Preserving friendships in school contacts: An algorithm to construct synthetic temporal networks for epidemic modelling

Abstract

High-resolution temporal data on contacts between hosts provide crucial information on the mixing patterns underlying infectious disease transmission. Publicly available data sets of contact data are however typically recorded over short time windows with respect to the duration of an epidemic. To inform models of disease transmission, data are thus often repeated several times, yielding synthetic data covering long enough timescales. Looping over short term data to approximate contact patterns on longer timescales can lead to unrealistic transmission chains because of the deterministic repetition of all contacts, without any renewal of the contact partners of each individual between successive periods. Real contacts indeed include a combination of regularly repeated contacts (e.g., due to friendship relations) and of more casual ones. In this paper, we propose an algorithm to longitudinally extend contact data recorded in a school setting, taking into account this dual aspect of contacts and in particular the presence of repeated contacts due to friendships. To illustrate the interest of such an algorithm, we then simulate the spread of SARS-CoV-2 on our synthetic contacts using an agent-based model specific to the school setting. We compare the results with simulations performed on synthetic data extended with simpler algorithms to determine the impact of preserving friendships in the data extension method. Notably, the preservation of friendships does not strongly affect transmission routes between classes in the school but leads to different infection pathways between individual students. Our results moreover indicate that gathering contact data during two days in a population is sufficient to generate realistic synthetic contact sequences between individuals in that population on longer timescales. The proposed tool will allow modellers to leverage existing contact data, and contributes to the design of optimal future field data collection.

Fig 1. Schematic representation of the friendship-based approach to synthetic contacts generation.

(A) Daily high-resolution empirical contact networks are schematically represented. They form the starting point (input) of the method. Friendship links are highlighted in orange, while blue links correspond to empirical links occurring only on a single day. (B) Different lists required for the algorithm are shown with examples of entries. Note that the timelines (lists of timestamps) are expressed in seconds from midnight on the initial day of the deployment, and the weights (“w”) are expressed in seconds. These lists are specific for each class and pair of classes, and each base day. (C) Description of the different steps of the algorithm. When assigning randomly a timeline and weight to contact links (step 3 of (C)), an entry of the third table is drawn. The algorithm operates class by class, and pair of classes by pair of classes, to generate synthetic contact networks. (D) Schematic representation of the generated synthetic contact networks. These networks inherit properties from the empirical contact networks, such as the number of links per class and between each pair of classes, a fraction of the friendship links (depicted in orange) and of the non-repeated links (depicted in blue). These links are complemented by random links (depicted in green) that were not necessarily observed in the base day.

Fig 2. Construction of infection pathways.

(A) Examples of simulated transmission chains between individuals are shown for a given seed. (B) An infection network ${\mathcal{G}}_{inf}(s,c{t}_x)$ built from 135 realisations of the model initialised with the same seed is shown. (C) Maximum spanning tree ${\mathcal{T}}_{inf}(s,c{t}_x)$ extracted from the infection network of (B). All results are obtained with ct_x = “Friendship 4d” contacts. Darker edges in (A) correspond to transmission events between different classes. Edge widths in (B) and (C) are proportional to their probability of occurrence p_ℓ(s, ct_x). Edges with probability of occurrence <0.01 are omitted for readability in (B). Nodes of the same class share the same color, and the seed is highlighted in black. Visualisations generated with Gephi [51].

Fig 3. Comparison between the different contact sequences.

(A) Daily total time measured in contact within and between classes for the recorded contacts on day 2. (B) Same as (A) for the corresponding friendship-based contacts. (C) Same as (A) for the corresponding class-mixing-based contacts. (D) Total time measured in contact between all individuals in the school on successive 15 minutes time steps on days 2 and 3 for the three types of contacts (empirical and two types of synthetic data). (E) Distribution of students’ local cosine similarities for each pair of days observed in the empirical contacts (black), together with the same distribution obtained with the friendship-based algorithm with optimised parameters, averaged over 10 realisations. (F) Same as (E) obtained instead from 10 realisations of the class-mixing-based approach. (G) Global similarities between the daily contact networks of consecutive days (computed by applying Eq 2 to the contact networks), for contact sequences obtained with different versions of the algorithm (each color corresponds to one single iteration of the contact sequence).

Fig 4. Epidemic size distributions.

(A) Distributions (Gaussian kernel density estimations) of the final epidemic sizes obtained with friendship-based contact sequences. (B) Same as (A) for class-mixing-based contact sequences. (C) Same as (A) for looped contact sequences. The distributions are computed over simulations leading to a fraction of infected individuals larger than 20% (over 120 days) in order to better highlight differences between the distributions. Results including all simulations are shown in the S1 Text, Fig S and T. The first and third quartiles (25% and 75%) are indicated with dotted lines while the median is shown with a dashed line. (D) Jensen-Shannon distance between all pairs of distributions. For each contact sequence, 150 simulations are conducted for each of the 325 students as seed (48, 750 simulations for each contact sequence).

Fig 5. Pairwise comparisons of simulated infection networks between students and classes.

(A) Distributions over all seeds of the global cosine similarities $GCS({\mathcal{G}}_{inf}(s,c{t}_a),{\mathcal{G}}_{inf}(s,c{t}_b))$ are shown for infection networks obtained from pairs of contact sequences ct_a and ct_b in “Friendship 2d”, “Friendship 3d” and “Friendship 4d” for infection networks between students. (B) Distributions of $GCS({\mathcal{G}}_{inf}(s,c{t}_a),{\mathcal{G}}_{inf}(s,c{t}_b))$ over all seeds s for different sequences ct_b (class-mixing-based and looped contacts) with ct_a fixed to “Friendship 2d” for infection networks between students. (C) Same as (B) with ct_a fixed to “Friendship 4d”. (D) Same as (A) for infection networks between classes. (E) Same as (B) for infection networks between classes. (F) Same as (C) for infection networks between classes. For each contact sequence, infection networks are obtained from 150 simulations for each seed, and each of the 325 students are successively considered as seed s.