Contact networks

We use traces for three different places, that are available from the SocioPatterns project (http://www.sociopatterns.org). The project collected longitudinal data on physical proximity and face-to-face contacts between individuals in several real-world environments.

1. A primary school (see [16, 17]) where 242 persons participated in 2009 over 2 days (coverage of 96% among children and 100% among teachers).
2. A workplace Institut de Veille Sanitaire (see [9]) where 232 employees participated in 2015 over two weeks (10 working days, coverage around 70% of the employees according to a previous deployment [18]).
3. A high school (see [19]) where 329 individuals (students) participated in 2013 over 5 days (coverage of 86% of the students in the 9 participating classes).

For each day on which data was gathered we extract a graph aggregating the data for that day: a node corresponds to an individual, an edge corresponds to a face to face contact within 1.5 meters within a 20 seconds time interval (interactions were measured using active radio-frequency identification devices (RFID)), and the weight of the edge is the number of such short contacts during the day. For comparison, we also generate a synthetic random graph, calibrated so that its main parameters (total number of nodes, of edges, and of contacts) match those of the high school contact network: more precisely, each edge is generated by selecting uniformly at random two nodes with one associated contact (rejecting loops and already generated pairs) and each of the remaining contacts is associated to an edge selected uniformly at random among the previously generated edges. Table 1 lists the main parameters of the graphs obtained by averaging over all days on which data was gathered. Fig 1 displays the three contact graphs on their first day, together with the synthetic random graph obtained. The node colors correspond to groups (classes or work departments) for the real-world contact graphs, and are chosen uniformly at random among 9 colors for the synthetic random graph. All other graphs and average graphs are depicted in S1–S4 Figs.

SARS-CoV-2 transmission model

We model the introduction of the virus in a network by randomly sampling an index case uniformly among all the nodes to determine the patient initially infected. Similarly to related works like [20], we assume a natural history that is an agent-based model that refines classical SEIR transmission models (see Fig 2): initially individuals are susceptible (S); once contaminated, having been exposed (E), they go through an incubation period, after which they become infectious (I) after which they are assumed to recover (R) and develop immunity. An individual may be symptomatic or asymptomatic. In the former case, before developing symptoms she/he goes through a pre-symptomatic phase that is already infectious. We assume that transmission between an infectious and a susceptible individual happens through proximity contacts as the ones recorded in the contact network. To every 20-second contact is associated an independent small probability of transmission, p, so that the transmission risk increases with the duration of contact. The time step of the simulation is one day, which is consistent with the daily rhythm of commuting: as an example, if an infectious person is in contact with a susceptible person for 15 minutes during the day (and therefore through 45 “contact events”), then the probability of transmission during that day equals 1 − (1 − p)⁴⁵, which for p = 0.001 is approximately 4.4%.

Stochastic simulation engine

We use an agent-based model with discrete time where the time step corresponds to a day. Each person is an agent whose state is either S, E, I, or R; according to the SEIR transmission model described above. Every day, the state of each agent can change according to the number of contacts with other agents in the contact network for that day, the states of these agents, and random coin flips based on the transmission probability. The state of an agent also depends on when it had a previous state change and on what random value was obtained for the duration of the current state. A dedicated C++ program was developed for that purpose and is made available.

The model parameters are summarized in Table 2. For each infected individual, the duration of the incubation period is randomly drawn into a Gamma distribution with mean 5.2 days and shape 5 [12, 13]. The duration of the pre-symptomatic period is then uniformly drawn in {1; 2} days, consistently to published studies [11] (Table 2). The remaining duration of infectiousness follows a gamma distribution with mean 8 [14] and shape 10. We assume that the fraction of asymptomatic individuals equals 40%, within the range of [21, 22]. Symptomatic individuals are assumed to self-isolate after one day of symptoms and therefore do not cause further contamination in the studied setting; on the contrary, asymptomatic individuals stay in the system and potentially transmit the virus throughout their infectious period. The choice of the rate of instantaneous transmission is described next.

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Table 2. The top table gives the parameter values used in our simulations, with the supporting references.

The bottom table summarizes some relevant quantities that can be computed from our simulations: their consistency with the literature is argued in the Discussion section.

https://doi.org/10.1371/journal.pcbi.1009264.t002

Super-spreaders

In the COVID-19 epidemic, the number of persons contaminated by an infectious person has been suggested to show a large variance [10, 24–28]: several studies have shown that many infected individuals do not contaminate anyone, whereas a small fraction of the infected population, termed ‘super-spreaders’, are responsible for the majority of the transmissions. Such super-spreading events may be due to several factors including a higher viral load or infectiousness of the super-spreader, a particularly high number of contacts, and whether those contacts occur in a confined space with poor ventilation [29]. Here, we model super-spreading as follows: on each day, and for each infectious individual i, a random super-spreading factor p_super is chosen independently from a Gamma distribution where E[p_super] = 1. Then, the transmission probability for each short contact with a susceptible individual on that day, is p₀p_super if i is symptomatic and p₀p_super/2 if i is asymptomatic, where p₀ is the baseline transmission parameter.

Calibration of the transmission parameter

The contamination parameter p₀ is calibrated so that the baseline local reproduction number , defined as the average number of persons infected by the index case, equals 1.25. The latter value is chosen to implicitly take into account the adoption of barrier measures including social and physical distancing, mask usage and hand hygiene. The idea of inferring p₀ from the model is inspired by [30]. We find that p₀ = 0.001 in the primary school contact graph, p₀ = 0.004 in the high school contact graph, and p₀ = 0.010 in the workplace contact graph. Several values of were investigated, ranging from 0.5 to 2.5, corresponding (for high schools) to p₀ ranging from 0.001 to 0.010. The quantitative relations between p₀ and in the different graphs are reported in Fig 7.

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Fig 7. Relation between the baseline transmission parameter p₀ and the network-dependent local reproduction number for our four contact networks.

We see that each curve is monotone increasing, as expected, but that the dependency is not quite linear.

https://doi.org/10.1371/journal.pcbi.1009264.g007

Persistent contacts

All simulations are initialized with an index case in the graph assumed to have been contaminated by the outside world. Importantly, we then focus on transmissions occurring within the contact graph. Since our proposed strategies act on the school or work place social networks and aims at limiting transmission clusters occurring in these specific locations, we do not model contagion of/from people who are not in the contact network. This choice is consistent with studies with similar focus [3].

Nevertheless, contacts with friends or colleagues who belong to the same social network may also happen outside the direct school/workplace environment. To model such interactions, we assume that there exists a background external graph G_ext of persistent contacts that take place every day, whether workday or weekend, whether telecommuting or not. We define this external graph from the contact network by applying a dampening factor of 25% to all contacts. This factor stems from imagining a scenario in which someone would invite colleagues or fellow students to come and interact for roughly two hours during the day instead of eight hours of interaction at work (hence the 25%); and those persons would be selected from among their usual school/work colleagues, proportionally to their contacts.

Strategies

Several non-pharmaceutical strategies were used or recommended across the world depending on activity type (school, workplace, university) or country. Here we concentrate on strategies at the level of the work/school environment which focuses on presence-sheet organization and promotion of hybrid telecommuting with partial use or partial closure of school or work environments. First, we consider on-off strategies, in which alternatively, either 100% of employees or students do face-to-face work, or 100% do telecommuting (distance learning). Such a strategy has, for example, been recommended as a way to exit the lockdown by alternating 4 days on and 10 days off [31]. Venezuela had for example a temporary exit strategy in which businesses were allowed to reopen on a week-on-week-off basis [32]. Second, we consider rotating strategies, in which 50% of employees or students do face-to-face while the other 50% do distance learning, periodically switching between the two groups. Organizing work with Rotating shifts was for example one of the actions recommended by the CDC [33]. We implement both types of strategies with different alternations: daily alternation (even day, odd day, not counting weekends) and weekly alternation (even week, odd week). Finally, we consider a full telecommuting strategy. This results in five strategies, which we compare in their ability to reduce the likelihood and intensity of epidemic outbreaks.

Evaluation criteria

More precisely, strategies are evaluated based on three criteria: the probability of outbreak, defined as the percentage of simulations for which at least 5 secondary cases were infected besides the index case; the velocity of outbreak (average delay until five persons are infected); and the average cumulative number of infections until extinction of the epidemics, within outbreaks.

Varying population immunity and cross immunity

In the current context (July 2021) vaccinations are being widely deployed in Europe and new, more transmissible, viral variants are spreading. This situation is attracting the attention of researchers: for instance, the effect of various level of immunity on the death toll in Italy has been studied in [34]. Therefore, we further evaluated the impact of the different strategies on the probability of outbreak assuming a portion of the population was already partially or fully immunized. Immunity could arise from previous natural infection or from vaccination. Since several vaccines have been introduced simultaneously and may differ in their properties, we analyse the impact of strategies under different assumptions of vaccine efficacy in terms of preventing acquisition and/or transmission. We finally assessed, in this context of a partially immune population, the remaining epidemic potential for a new strain or variant with a higher transmissibility potential.

Summary

By simulating SARS-CoV-2 transmission over a diversity of contact networks, we showed how (hybrid) telecommuting reduces the virus transmission in schools and workplaces. We focused on three types of strategies: On-Off, Rotating, and Full telecommuting. Our results highlight that, whatever the contact network, these measures significantly reduce the risk of outbreak, delay the time when the outbreak occurs, and reduce the overall attack rate. This conclusion holds even though we assume some persistent contacts between individuals and a fraction of their workplace contacts, when they are not at the work location (for example, colleagues meeting outside work).

The rankings of the strategies are consistent (Fig 3): Full telecommuting (maintaining persistent contacts) significantly dominates the Rotating strategies which significantly dominate the On-Off strategies which in turns significantly dominate the absence of any policy. This strategies classification was not affected by the introductions of assumptions on vaccine- or naturally-induced immunity; nor by the modelling of an hypothetical more epidemic variant circulating in a partially immune population assuming cross-immunity.

Interestingly, despite strategies strongly differed in their probability of occurrence and size of outbreak (Fig 3), the average delay before observing an outbreak did not vary much across strategies. As a reminder, that delay-related criteria was defined as the time between index case’s virus acquisition and a cumulative incidence of at least 5 secondary infected cases, conditioned on the occurrence of an outbreak. The time until that happens, conditioned on the fact that it does happen, is then largely determined by the model parameters related to virus natural history of infection, rather than by the choice of strategy. For example, considering a single index case, and an average reproductive number of 1.25, at least 2 generations of transmissions are required to reach a cumulative incidence of 5. Considering a generation time between 5 (if symptomatic) and 8 days (if asymptomatic), this would represent between 10 and 15 days. It is then expected that that delay is slightly increased when the effective reproductive number is reduced, but that it impacts more strongly the probability of outbreak.

These results are consistent with intuition. A back-of-the-envelope calculation suggests that if we order strategies according to how much they reduce contacts, the ranking is: On-off, Rotating, Full Telecommuting. Indeed, Rotating strategies always induce fewer contacts overall than On-Off strategies, because they involve the presence of smaller groups, but that does not necessarily imply less epidemic propagation because of non linear effects (in Fig 4 the relationship between R_e as a function of are not quite linear). It is important to note that we focused here on the dissemination risk of the pathogen in a location from a single introduction by an infected individual. We therefore simulate here the beginning of an outbreak only and do not consider multiple introductions from the community.

Our results are also consistent with several studies that argue the advantage of Rotating strategies, either based on deterministic compartmental models [3, 35] or on agent-based simulations [20]. Compared with the daily alternation, the weekly alternation is naturally in phase with the duration of the incubation period and generation time of the disease. As a consequence, one can expect that it more effectively breaks the contact chains. Consistently, Fig 3 (as well as S7 and S13 Figs) show that weekly alternations are better than daily alternations.

Empirical contact data have been extensively used in infectious diseases epidemiology to realistically simulate epidemic outbreaks and assess surveillance or mitigation measures in specific environments like schools and hospitals [36–39], including the issue of school closures [16]. Regarding COVID-19 outbreak though, most previous work has focused on synthetically generated populations, both at large scale (e.g. a whole country [40, 41]) and for smaller communities like schools [20]. Here, we build our simulations on publicly available empirical data collected in schools and workplaces [9]. Therefore, our work has some parallelism with [42], where contact data informed the agent-based simulation of COVID-19 outbreaks in a long-term care facility. Simulations on real small-scale networks enable analysing the evolution of the epidemics using realistic contact patterns, without making strong hypotheses regarding the structure of contacts and heterogeneity associated with it. As illustrated on Fig 5, transmission in such small populations is characterized by a strong stochasticity, with high chance of natural extinctions, which are less probable in epidemic models from the general population. Fig 5 also shows that due to the strong clustering, most of transmission events occur within groups rather than between group (in highschool, 93% of contacts and 86% of contaminations are within a group). Visualizing the detailed evolution of the epidemics in these environments (see Fig 5) and leveraging this fine understanding yields explicit recommendations about the effectiveness of the strategies.

A key feature of the COVID-19 pandemic is the role played by asymptomatic cases in the transmission. From the transmission trees shown on Fig 5 and S19–S22 Figs, one can note that an important proportion of the transmissions arise from asymptomatic cases. Indeed, in our baseline simulations over the high school contact network, we estimate that 56% of transmissions are due to asymptomatic individuals on average. Compared to symptomatic individuals, asymptomatic individuals are less infectious but do not self-isolate, so they have a reduced rate of transmission but over a longer period of time. The assumption that symptomatic cases would isolate relatively quickly is consistent with current recommendations in school and workplaces where individuals are asked to stay at home when they have any suspect symptoms. Imperfect compliance with isolation recommendations (which may be more realistic) would result in an increased risk of outbreak in all settings: see S14(c) Fig.

Validity of the parameter values and robustness of the results

A lot of uncertainty exists regarding SARS-CoV-2 epidemiology and natural history and our choice of parameters, despite based on published data, deserves to be discussed. From the analysis of our simulations, we show below that our estimates are consistent with other studies.

In order to compare the risk associated with the graph structure, and not the number of contacts, we chose to analyse the three graphs by considering the same local reproduction number . We calibrated the baseline probability of transmission to ensure a baseline . Consequently, our estimates of the transmission risk over a contact strongly varied according to the analysed graph: p = 0.001, 0.004, 0.010 for the three graphs analysed here. Limited data is available regarding the transmission of SARS-CoV-2 in such environments and populations (here children, teenagers and adults). However, previous study on influenza transmission estimating the transmission risk from an infectious to a susceptible individual in similar contact records of 20 seconds [39] found consistent values, p = 0.003. The baseline value was set to simulate a significant but moderate epidemic risk within the network. This value may vary due to deployment of NPIs as frequent hand washing, mask wearing and social distancing. We ran simulations for values of ranging 0–2.5. Higher values led to higher risks of outbreaks and bigger outbreak sizes (reaching 135 for ), but regardless of the value of , simulations confirm that the investigated strategies reduce the global risk compared with no strategy: all investigated strategies are able to reduce R below 1 for (Fig 4). Additionally variations of did not affect strategies classification. Overall, our results for R_e (Fig 4) were comparable to [31]. Indeed, if we set , assuming that the local reproduction number is the same on weekends as on weekdays would yield R_W = 1.48; we then obtain that for full telecommuting R_L = R_e ∼ 0.53, and simulating the “4 days on, 10 days off” On-Off strategy from [31] yields R_e ∼ 0.82. Thus, this is consistent with the findings from [31]: for R_W = 1.5 and R_L = 0.6 they find that R_e = 0.86. We also note that from the above calculation, our baseline intensity set at 25% for persistent contacts happens to yield a ratio R_W/R_L that is almost the same as in [31], further confirming our choice of 25%.

Another key characteristics of SARS-CoV-2 dynamics is the generation time, that is, the average number of days until the secondary generation of cases are infected by the index case. Here, the effective generation time, recovered from baseline simulations of our model, is 7.3 days: this value is a weighted average of the generation times resulting from transmissions from asymptomatic individuals (8.8 days) and transmissions from symptomatic individuals (5.5 days). The latter is consistent with an estimate of 5.2 for the Singapore cluster (and a little higher than for the Tianjin cluster) [23].

Several studies have stressed the high heterogeneity in transmissions across individuals, suggesting that about 80% of transmission events are caused by only about 10% of the total cases (see [25] for example). We therefore integrated the possibility of super-spreading events in our model: even though we do not reach such high levels of dispersion, our baseline model for high schools already shows much dispersion: among all simulations, the 20% with the most secondary infections accounted for 68% of secondary infections (see S1 Text and S13 Fig).

Besides the above consistency checks, extensive sensitivity analyses were carried out to assess the robustness of our results with respect to model assumptions and parameter values: graph of persistent contacts, asymptomatic probability, 20-second mean transmission probability, asymptomatic relative infectiousness, super-spreading transmission factor, length and dispersion of exposed period, of presymptomatic period, or symptomatic period, and number of days of symptoms before isolation. They are presented in S13–S17 Figs. These sensitivity analyses show that although the evolution of the epidemic varies greatly with the parameters, corresponding variations are smooth and the ranking of the strategies is always respected. We observe that the duration of the epidemic until outbreak is the least sensitive measure, whereas the most sensitive measure is the total number of infected people when there is an outbreak (attack rate). This quantity increases when the graph of persistent contacts is replaced by a (calibrated) complete graph (S9 Fig); when increases (S7 Fig); and when the shape parameter of the transmission probability distribution increases, due to super-spreaders (S13 Fig). Interestingly, the attack rate also becomes much larger when the contact graph is replaced by a (calibrated) homogeneous graph (S5 Fig).

Our simulations are based on empirical data collected in three specific locations (French primary school, high school and a workplace) and are therefore specific to these locations. Nevertheless, both our results on the random network and our extended sensitivity analysis support the generality of our finding that all strategies globally reduce the dissemination risk and Rotating strategies give the best effect. Comparison with real epidemiological data in France was not possible. Indeed, in May-June 2020 variations of the Rotating and On-Off strategies were implemented in most schools, but at that time tests were not readily available to measure the impact of those measures and the virus was nearly absent. However, assessing the impact of these strategies on transmission risk in schools should be possible in the near future, because tests are now widely available and used while some regions have high levels of viral circulation.

Limitations of our study

The results presented here should be interpreted in the light of our rather simple assumptions.

Firstly, virus transmission is assumed to occur within the contact network only, thereby neglecting potential acquisitions through external contacts, such as family members or friends are not considered, with the exception of the index case. As a matter of fact, when the levels of community circulation of the virus are high, individuals can also be exposed to the virus outside school or work place, this chance being potentially increased over telecommuting periods. Our objective was not to provide predictions about the expected prevalence in schools or workplaces but rather to evaluate the virus dissemination risk within the network, or in other words, the network vulnerability: we therefore focused on the quantification of this risk following a single introduction of the virus by an index case. Other studies [20] that include repeated acquisition from the community and simulate the epidemics evolution over longer time spans have reached conclusions that are consistent with ours, namely, on the advantage of rotating strategies.

Conversely, infected people within the social network might in turn infect members of other social groups, but those are outside the reach of the proposed strategies: this effect was not analyzed here. Nevertheless, in order to more realistically model contact patterns in a situation where telecommuting recommendations are not strictly enforced or complied with, we assumed that telecommuting individuals maintain a fraction of their in-network contacts. This could depict a situation where compliance with telecommuting recommendations is low. In a case of a strict lockdown or curfew, individuals would have no contacts over their telecommuting periods and may not be able to visit colleagues or school friends. In terms of modelling, this scenario simply requires removing persistent contacts and leads to a lower risk of outbreak than the one presented in our baseline model. In order to assess how changes in persistent contacts network affect the reported results, we simulated the model for different assumptions related to the external persistent graphs. The top left panel of S9 Fig shows that when a strict lockdown removes all persistent contacts (that situation is obtained in the simulation by multiplying the external contact graph by a factor of 0%), the outbreak probability drops from 27% to 17%. Moreover, in that case, all of our strategies cause the reproduction factor to drop below 1 for all values of and for all contact networks (see S11 and S12 Figs). Thus, adding a curfew on top of a hybrid telecommuting strategy leads to significant improvements.

Secondly, we performed our simulations on a small set of empirical contact graphs that were built from publicly available data about just three schools and workplaces. Extrapolating from such a small set should be done cautiously. However, we are comforted by two facts. First, the main features of those networks, such as their degree distribution and their community structure (more than 70% of contacts within classes or departments; see Table 1), are representative of the social groups that we are interested in. These features are indeed key to shaping the progression of the epidemics: our simulations therefore allow studying mitigation in networks with such community structure. Second, despite significant differences between the three empirical graphs, our results are consistent across all of them and are robust to variations in the simulation parameters. Furthermore, our qualitative conclusions also extend to synthetic random graphs as described in Materials and Methods. However, quantitative results for the random graph are rather different from the original graph that has been used to tune its parameters (see S5 Fig): for instance, estimated risks were significantly worse in terms of total attack rate but better in terms of outbreak explosion. This discrepancy cautions against deriving quantitative predictions from homogeneous random models and highlights the importance of using empirical data and heterogeneous graphs that take into account real contact structures when assessing virus transmission. Another important aspect is that population coverage was imperfect in the Sociopatterns experiments. Indeed, information is missing for 4% (children), 14% (students), and 30% (employees) of the nodes of the contact networks. In order to assess to which extent data incompleteness could affect our results, we run a sensitivity analysis in which we artificially decreased the number of participants in each graph to remove a random fraction (their connections are lost and they do not participate anymore to the transmission). Corresponding results are presented in S18 Fig. Whatever the incompleteness level, strategies order is preserved. Lower participation rates lead to underestimating the epidemic risk and the epidemic size while overestimating the delay. This analysis confirms that, up to a certain threshold, the fact that Sociopatterns contacts graphs were not perfectly observed should not affect our main conclusions.

Thirdly, for simplification, we assumed in our baseline model a fixed probability of being asymptomatic, equals to 40% whatever the population used. Studies have suggested that children are likely to be more frequently asymptomatic than older individuals. To address that point, we ran a sensitivity analysis varying the probability of being asymptomatic in the different networks. Results suggest that our main results were not affected by this hypothesis, see S15 Fig.

Implementation and choice in practice

Of course, the choice of strategies also crucially depends on criteria such as the feasibility in practice, ease of implementation, etc. For example, hybrid teaching, in which teachers have to manage distance-teaching for half their group and onsite-teaching for the other half, has been used in many French universities since the beginning of the epidemic, but it may be more convenient for an instructor to teach on and off, having the full group either online or onsite. On the other hand, in sectors such as manufacturing a minimum of workers on site may be essential to maintain production, and then Rotating will be the most appealing strategy. We note that the main ingredient differentiating Rotating from On-Off is the breaking of connections between groups (except for the persistent contacts described above). A strategy in a school that would, for example, bring in all students of one level on even weeks and all students of another level on odd weeks would resemble On-Off more than Rotating, because it would not break the groups of students who are in contact.

It is also important to note that modifications in the implementation of the proposed strategies could result in potentially different dynamics. Let’s discuss possible consequences of two natural variations of the proposed strategies. In the first one, a daily Rotating strategy is implemented but schedule is set such that every employee meets each colleague at least once a week. This strategy generates leaky isolation of subgroups, and is therefore expected to limit control efficiency. In the second one, Rotating is planned so that collaborators are grouped in the same group of the colleagues with whom they interact the most. It is likely that such a partition would erase the advantage of Rotating over On-Off, as compared to our simulations where individuals are randomly partitioned.

Conclusion

In this paper, we simulate SARS-CoV-2 transmission and assess the epidemiological impact of various telecommuting strategies. Our study goes beyond previous work by modeling the fine-grained spreading effects of Sars-Cov2, using real-world contact networks at a workplace, a primary school and a high school. To summarize, our results highlight that (1) when is moderately high, all the hybrid telecommuting strategies considered reduce it to less than 1, and the choice between them should primarily be done on the basis of practical considerations. (2) To help prevent dissemination of the disease, it is preferable to alternate over longer periods (weekly rather than daily), but the difference is so slight that practical, psychological, and other considerations should determine the alternation time. In future work, it might be interesting to incorporate the real-life networks as blocks within larger synthetic networks for simulations at larger scales of society.