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Projecting social contact matrices to populations stratified by binary attributes with known homophily

Claus Kadelka

TL;DR

A new method is introduced, which uses linear algebra and non-linear optimization, to expand a given contact matrix to populations stratified by binary attributes with a known level of homophily, and enables any modeler to account for the presence ofhomophily with respect to binary attributes in contact patterns, ultimately yielding more accurate predictive models.

Abstract

Contact networks are heterogeneous. People with similar characteristics are more likely to interact, a phenomenon called assortative mixing or homophily. While age-assortativity is well-established and social contact matrices for populations stratified by age have been derived through extensive survey work, we lack empirical studies that describe contact patterns of a population stratified by other attributes such as gender, sexual orientation, ethnicity, etc. Accounting for heterogeneities with respect to these attributes can have a profound effect on the dynamics of epidemiological forecasting models. Here, we introduce a new methodology to expand a given e.g. age-based contact matrix to populations stratified by binary attributes with a known level of homophily. We describe a set of linear conditions any meaningful social contact matrix must satisfy and find the optimal matrix by solving a non-linear optimization problem. We show the effect homophily can have on disease dynamics and conclude by briefly describing more complicated extensions. The available Python source code enables any modeler to account for the presence of homophily with respect to binary attributes in contact patterns, ultimately yielding more accurate predictive models.

Projecting social contact matrices to populations stratified by binary attributes with known homophily

TL;DR

A new method is introduced, which uses linear algebra and non-linear optimization, to expand a given contact matrix to populations stratified by binary attributes with a known level of homophily, and enables any modeler to account for the presence ofhomophily with respect to binary attributes in contact patterns, ultimately yielding more accurate predictive models.

Abstract

Contact networks are heterogeneous. People with similar characteristics are more likely to interact, a phenomenon called assortative mixing or homophily. While age-assortativity is well-established and social contact matrices for populations stratified by age have been derived through extensive survey work, we lack empirical studies that describe contact patterns of a population stratified by other attributes such as gender, sexual orientation, ethnicity, etc. Accounting for heterogeneities with respect to these attributes can have a profound effect on the dynamics of epidemiological forecasting models. Here, we introduce a new methodology to expand a given e.g. age-based contact matrix to populations stratified by binary attributes with a known level of homophily. We describe a set of linear conditions any meaningful social contact matrix must satisfy and find the optimal matrix by solving a non-linear optimization problem. We show the effect homophily can have on disease dynamics and conclude by briefly describing more complicated extensions. The available Python source code enables any modeler to account for the presence of homophily with respect to binary attributes in contact patterns, ultimately yielding more accurate predictive models.
Paper Structure (7 sections, 4 theorems, 34 equations, 3 figures, 4 tables)

This paper contains 7 sections, 4 theorems, 34 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Contact matrices
  3. Homophily
  4. Extensions of contact matrices
  5. Application
  6. More complicated extensions of contact matrices
  7. Conclusion

Key Result

Theorem 2.8

Given a combined attribute space $A$, a corresponding distribution $N$ of a population, and a contact matrix $C\in [0,\infty)^{A\times A}$, the transformed contact matrix $\tilde{C}$ defined by is reciprocal.

Figures (3)

  • Figure 1: For different combinations of the prevalence $P$ of the added binary attribute (axes), the maximal homophily value ($h_{\max}$) that yields a non-negative solution of Equation \ref{['eq_linear_system']} is shown. $C$ and $N$ are defined as in Table \ref{['fig:running_ex_v1']}a.
  • Figure 2: Proportion of random prevalence vectors $P \in [0,1]^4$ with fixed $P_i$ ($i=1,\ldots,4$ in sub panels), for which $h_{\max}<1$ for $C$ and $N$ defined for the U.S. population (see Example \ref{['ex_real3']}).
  • Figure 3: Effect of homophily on disease dynamics. For different levels of homophily (sub panels), the proportion of currently infected people is shown for all four sub-populations considered in Example \ref{['ex_sir']}. Color distinguishes the value of the added binary attribute, while line style differentiates age.

Theorems & Definitions (38)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8
  • proof
  • Corollary 2.9
  • ...and 28 more