them, summarize key properties of contact matrices, examine the problem of restratifying contact matrices, and develop a method to incorporate age-stratified contact data and geographic mobility data. Throughout the chapter, we illustrate the concepts and methods discussed using a motivating example of developing contact matrices for SARS-CoV-2 transmission modelling. Overall, the chapter aims to be a practical introduction to contact matrices that will support the reader to construct and apply contact matrices in their own work.

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Change history

  • 08 February 2024

    A correction has been published.


  1. 1.

    Equation (4.1) assumes “frequency-dependent” transmission, which does not scale with population density; an alternate assumption is called “density-dependent” transmission, which replaces the population size N with an area A in (4.1), and changes the interpretation of C [5]. We assume frequency-dependent transmission throughout this chapter, because the survey-derived contact data used to inform C (see Sect. 4.2) are taken as fixed, as opposed to scaling with population density \(N/A\).

  2. 2.

    The distinction between aerosols and droplets falls along a spectrum but is mainly defined by the ability of aerosols to, unlike droplets, remain suspended in the air for several seconds or more.

  3. 3.

  4. 4.

    The intrinsic connectivity matrix \(\Gamma \) serves a similar role to the log-odds matrix \(\theta \) from Sect. 4.3.3, but the two matrices are not the same.


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The study was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC CGS-D); the Canadian Institutes of Health Research (VR5-172683); and the 2020 COVID-19 Centred Research Award from the St Michael’s Hospital Foundation Research Innovation Council.

From Unity Health Toronto, we thank Mackenzie Hamilton for helpful discussions and support in conceptualizing the chapter; Linwei Wang, Korryn Bodner, and Huiting Ma for helpful discussions; Kristy Yiu for research coordination support; and Gary Moloney for support with geographic data processing.

The data and R code used for the motivating example are available at

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Knight, J., Mishra, S. (2023). Contact Matrices in Compartmental Disease Transmission Models. In: David, J., Wu, J. (eds) Mathematics of Public Health. Fields Institute Communications, vol 88. Springer, Cham.

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