article: Social interactions shape the patterns of spreading processes in a population. Techniques such as diaries or proximity sensors allow to collect data about encounters and to build networks of contacts between individuals. The contact networks obtained from these different techniques are however quantitatively different. Here, we first show how these discrepancies affect the prediction of the epidemic risk when these data are fed to numerical models of epidemic spread: low participation rate, under-reporting of contacts and overestimation of contact durations in contact diaries with respect to sensor data determine indeed important differences in the outcomes of the corresponding simulations with for instance an enhanced sensitivity to initial conditions. Most importantly, we investigate if and how information gathered from contact diaries can be used in such simulations in order to yield an accurate description of the epidemic risk, assuming that data from sensors represent the ground truth. The contact networks built from contact sensors and diaries present indeed several structural similarities: this suggests the possibility to construct, using only the contact diary network information, a surrogate contact network such that simulations using this surrogate network give the same estimation of the epidemic risk as simulations using the contact sensor network. We present and compare several methods to build such surrogate data, and show that it is indeed possible to obtain a good agreement between the outcomes of simulations using surrogate and sensor data, as long as the contact diary information is complemented by publicly available data describing the heterogeneity of the durations of human contacts.
  Knowledge of the structure of human interactions is crucial for the study of infectious diseases spread and the design and evaluation of adequate containment strategies. The structure of contact networks, [1], the presence of communities [2], bridges or “linkers” between communities [3–5], “super-spreaders” [6–8], the heterogeneity of contact durations [9], are all important characteristics that determine potential transmission patterns. The study of human contacts is particularly relevant in contexts such as schools, working places, hospitals where individuals might spend several hours in close proximity [5,10–22]. Interactions and contacts between individuals are conveniently seen within the framework of networks in which nodes represent individuals and (weighted) links correspond to the occurrence of contacts (the weight giving the duration of the contacts). Measuring directly such networks represents an important challenge [20]. Many studies have relied on contact diaries or surveys [10,12,23–32], while technological advances have led to a strong increase in the use of wearable sensors in the recent years [9,13–15,17–20,33–35]. Quantitative comparisons between datasets obtained from sensors and self-reported diaries, in terms of the numbers and durations of contacts between individuals and of the contact network statistics, are however scarce, mainly because very few studies have combined these two data collection means [21,36]. These investigations have shown that diaries suffer from small participation rates, under-reporting of contacts, and over-estimation of the contact durations. Under-reporting is particularly strong for short contacts, while long ones are better reported, and some studies have put forward methods to estimate its magnitude and to correct for it [29,37]. Interestingly, and despite the much lower number of nodes and links in contact networks inferred from contact diaries, the overall structure of these networks is very similar to the one obtained from wearable sensors. Moreover, the links with largest weights (as measured by sensors), which might play a major role in propagation processes, are reported with high probability in the contact diaries. In this paper, we go beyond the comparison of the contact networks obtained by these methodologies and explore the impact of their differences on the evaluation of the epidemic risk when such datasets are used in numerical simulations of infectious disease propagation. Our goal is to understand to what extent and how the information gathered from contact diaries can be used in such simulations in order to yield an accurate description of the epidemic risk, despite the biases mentioned above. We first compare the outcomes of spreading simulations performed using data coming from wearable sensors and from contact diaries that describe the contacts between students in the same context (a high school) and on the same day. Although the two networks are supposed to describe the same reality, we observe important differences in the simulations, due to the low participation rate in the diaries and to a stronger community structure in the contact diaries network than in the contact sensors network. We then design and evaluate a set of methods to use the information contained in the contact diaries to build surrogate versions of the contacts that yield, when used in the simulations, a better estimation of the real epidemic risk as quantified by the distribution of epidemic sizes (considering as ground truth the dataset from the sensors). We show that good results are obtained when the contact diary information is complemented by known stylized facts characterizing human interactions, in particular the heterogeneity of contact durations.
  We use two datasets collected in a French high-school in 2013 and made publicly available in [21]. The data describe face-to-face contacts between students of 9 classes as collected by (i) the SocioPatterns infrastructure [38] based on wearable sensors, during one week and (ii) self-reported contact diaries filled on a specific day of the same week (Dec. 5th, 2013). In the diaries, contact was explicitly defined as close (less than 2 m) face-to-face proximity, in order to match as much as possible this definition to the contacts detected by sensors. Using these data, we build two distinct contact networks for the day in which the diaries were collected: the Contact Sensors Network (CSN) and the Contact Diaries Network (CDN). In each network, nodes represent students and a link is drawn between a pair of nodes (i, j) if at least one contact between students i and j is present in the corresponding dataset during the considered day. We present and compare in the Supporting Information the main networks’ characteristics. Note that, in the diaries, some participants reported contacts with non-participants. One could a priori use this information and build a contact network including both participants and non-participants. However, since by definition the contacts of non-participants are unknown, this would introduce a potentially strong and most importantly completely uncontrolled bias in the measures of the network’s structural properties such as, e. g. , its clustering or the node degrees. A weight can moreover be assigned to each link (i, j): for the CSN, the weight wij is given by the cumulative duration of the contacts registered by the sensors on that day between i and j; for the CDN we can use the duration reported by the students in the diaries, building the network CDND, or use for each link a duration taken at random from the list of durations registered by the sensors, obtaining the network CDNS (see Methods for details). The rationale behind building CDNS comes from the results of [21,36] that show that durations reported by students tend to be strongly overestimated. Since, on average, contacts reported in the diaries as long tend also to be long according to the sensor data, we will also consider a different assignment of links to the CDN, in which we still take durations at random from the list of durations registered by the sensors, but assign the longer durations to the links of CDN with longer reported durations: we denote the resulting network by CDNS’. The contact sensor network counts 295 nodes (participation rate 77. 8%) and 2162 links, while the contact diaries network has 120 nodes (participation rate 31. 6%) and 348 links. Incomplete participation, even in the case of the sensor data, leads to biases in the simulations using the CSN with respect to what would be obtained if the whole population had participated, due to the fact that contacts with and among non-participants are not detected. This point has been discussed in [40], together with methods to build surrogate data and obtain estimate of the epidemic risk in the case of such population sampling. In order not to confuse the issues of population sampling and comparison between diaries and sensors, we consider here as ground truth the CSN, collected by wearable sensors for which the definition of contact does not depend on a possible interpretation of the diary question by the students nor on the fact that they might not recall contact events. In the following, we perform simulations of the spread of infectious diseases in the considered population, using as substrate for propagation events the contact networks described above. It is important to note here that we consider propagation processes on static networks. Indeed, the CDN does not contain information on the timing of the contacts, so that it is natural to compare the outcome of simulations performed on such a static network with simulations performed on a static version of the CSN. Moreover, when modeling the propagation of infectious diseases with realistic timescales of several days, it has been shown in [39] that a daily weighted contact network contains enough information to obtain a good estimate of the process outcome. When dealing with faster processes, the temporal evolution of the network would become relevant; in that case, it would be possible to use the techniques put forward in [40] to build realistic surrogate timelines of contacts on weighted networks, using the robustness of the distributions of the durations of single contact events and of the intervals between successive contacts measured in different contexts. Note also that, even if the networks do not take into account the timing of the contact events, they still include information on the aggregate contact durations through the weights, which are known to play a crucial role in the outcome of spreading processes [39,41–43]. For simplicity, we consider the paradigmatic Susceptible-Infected-Recovered (SIR) model of epidemic propagation. In this model, each Susceptible node i can be infected by an Infected neighbour j with probability β*wij*dt for each small time step dt. Infected people recover with rate μ and enter in the Recovered category. Recovered individuals cannot be infected again. The process starts with a single Infected individual chosen at random (the seed) and ends when there are no more Infected nodes. The epidemic risk in the population, which depends on the interplay of the ratio β/μ and the network’s structure and weights, is measured by the distribution of the final size of epidemics (i. e. , of the fraction of individuals in the Recovered category at the end of the process), obtained by repeating the simulations with randomly chosen seeds. Note that, since we consider static networks, only the ratio β/μ is relevant, and multiplying both by a certain factor only changes the timescale on which the epidemic unfolds. The shape of the distribution of epidemic sizes depends on the features of the underlying network structure in terms of possible patterns of contagion. The comparison of these distributions gives hints about similarities and discrepancies of various datasets for the evaluation of the epidemic risk. We first compare in Fig 1 the outcome of simulations of the SIR model performed on the CSN and on the two versions of the CDN described above (CDND with weights reported by students and CDNS with weights registered by sensors assigned randomly to the links), for one specific value of β/μ = 30. The three distributions of epidemic sizes are very different from each other. The outcome of simulations performed using CSN is quite standard, with a fraction of small outbreaks that reach only a small fraction of the population and another peak corresponding to large outbreaks. As shown in the Supporting Information, the outcome does not depend on the class of the initial seed. The shape of the distribution obtained when using the CDND is more peculiar, with a series of peaks, including one at very large epidemic sizes. Such structure is typical of spreading processes on networks with a strong community structure [4], which corresponds to the results of [21]: (i) due to the low participation rate and the under-reporting, the community structure of the CDN is stronger than the one of the CSN, with few links between classes; depending on the seed, the simulated disease can thus remain confined in one class or in a group of few classes, leading to the peaks at intermediate values of the epidemic size; we moreover show in the SI that the outcome depends on the class of the initial seed for the CDN but not for the CSN; (ii) on the other hand, as contact durations are overestimated, the propagation probability on each link is also overestimated and, if the disease manages to spread between classes, almost all individuals are affected, leading to the peak at large epidemic sizes. The CDNS case shows a different result: no more than half of the whole population is affected by the spread. As the weights have in this case the same statistics as the CSN, this is simply due to the low participation rate [40] and the much smaller average degree in the CDN with respect to the CSN. We also note that, since the weights are assigned randomly to the links between students, the structure of the contact matrix giving the average durations of contacts between students of different classes can strongly differ between the CDNS and both the CSN and the CDND, leading to different patterns of propagation between classes (see Supporting Information). We finally note that the simulations on the CDNS’, which keeps the distribution of the weights from CSN and in which larger weights are assigned to links with longer reported durations, yield even smaller outbreaks. This is probably due to the fact that the large weights reported in the diaries tend to be within classes, so that the links bridging classes and favoring the spread tend to have smaller weights in the CDNS’ than in the CDNS. We also show in the SI the temporal evolution of the density of infectious individuals for the various cases considered here.
  Data on the contact patterns of individuals collected by different methods lead to different contact network structures, and some studies have started to investigate this issue through detailed quantitative comparisons [21,36]. In the present paper, we have gone further by comparing the outcome of simulations of spreading processes performed on contact networks gathered either through wearable sensors or through contact diaries. Not surprisingly, we have shown that the results differ strongly, due to the low participation rate to the diaries, the under-reporting of contacts and the overestimation of contact durations in diaries. In particular, the direct use of the links and durations reported in the diaries yields a peculiar distribution of epidemic sizes suggesting a very strong community structure that might lead to the design of inadequate containment strategies. On the other hand, using the links reported in the diaries but more realistic weights yields a strong underestimation of the epidemic risk. In a second and more important step, we have asked if, despite this low participation rate and these biases, the information contained in the contact network built from the contact diaries can be used to build a surrogate contact network whose properties are close enough to the real contact network (considered here to be the one obtained from the wearable sensors) to yield a correct estimation of the epidemic risk when used in simulations of spreading processes. The rationale comes from the structural similarities found in the contact matrices giving the densities of links between individuals of different classes obtained using both sensors and diaries [21]. These similarities suggest to build a surrogate contact network starting from the contact diary network, adding nodes and links in order to maintain this matrix fixed, and assigning weights to the links. We note that two recent works [37,40] have considered related but different issues. In [37], only diary data is available, and the authors present a synthetic network model based on data and adjusting for under-reporting. This adjustment for reporting errors leads in this case only to a small difference in epidemic predictions. In [40] on the other hand, only sensor data is considered, and the authors assume to have an incomplete information on the contact network registered by sensors due to an uniform population sampling (hence, all contacts between participating individuals are assumed to be known). Here on the other hand, the available dataset is given by diaries, in which population sampling is not uniform (actually, the students who filled in diaries tend to have more contacts than the others) and in which under-reporting implies that many links between participating individuals are also missing. Moreover, we face two additional issues (i) the low sampling rate yields a contact matrix of link densities which contains zeros, in an unrealistic way, and (ii) various possibilities can be considered when assigning weights to the links of the surrogate networks as weights reported in the diaries are strongly overestimated. Despite these issues, the surrogate network we build yields, when used in simulations, a good agreement with simulations performed on the whole contact sensor network in terms of epidemic risk prediction, under the condition of using the list of weights (cumulative contact durations) gathered by the wearable sensors. In order to get rid of this condition, we argue that this list comes from a distribution that has been shown in previous works to be very robust across contexts [9]. We therefore consider weights taken at random from a pool of publicly available datasets, and show that using these weights gives also satisfactory results. Overall, we thus have presented a procedure that uses only the information contained in the contact diaries and in public data, which allows to obtain a good prediction of the epidemic risk, as measured by the distribution of epidemic sizes, when used in simulations of a spreading process. In the Supporting Information, we moreover consider the issue of using, instead of contact diaries, data coming from friendship surveys, in order to build the surrogate contact network used in the simulations. We show that the epidemic risk prediction obtained through this procedure is not accurate. This could be expected as daily encounters in the school are not necessarily related to the existence of a relationship between students: contacts occur between non-friends due to daily activities, while friends do not meet necessarily every day. This outcome highlights the importance of taking into account the different nature of social ties [21,46], which can each be relevant for specific processes. Some limitations of our work are noteworthy. First, our results rely on an assumption made in replacing the zeros observed in the contact matrix of link densities by random values. In the context under scrutiny, zero values can indeed easily be considered as unrealistic. In other contexts, it might however happen that different groups in the population really do not mix. In such a case, one might expect that this kind of information could be gathered from other sources (schedules, location of classrooms or offices, etc) [47] and thus integrated into the procedure. Second, we have considered as ground truth the contact sensor network. On the one hand, this network in fact suffers from an incomplete participation, so that the outcome of spreading processes is underestimated with respect to hypothetical data containing information on the whole population. However, such underestimation can be compensated through the procedure presented in [40]. On the other hand, it is important to note that it is not yet completely clear whether the contacts measured by wearable sensors are the best proxy for potentially infectious contacts. We work therefore under this hypothesis, which is indeed quite widely used but should be kept in mind. Third, we have considered here static networks. As discussed in [39], the outcome of simulations is then close enough to the one of simulations taking into account the full contact dynamics if we consider slow enough processes. For fast processes, the burstiness of contacts becomes very relevant; in this case, it would be crucial to supplement our procedure by the construction of surrogate timelines of contacts and intervals between contacts at high temporal resolution, as done in [40]. Finally, we cannot at this point investigate the efficiency of our procedure in other contexts, for lack of datasets reporting contacts measured by both wearable sensors and contact diaries in the same context and at the same date. Hopefully such datasets will become more available in the future, yielding new testing grounds for our method. Many populations of interest can indeed be divided into groups or categories that do not mix homogeneously, often with more contacts within groups than between groups, and for which the contact matrix formalism and the procedures we present to construct surrogate networks are therefore relevant [40]. We conclude by mentioning that future work could also investigate other dynamical processes on networks, such as information spreading or opinion formation processes.
  The datasets we use have been presented and made publicly available in [21]. They correspond to contacts between students of 9 classes in a high school in France, collected through wearable sensors on the one hand and contact diaries on the other hand. The sensors registered contacts with a temporal resolution of 20s for 327 participating students (out of 379 in the 9 classes, i. e. , a 86. 3% participation rate) during the week of Dec. 2–6,2013. Contact diaries contain data reported by students about encounters and their cumulative durations for Dec. 5,2013. In these diaries, students were asked to report the cumulative durations of their contacts choosing among four intervals: at most 5 minutes, between 5 and 15 minutes, between 15 minutes and 1 hour, more than one hour. The students belong to 9 classes with different specializations: “MP” classes focus more on mathematics and physics, “PC” classes on physics and chemistry, “PSI” classes on engineering studies and “BIO” classes on biology. We collected data among students of nine classes corresponding to the second year of such studies: 3 classes of type “MP” (MP, MP*1, MP*2), two of type “PC” (PC and PC*), one of type “PSI” (PSI*) and 3 of type “BIO” (2BIO1,2BIO2,2BIO3). Using these datasets, we build two networks of contacts among students for the same day (Dec 5,2013): the Contact Sensors Network (CSN) and the Contact Diaries Network (CDN). In each network, nodes represent students, and a links is drawn between two students if: The resulting networks have 295 nodes for the CSN (other students were absent or did not wear the sensors on that day) and 120 nodes for the CDN. In particular, no student from classes PC* and PSI* filled a diary, and only one from MP*1. We thus discarded these classes in most of the analysis and in particular in the contact matrices, remaining with 6 classes. Each link carries a weight. In the CSN it represents the cumulative duration of contacts registered by sensors during the day. For the CDN we consider several possibilities. In the CDND we use weights reported in the diaries: we associate to each time-interval its maximum possible value (5,15,60 minutes respectively for the first three intervals and 4 hours for the last one. This choice takes into account data reported and registered and the school schedule) and, if two students reported different durations for their encounter, we use the average of the reported values. In the CDNS on the other hand, we consider weights randomly drawn from the distribution of contact durations registered by sensors. Results are averaged over 1000 such weight assignments. For the CDNS’ finally, we start from CDN and rank the E links in decreasing order of their reported weights (assigned as in CDND, and with random order for equal weights). We extract E weights from the distribution of contact durations registered by sensors, rank them as well in decreasing order, and assign the weights to the links of CDN in such a way to match the two orderings (i. e. , assigning the largest weights to the links with largest reported weights). The CSN, the CDND and the CDNS are matched to retain only nodes appearing in both CSN and CDN (see Table 1 for details about classes size before and after matching). We refer to them as the matched networks: CSNm, CDN D m and CDN S m. The CMDN is built by using a Contact Matrix Distribution (CMD). Following [44], we consider a CMD where each entry, (X, Y), is the empirical distribution of durations reported by diaries for contacts between all students in class X and class Y, including zero durations (corresponding to an absence of link between two students). We fit each such distribution by a negative binomial functional form. Then, for each pair of nodes, we draw at random a weight using the corresponding negative binomial fit. Note that in this way we do not maintain fixed the link structure of the CDN. We however keep on average the same density of links between different classes. The basic steps for building a binary surrogate contact network CDNs for the six considered classes, starting from the matched contact diaries network, are: Results are averaged over 500 realisations of this procedure. As explained in the main text, we moreover deal in two different ways with the zero values of the link densities between several class-pairs in the CDN. We either keep these densities or replace them with values drawn at random from a uniform distribution of values between the minimum and maximum values (diagonal excluded) of the contact matrix of Fig 3 (b). In this way the contact matrix structure is preserved, with more interactions within than between classes. To assign weights to the links of the surrogate networks, we consider several possibilities. We first assume homogeneous contact durations and assign to each link a weight equal to the average of cumulative durations registered by sensors. This yields two versions of the surrogate contact networks: We refer to the contact sensors network under the homogeneous duration hypothesis by CSNH. If instead we assume heterogeneous contact durations, we obtain two possible surrogate contact diaries networks: we assign weights at random to the links of CDNs, drawn at random with replacement from the list of durations either reported by students or registered by sensors. We thus obtain four versions of the surrogate contact networks: Finally, the surrogate contact network obtained by assigning weights randomly drawn from the negative binomial fit of the distribution of publicly available contact durations registered by sensors is indicated by the acronym CDN nz, NB s.
summary: Schools, offices, hospitals play an important role in the spreading of epidemics. Information about interactions between individuals in such contexts can help understand the patterns of transmission and design ad hoc immunization strategies. Data about contacts can be collected through various techniques such as diaries or proximity sensors. Here, we first ask if the corresponding datasets give similar predictions of the epidemic risk when they are used to build a network of contacts among individuals. Not surprisingly, the answer is negative: indeed, if we consider data from sensors as the ground truth, diaries are affected by low participation rate, underreporting and overestimation of durations. Is it however possible, despite these biases, to use data from contact diaries to obtain sensible epidemic risk predictions? We show here that, thanks to the structural similarities between the two networks, it is possible to use the contact diaries to build surrogate versions of the contact network obtained from the sensor data, such that both yield the same epidemic risk estimation. We show that the construction of such surrogate networks can be performed using solely the information contained in the contact diaries, complemented by publicly available data on the heterogeneity of cumulative contact durations between individuals.
keywords: complement system
  medicine and health sciences
  immune physiology
  education
  infectious disease epidemiology
  binomials
  sociology
  immunology
  social sciences
  epidemiological methods and statistics
  simulation and modeling
  mathematics
  algebra
  polynomials
  research and analysis methods
  immune system proteins
  infectious diseases
  proteins
  epidemiology
  schools
  immune system
  epidemiological statistics
  biochemistry
  physiology
  biology and life sciences
  physical sciences
title: How to Estimate Epidemic Risk from Incomplete Contact Diaries Data?