Contact patterns among high school students

Face-to-face contacts between individuals contribute to shape social networks and play an important role in determining how infectious diseases can spread within a population. It is thus important to obtain accurate and reliable descriptions of human contact patterns occurring in various day-to-day life contexts. Recent technological advances and the development of wearable sensors able to sense proximity patterns have made it possible to gather data giving access to time-varying contact networks of individuals in specific environments. Here we present and analyze two such data sets describing with high temporal resolution the contact patterns of students in a high school. We define contact matrices describing the contact patterns between students of different classes and show the importance of the class structure. We take advantage of the fact that the two data sets were collected in the same setting during several days in two successive years to perform a longitudinal analysis on two very different timescales. We show the high stability of the contact patterns across days and across years: the statistical distributions of numbers and durations of contacts are the same in different periods, and we observe a very high similarity of the contact matrices measured in different days or different years. The rate of change of the contacts of each individual from one day to the next is also similar in different years. We discuss the interest of the present analysis and data sets for various fields, including in social sciences in order to better understand and model human behavior and interactions in different contexts, and in epidemiology in order to inform models describing the spread of infectious diseases and design targeted containment strategies.Comment: Supplementary Information at http://s3-eu-west-1.amazonaws.com/files.figshare.com/1677807/File_S1.pd

Estimating the epidemic risk using non-uniformly sampled contact data

Many datasets describing contacts in a population suffer from incompleteness due to population sampling and underreporting of contacts. Data-driven simulations of spreading processes using such incomplete data lead to an underestimation of the epidemic risk, and it is therefore important to devise methods to correct this bias. We focus here on a non-uniform sampling of the contacts between individuals, aimed at mimicking the results of diaries or surveys, and consider as case studies two datasets collected in different contexts. We show that using surrogate data built using a method developed in the case of uniform population sampling yields an improvement with respect to the use of the sampled data but is strongly limited by the underestimation of the link density in the sampled network. We put forward a second method to build surrogate data that assumes knowledge of the density of links within one of the groups forming the population. We show that it gives very good results when the population is strongly structured, and discuss its limitations in the case of a population with a weaker group structure. These limitations highlight the interest of measurements using wearable sensors able to yield accurate information on the structure and durations of contacts

Random inelasticity and velocity fluctuations in a driven granular gas

We analyze the deviations from Maxwell-Boltzmann statistics found in recent experiments studying velocity distributions in two-dimensional granular gases driven into a non-equilibrium stationary state by a strong vertical vibration. We show that in its simplest version, the ``stochastic thermostat'' model of heated inelastic hard spheres, contrary to what has been hitherto stated, is incompatible with the experimental data, although predicting a reminiscent high velocity stretched exponential behavior with an exponent 3/2. The experimental observations lead to refine a recently proposed random restitution coefficient model. Very good agreement is then found with experimental velocity distributions within this framework, which appears self-consistent and further provides relevant probes to investigate the universality of the velocity statistics.Comment: 5 pages, 5 eps figure

Can co-location be used as a proxy for face-to-face contacts?

Technological advances have led to a strong increase in the number of data collection efforts aimed at measuring co-presence of individuals at different spatial resolutions. It is however unclear how much co-presence data can inform us on actual face-to-face contacts, of particular interest to study the structure of a population in social groups or for use in data-driven models of information or epidemic spreading processes. Here, we address this issue by leveraging data sets containing high resolution face-to-face contacts as well as a coarser spatial localisation of individuals, both temporally resolved, in various contexts. The co-presence and the face-to-face contact temporal networks share a number of structural and statistical features, but the former is (by definition) much denser than the latter. We thus consider several down-sampling methods that generate surrogate contact networks from the co-presence signal and compare them with the real face-to-face data. We show that these surrogate networks reproduce some features of the real data but are only partially able to identify the most central nodes of the face-to-face network. We then address the issue of using such down-sampled co-presence data in data-driven simulations of epidemic processes, and in identifying efficient containment strategies. We show that the performance of the various sampling methods strongly varies depending on context. We discuss the consequences of our results with respect to data collection strategies and methodologies

Epidemic risk from friendship network data: an equivalence with a non-uniform sampling of contact networks

Contacts between individuals play an important role in determining how infectious diseases spread. Various methods to gather data on such contacts co-exist, from surveys to wearable sensors. Comparisons of data obtained by different methods in the same context are however scarce, in particular with respect to their use in data-driven models of spreading processes. Here, we use a combined data set describing contacts registered by sensors and friendship relations in the same population to address this issue in a case study. We investigate if the use of the friendship network is equivalent to a sampling procedure performed on the sensor contact network with respect to the outcome of simulations of spreading processes: such an equivalence might indeed give hints on ways to compensate for the incompleteness of contact data deduced from surveys. We show that this is indeed the case for these data, for a specifically designed sampling procedure, in which respondents report their neighbors with a probability depending on their contact time. We study the impact of this specific sampling procedure on several data sets, discuss limitations of our approach and its possible applications in the use of data sets of various origins in data-driven simulations of epidemic processes

Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys

Given their importance in shaping social networks and determining how information or diseases propagate in a population, human interactions are the subject of many data collection efforts. To this aim, different methods are commonly used, from diaries and surveys to wearable sensors. These methods show advantages and limitations but are rarely compared in a given setting. As surveys targeting friendship relations might suffer less from memory biases than contact diaries, it is also interesting to explore how daily contact patterns compare with friendship relations and with online social links. Here we make progresses in these directions by leveraging data from a French high school: face-to-face contacts measured by two concurrent methods, sensors and diaries; self-reported friendship surveys; Facebook links. We compare the data sets and find that most short contacts are not reported in diaries while long contacts have larger reporting probability, with a general tendency to overestimate durations. Measured contacts corresponding to reported friendship can have durations of any length but all long contacts correspond to reported friendships. Online links not associated to reported friendships correspond to short face-to-face contacts, highlighting the different nature of reported friendships and online links. Diaries and surveys suffer from a low sampling rate, showing the higher acceptability of sensor-based platform. Despite the biases, we found that the overall structure of the contact network, i.e., the mixing patterns between classes, is correctly captured by both self-reported contacts and friendships networks. Overall, diaries and surveys tend to yield a correct picture of the structural organization of the contact network, albeit with much less links, and give access to a sort of backbone of the contact network corresponding to the strongest links in terms of cumulative durations

Mitigation of infectious disease at school: targeted class closure vs school closure

School environments are thought to play an important role in the community spread of airborne infections (e.g., influenza) because of the high mixing rates of school children. The closure of schools has therefore been proposed as efficient mitigation strategy, with however high social and economic costs: alternative, less disruptive interventions are highly desirable. The recent availability of high-resolution contact networks in school environments provides an opportunity to design micro-interventions and compare the outcomes of alternative mitigation measures. We consider mitigation measures that involve the targeted closure of school classes or grades based on readily available information such as the number of symptomatic infectious children in a class. We focus on the case of a primary school for which we have high-resolution data on the close-range interactions of children and teachers. We simulate the spread of an influenza-like illness in this population by using an SEIR model with asymptomatics and compare the outcomes of different mitigation strategies. We find that targeted class closure affords strong mitigation effects: closing a class for a fixed period of time -equal to the sum of the average infectious and latent durations- whenever two infectious individuals are detected in that class decreases the attack rate by almost 70% and strongly decreases the probability of a severe outbreak. The closure of all classes of the same grade mitigates the spread almost as much as closing the whole school. Targeted class closure strategies based on readily available information on symptomatic subjects and on limited information on mixing patterns, such as the grade structure of the school, can be almost as effective as whole-school closure, at a much lower cost. This may inform public health policies for the management and mitigation of influenza-like outbreaks in the community

Glass transition and random walks on complex energy landscapes

We present a simple mathematical model of glassy dynamics seen as a random walk in a directed, weighted network of minima taken as a representation of the energy landscape. Our approach gives a broader perspective to previous studies focusing on particular examples of energy landscapes obtained by sampling energy minima and saddles of small systems. We point out how the relation between the energies of the minima and their number of neighbors should be studied in connection with the network's global topology, and show how the tools developed in complex network theory can be put to use in this context

Effect of risk perception on epidemic spreading in temporal networks

Many progresses in the understanding of epidemic spreading models have been obtained thanks to numerous modeling efforts and analytical and numerical studies, considering host populations with very different structures and properties, including complex and temporal interaction networks. Moreover, a number of recent studies have started to go beyond the assumption of an absence of coupling between the spread of a disease and the structure of the contacts on which it unfolds. Models including awareness of the spread have been proposed, to mimic possible precautionary measures taken by individuals that decrease their risk of infection, but have mostly considered static networks. Here, we adapt such a framework to the more realistic case of temporal networks of interactions between individuals. We study the resulting model by analytical and numerical means on both simple models of temporal networks and empirical time-resolved contact data. Analytical results show that the epidemic threshold is not affected by the awareness but that the prevalence can be significantly decreased. Numerical studies highlight however the presence of very strong finite-size effects, in particular for the more realistic synthetic temporal networks, resulting in a significant shift of the effective epidemic threshold in the presence of risk awareness. For empirical contact networks, the awareness mechanism leads as well to a shift in the effective threshold and to a strong reduction of the epidemic prevalence

How memory generates heterogeneous dynamics in temporal networks

Empirical temporal networks display strong heterogeneities in their dynamics, which profoundly affect processes taking place on these networks, such as rumor and epidemic spreading. Despite the recent wealth of data on temporal networks, little work has been devoted to the understanding of how such heterogeneities can emerge from microscopic mechanisms at the level of nodes and links. Here we show that long-term memory effects are present in the creation and disappearance of links in empirical networks. We thus consider a simple generative modeling framework for temporal networks able to incorporate these memory mechanisms. This allows us to study separately the role of each of these mechanisms in the emergence of heterogeneous network dynamics. In particular, we show analytically and numerically how heterogeneous distributions of contact durations, of inter-contact durations and of numbers of contacts per link emerge. We also study the individual effect of heterogeneities on dynamical processes, such as the paradigmatic Susceptible-Infected epidemic spreading model. Our results confirm in particular the crucial role of the distributions of inter-contact durations and of the numbers of contacts per link