We use a high-resolution contact network measured by the SocioPatterns collaboration [
34] using wearable proximity sensors in a primary school. The sensors detect the face-to-face proximity relations (“contacts”) of individuals with a 20-seconds temporal resolution [
35]. The time-resolved contact network considered here, analyzed in Ref. [
38], describes the contacts among 232 children and 10 teachers in a primary school in Lyon, France, and covers two days of school activity (Thursday, October 1
s
t
and Friday, October 2
n
d
2009). The school is composed by 5 grades, each of them comprising two classes, for a total of 10 classes. Contacts events are individually resolved, and their starting and ending times are known up to the 20-second resolution of the measurement system.
The French national bodies responsible for ethics and privacy, the Commission Nationale de l’Informatique et des Libertés (CNIL,
http://www.cnil.fr) and the ‘Comité de Protection des personnes’ (
http://www.cppsudest2.fr/), were notified of the study, which was approved by the relevant academic authorities (by the ‘directeur de l’enseignement catholique du diocèse de Lyon’, as the school in which the study took place is a private catholic school). In preparation for the study, parents and teachers were informed through an information leaflet, and were invited to a meeting in which the details and the aims of the study were illustrated. Verbal informed consent was then obtained from parents, teachers and from the director of the school. As no personal information of participants was collected, the relevant academic authorities considered that written consent was not needed. Special care was paid to the privacy and data protection aspects of the study: The communication between the sensors and the computer system used to collect data were fully encrypted. No personal information of participants was associated with the identifier of the corresponding sensor.
Daily-aggregated contact networks were published in the context of the original paper [
38]. Here we make available to the public the full high-resolution dataset. We publish here as Additional file
1 the full time-resolved contact list, with node metadata on school role (students vs teachers) and class/grade affiliation of each individual.
Extending the temporal span of the empirical data
Realistic parameters for the infectious and latent periods of influenza-like disease are of the order of days. Since the dataset we use only spans two school days, our numerical simulations will unfold over time scales longer than the duration covered by the contact dataset. To address this problem, several possibilities to extend in time the empirical contact data have been explored [
41]. Here we consider a simple periodic repetition of the 2-day empirical data, modified to take into account specific features of the school environment under study. First, since our data only describes contacts during school hours, we assume that children are in contact with the general community for the rest of the day. Moreover, children in France do not go to school on Wednesday, Saturday and Sunday: on these days, therefore, children are also considered in contact with the general community. Overall the temporal contact patterns we use have the following weekly scheme:
i)
Monday and Tuesday correspond to the first and second day of the empirical dataset: between 8.30am and 5:00pm contacts within the school are described by the empirical data. Outside of this interval, children are assumed to be isolated from one another and in contact with the community.
ii)
Wednesday: children are in contact with the community for the entire day.
iii)
Thursday and Friday: the first and second day of the empirical dataset are repeated as in i).
iv)
Saturday and Sunday: children are in contact with the community for the entire weekend.
The above weekly sequence is repeated as many times as needed. Other extension procedures include partial reshuffling of the participants’ identities across days [
41], to model the partial variability of each individual’s contacts from one day to the next. Here we limit our investigation to the simple scheme outlined above, because a repetition procedure is appropriate to model a school environment, where activities follow a rather repetitive daily and weekly rhythm, and each child is expected to interact every day with approximately the same set of individuals, namely the members of her/his class and her/his acquaintances in other classes.
Epidemic model
To simulate the spread of an influenza-like disease we consider an individual-based stochastic SEIR model with asymptomatic individuals, with no births, nor deaths, nor introduction of individuals [
42]. In such a model each individual at a given time can be in one of five possible states: susceptible (S), exposed (E), infectious and symptomatic (I), infectious and asymptomatic (A), and recovered (R). Whenever a susceptible individual is in contact with an infectious one, she/he can become exposed at rate
β if the infectious individual is symptomatic, and
β/2 if the infectious individual is asymptomatic [
43]-[
47]
b. Exposed individuals, who cannot transmit the disease, become infectious after a latent period of average 1/
μ. Exposed individuals becoming infectious have a probability
p
A
of being asymptomatic (A) and a probability 1−
p
A
of being symptomatic (I). Both symptomatic and asymptomatic infectious individuals recover at the end of the infectious period of average duration 1/
γ, and acquire permanent immunity to the disease.
As mentioned above, our data describe human contacts only within the school premises. During the spread of an epidemic in the community, however, exposure to infectious individuals also occurs outside of school. Accordingly, we consider that individuals have a generic risk of being contaminated by infectious individuals outside of the school. For simplicity, here we assume that this risk is uniform and we introduce it into the model through a fixed rate of infection β
com
. That is, the probability that a susceptible individual, during a small time interval dt, becomes exposed due to random encounters outside of school is β
com
d
t.
Finally, we assume that symptomatic individuals are detected at the end of each day. They are subsequently isolated until they recover and therefore cannot transmit the disease anymore. Asymptomatic individuals, on the other hand, cannot be detected and thus are not isolated. Each simulation starts with a completely susceptible population, except for a single, randomly chosen infectious individual, chosen as symptomatic with probability 1−p
A
and asymptomatic with probability p
A
.
We consider the following parameter values for the SEIR model:
β=3.5·10
−4
s
−1 (1/
β≈48
m
i
n)
c,
β
com
=2.8·10
−9
s
−1, 1/
μ=2 days, 1/
γ=4 days. As in many previous studies [
43]-[
47],[
49] and in a way compatible with empirical results [
49], the fraction of infected asymptomatic individuals is set to
p
A
=1/3. These parameter values are in line with those commonly used in models of influenza-like illnesses [
41],[
44]-[
46],[
50],[
51]. Moreover, for each infected individual, we extract at random the durations of her/his latency and infectious periods from Gaussian distributions of respective averages 1/
μ and 1/
γ and standard deviations equal to one tenth of their average. We perform simulations with a time step
dt determined by the temporal resolution of the data set considered, namely 20 seconds.
We carry out sensitivity analyses with respect to our modelling choices and parameters. First, we consider a larger value of
β
com
while keeping fixed the values of the other parameters, to investigate the role of the generic risk of infection in the community. Second, we report in the Additional file
2 the results obtained with two different sets of parameters corresponding to faster spreading processes, namely: (i)
β=6.9·10
−4
s
−1 (1/
β≈24
m
i
n);
β
com
=2.8·10
−9
s
−1; 1/
μ=1 day; 1/
γ=2 days,
p
A
=1/3, and (ii)
β=1.4·10
−3
s
−1 (1/
β≈12
m
i
n),
β
com
=2.8·10
−9
s
−1, 1/
μ=0.5 day, 1/
γ=1 day,
p
A
=1/3. Moreover, we also show in the Additional file
2 results obtained by assuming a larger fraction of asymptomatic individuals, namely
p
A
=1/2. Third, we consider in the Additional file
2 a different shape for the distributions of the latent and infectious periods, namely Weibull distributions of average values 1/
μ and 1/
γ and various shape parameters, corresponding to broader distributions.
Mitigation measures
The baseline mitigation measure is given by the isolation of symptomatic children at the end of each day. We consider the three following additional strategies: whenever the number of symptomatic infectious individuals detected in any class reaches a fixed threshold,
(i)
the class is closed for a fixed duration (“targeted class closure” strategy);
(ii)
the class and the other class of the same grade are both closed for a fixed duration (“targeted grade closure” strategy);
(iii)
the entire school is closed for a fixed duration (“whole school closure” strategy).
In all cases, the children affected by the closure are considered to be in contact with the community during the closure period – with the exception of detected infectious cases – and therefore they have a probability per unit time β
com
of acquiring the disease. When the closure is over, the class (or grade) is re-opened and the corresponding children go back to school.
For benchmarking purposes, in the Additional file
2 we also consider strategies based on random class closures: whenever the number of symptomatic infectious individuals detected in any class reaches a fixed threshold,
(iv)
one random class, different from the one in which symptomatic individuals are detected, is closed (“random class closure” strategy)
(v)
the class and a randomly chosen one in a different grade are closed (“mixed class closure” strategy).
Note that during the course of an epidemic, in principle, several classes can be closed at the same time or successively, but once a class (or grade) is re-opened, we do not allow it to be closed again. Similarly, when using the whole-school closure strategy, we assume for simplicity that once the school is re-opened it cannot be closed again.
All of the closure strategies describe above depend on two parameters: the closure-triggering threshold, i.e., the number of symptomatic individuals required to trigger the intervention, and the duration of the closure. We will explore thresholds of 2 or 3 symptomatic individuals and closure durations ranging from 24 to 144 hours (from 1 to 6 days). Closure durations are specified in terms of absolute time: for instance, a 72 hours closure starting on a Thursday night spans the following Friday, Saturday and Sunday and ends on the next Monday morning.