Estimating household contact matrices structure from easily collectable metadata

Abstract

Contact matrices are a commonly adopted data representation, used to develop compartmental models for epidemic spreading, accounting for the contact heterogeneities across age groups. Their estimation, however, is generally time and effort consuming and model-driven strategies to quantify the contacts are often needed. In this article we focus on household contact matrices, describing the contacts among the members of a family and develop a parametric model to describe them. This model combines demographic and easily quantifiable survey-based data and is tested on high resolution proximity data collected in two sites in South Africa. Given its simplicity and interpretability, we expect our method to be easily applied to other contexts as well and we identify relevant questions that need to be addressed during the data collection procedure.

Fig 1. Data collection schedule for the 60 selected households.

Each row corresponds to a household with the rural site on the left and the urban site on the right. Time is displayed on the x axis and dates are reported in the day/month format. Vertical gray lines correspond to the beginning and end of each deployment. A black dot indicates that at least one contact was measured, while a white one that no contact was recorded on that day.

Fig 2. Properties of the measured data.

a: normalized contact matrix across the three deployments. The color code refers to the values of the logarithm of R_counts whose entries are proportional to the ratio between the number of contacts and the number of possible interacting pairs, setting the mean of R_counts to 1. The two axis correspond to the age groups and the number reported indicates the highest age of each group. b: contact duration distribution expressed as number of seconds of interaction across the three deployments in logarithmic scale.

Fig 3. Test of the model for household interaction.

a: histogram of the cosine similarity between C and its estimators. The gray curve corresponds to the histogram over the 2500 realization of $\mathcal{X}$ using T as an estimator of C. The orange curve is obtained with the first order model of Eq (2), while the blue curve corresponds to the second order model of Section A second order model for household interaction. b, c: correlation between the fluctuations of the activity δ^(u), the group average degree δ^(η) and the presence of a major occupation outside the house δ^(y). The quantities δ_a,c are defined in Eq (3). The Pearson correlation coefficient r is reported in text.

Fig 4. Measured vs estimated normalized contact matrices in the two sites.

The first row, in blue, corresponds to Agincourt, the rural site, while the second, in purple, to Klerksdorp, the urban site. The first column shows the matrix C aggregated over the three deployments, as measured by the proximity sensors. The second column is the corresponding random encounter matrix T. The third and the fourth are the estimates obtained by our first and second order models, respectively. All matrices are normalized by the empirical average of their entries.