Halting SARS-CoV-2 by Targeting High-Contact Individuals

Two decades ago network scientists proposed that infectious diseases may be effectively halted by targeting interventions at a minority of highly connected individuals. Can this strategy be effective in combating a disease transmitted through physical proximity such as the contemporary SARS-CoV-2? The strategy's effectiveness critically depends on high between-person variability in close-range contacts. We analyze population survey data showing that indeed the frequency of close-range contacts across individuals is characterized by a small fraction of people reporting very high frequencies of close-range contacts. We simulate a population with empirical contact distributions coming out of lockdown. Simulations show that targeting a small fraction of high-degree individuals dramatically improves containment. Our results further suggest two concrete procedures for identifying high-contact individuals: acquaintance sampling and employment-based target.


Introduction
The world is currently plagued by the new coronavirus SARS-CoV-2 and the social and economic consequences of its containment. Many countries are in the process of gradually lifting a lockdown initially imposed to contain spread, reintroducing measures to contain a second wave. Most containment measures that are being lifted combine general interventions aimed at broad groups of citizens (children, elderly, contact professions) or categories of meeting places (schools, restaurants, airports) (Zhang et al. 2020). The economic costs of such measures are tremendous, expressed in percentages of GDP. At the same time, easing restrictions on interaction achieved through such measures could spur new waves of infections. effective method to reach high-contact individuals and can considerably help "flattening" the curve (section 5). We conclude that, possibly coupled with job-specific targeting, this is a feasible practical method by which governments could identify high-degree nodes in contact networks, thus efficiently coping with the epidemic on the long-run.

Data analysis
We draw on data from COMES-F, a survey conducted in 2012 with data on the number of close-range contacts for a representative sample of about two thousand French residents (Béraud et al. 2015). The leading epidemiological compartmental models of SARS-CoV-2 spread in France currently rely on these data (see, for instance, Di Domenico et al. 2020; Roux et al. 2020;Salje et al. 2020a; in a comparative perspective, see Patrick et al. 2020). In these models, COMES-F data are routinely employed to build social contacts matrices, i.e. average contacts between age groups by places (like school, public transportation, home, etc.) (for an example of this methodology more generally, see, for instance, Prem et al 2017). In contrast to prior use of the data, we rely on the entire cross-individual heterogeneity of the observed distribution of close-range contacts. We implement this distribution in a social network model of disease propagation (section 3), so that we can evaluate the effectiveness of interventions targeted at highconnectivity individuals (section 4).
COMES-F is a large-scale population survey conducted in France during the first half of 2012. An initial sample of 24,250 was drawn from the French population excluding overseas territories through random-digit dialing of landline and mobile numbers. Using quotas for age, gender, days of the week and school holidays, 3,977 people who accepted to participate were sent a diary to complete. 2,033 (51%) contact diaries were returned (participants' age and household size were used as sampling weights to maintain representativeness). In these diaries, participants were asked to keep track of all short-range contacts over the course of 2 full days, and report on sex and age of these contacts, meeting context, and contact duration. A short-range contact was defined as talking to someone at less than 2 meters (possibly including physical contact). COMES-F made a special effort to collect high-quality data also on social contacts of children, defined as respondents aged less than 15 (753 respondents). For them, an adult member of the household completed the diary.
To relieve the reporting burden, respondents were asked to record up to 40 close-range contacts. In addition, respondents currently in employment were asked whether they regarded their occupation as especially exposed to short-range contacts. This turned out to concern 257 respondents. These respondents had to indicate the average number of persons they estimated to meet every day because of their job. Should this number be higher than 20, those specific respondents were asked to enumerate, over the course of the number of reported close-range contacts varies as a function of respondents' socio-demographic features, meeting places and time can be found in Béraud et al. (2015). Here we descriptively focus on one specific, under-investigated aspect of the observed distribution of close-range contacts: their within-and betweengroup heterogeneity. The distribution of contacts is shown in figure 1. The 2,033 individuals reported a total of 19,728 per-day close-range contacts (left panel). The median number of contacts is 8 whereas the average is approximately 9.5, with strong right-skewness. Respondents reporting a number of close-range contacts greater than twice (n=175) or even three times (n=36) the mean are not rare. This is even more visible when considering respondents who declared having an occupation especially exposed to social contacts (n=257) (figure 1's right panel). Overall they reported 14,971 additional job-related contacts. For those contacts the median is 30 whereas the average is approximately 58. One still finds 20 and 10 cases respectively at twice and three times the mean. The central tendencies of both distributions are consistent with those found in other contact surveys (Hoang et al. 2019: 727-728). In both cases, averages are clearly driven by a small fraction of individuals reporting (extremely) high numbers of short-range contact.
Other studies tend to discard outliers as they may heavily impact statistical estimation of central tendencies (see Béraud et al. 2015: 5, 9). We explicitly leave in outliers, as they are characteristic of the distribution of contact and are a crucial characteristic of empirical networks. It is not unusual for a small fraction of people to have jobs with very high frequencies of contact. The feature of high distributional skew is also visible in recent smaller-scale contact surveys conducted in China (Zhang et al. 2019;Zhang et al. 2020) and in a recent six-country survey permitting high self-reported values that range into the 4 and 5 digits. There are three suspicious cases where 999 was recorded. We judged it best to leave these values unaltered. The survey documentation does not identify these cases as missing values nor as maximum values due to a limit of 3 on the number of digits. While these values greatly impact the average level of contact, they do not make or break the skewed nature of the distribution. Accordingly, analyses (available upon request) that exclude 999 cases and upwardly adjust per-contact transmission probabilities accordingly to achieve the same R0 show intervention effects that are qualitatively unchanged. Vice versa, analyses that substitute the 999 cases with higher numbers imputed following the shape of the distributional tail and downwardly adjust per-contact transmission probabilities produce similar intervention effects as well. To obtain a single measure of daily close contact for each respondent we simply combined the number of per-day contacts tracked in the two-day contact diary with the number of supplementary professional contacts estimated by respondents. This is the measure we use in section 3.1 to construct a synthetic network. Figure 1. Left: Fraction of cases (y-axis) reporting a given number of close-range contacts (averaged over the two days) (x-axis) (n=2,033); Right: Fraction of cases (y-axis) reporting a given number of daily job-related contacts (xaxis) among respondents regarding their occupation as especially exposed to social contacts (n=257) This high variability in the number of social contacts persists within major demographic categories that current interventions instead tend to treat as a whole. Figure 2's upper panel shows the distribution of per day self-reported close-range contacts by respondent's gender. Past multivariate analyses of these data found that women (mainly adult women) tend to have a higher average number of contacts than men (see Béraud et al. 2015: 6 and table 1). Figure 2 shows that, behind this main mean effect, there is a large degree of variation within genders. For both men and women numbers of contacts (far) higher than the median occur frequently.  Figure 2's bottom panel shows the distribution of close-range contacts per day by respondent's age. Age is the most recurrent variable used in epidemiological models to represent socially structured social interactions. Age assortativity (and dissortativity at home) is found to be one of the most robust empirical regularities in epidemiological social contact surveys, as also found in multivariate analyses of the COMES-F data (see Béraud et al. 2015: 7-8). This motivates the use of average contacts per (more or less disaggregate) age-groups in age-structured compartment models (see, for some recent examples, Di Domenico et al. 2020;Roux et al. 2020;Salje et al. 2020: 3-4; in a comparative perspective, see Patrick et al. 2020). Now, apart from positive (more or less linear) effects of age on the likelihood of having more social contacts (which is indeed found in these data, see Béraud et al. 2015: table 1), figure 2's bottom panel again shows high variability within age-groups. Rather than averaging this variability, we propose to exploit it using the existence of high-contact individuals as leverage for effective intervention in viral diffusion dynamics.
Focusing the analysis on adult respondents in employment, we find a similar pattern for broad occupational groups (see figure 3). Among both "normal" respondents (top panel) and those regarding their occupation as especially exposed to social contacts (bottom panel), there is a great deal of within-group variation across individuals in the number of self-reported close-range contacts. High-contact individuals seem especially concentrated among high (e.g. elementary school teacher, teaching assistant) and low routine non-manual workers (e.g. bank teller, teller in public administration) and service class (e.g. university professor, politician, journalist or doctor). Data (available upon request) on an eleven-category variable describing respondents' occupational sector also suggest that high-contact individuals tend to concentrate in the service sector and non-manual routine labor, specifically shops, services to persons, education, health, and administration. However, while these data suggest that social contacts within occupations are much more dispersed than one could expect under a distribution symmetrically centered around the mean, COMES-F does not provide a detailed list of jobs. The appendix shows that in data from another survey with less precise measures of close social contacts, more fine-grained distinctions in professional categories can indeed account for a substantial portion of person-to-person variability in closecontact frequency. category (y-axis); Bottom: Distribution of daily job-related contacts among respondents regarding their occupation as especially exposed to close-range contact (n=257) by occupation category (y-axis). Occupational categories (n in parenthesis, top panel first): F=farmers (nTOP=14, nBOTTOM=2), PB=petty bourgeoisie (craftsmen and shopkeepers) and entrepreneurs (nTOP=26, nBOTTOM=18), SC= Service class (managers, high-skilled administrators, intellectual, scientific and liberal professions) (nTOP=123, nBOTTOM=91), HNM=High routine non-manual worker (nTOP=47, nBOTTOM=41), LNM=Low routine non-manual workers (nTOP=183, nBOTTOM=93), MW=manual workers (43,nBOTTOM=12).

Model
Using COMES-F survey data we build an agent-based computational model in which the synthetic network through which the virus diffuses is calibrated on the French contact data (for other work using empirical network data in agent-based diffusion models, see Smith and Burrow 2018;Manzo et al. 2018). Our aim is to study the macroscopic consequences of cross-individual variability in close-range contact frequencies empirically observed in France and assess whether this variability can be exploited for effective intervention in the ongoing epidemic. As such, we simulate a population the size of the COMES-F sample from which we eliminate four respondents who reported no close-range contacts. (We find that populations much larger than our sample produce qualitatively similar results but naturally take much longer to simulate.) We now explain how we connect these 2,029 agents (section 3.1), present the transmission model (3.2), and describe the interventions we simulate (3.3).

Network construction and features
We connected the agents according to two social network models. The first, which we will refer to as the "empirical" model, is the focus of our simulation; the second, which we will refer to as the "random" model, constitutes a benchmark against which to compare dynamics and effects of interventions.
In the empirical network model, agents are first given a degree (number of network ties) precisely equal to the number of per day close contacts reported by each respondent in the survey (see figure 1).
Then, to connect agents to one another, we adapted the configuration model, an algorithm that was proposed to generate random networks with arbitrary degree distributions (for a discussion, see Jackson 2008: 83-85). To avoid duplicate links and self-links while ensuring an exact match between each virtual agent and an empirical respondent, we considered source agents in descending order of the to-be-generated degree rather than in random order, and then randomly picked available destination agents. Every time a connection was made, the degree of the two newly connected agents naturally increased by one. As soon as an agent reached the to-be-generated degree, it was excluded from the search algorithm. We found this procedure to always converge, achieving the intended empirical degree distribution.
Under the random mixing assumption common to epidemiological models, social contacts are assumed to happen at random within certain categories (usually age-groups): only the (within-category) average number of contacts is of interest. From a network perspective, this amounts to postulating a random network where contact probabilities across individuals have little variability and the degree of each node can be assumed being well approximated by the average degree (Newman 2002;Barthélemy et al. 2005).
We therefore also study an Erdős-Rényi random graph with the average degree observed in the survey, as a benchmark distribution. This random network is characterized by low variability in contact across agents. Table 1 shows network statistics computed over 100 realizations of the two networks. By construction, the empirical network has the precise mean, median and standard deviation of the degree distribution observed from the survey. The degree distribution of the empirical network has strong rightskewness, with the mean well above the median. By contrast, the degree distribution of the random network has essentially equal mean and median. The dispersion of the degree also strongly differs between the two networks, with the empirical network exhibiting variation in the nodal number of links that is five times higher than in the random network. Table 1 also shows several other network statistics. We cannot calibrate or verify these network features, as the survey data lack data on connections between respondents' contacts. Nonetheless, the empirical network generated by our modified configuration model shares various features with social networks studied elsewhere (for a detailed discussion, see Appendix A2).

Network
Average degree

SEIR model
We model disease propagation through the network by following the logic of compartmental models, which have been previously applied to the COVID-19 outbreak (Brethouwer et al. 2020;Kucharski et al. 2020;Qun et al. 2020;Prem et al. 2020). In particular, we adopt a SEIR model (Martcheva 2015) and follow recent French studies (and parametrization therein, see Salje et al. 2020a, b) to determine how agents unidirectionally move from being (S)usceptible, to (E)xposed, (I)nfected, and eventually (R)ecovered. In particular, agents are supposed to stay within a given state as many iterations as days current empirical studies show real-world individuals remain within those states. One iteration thus represents one day.
Upon infection, agents first enter E where they stay 4 days; during this period, they are not infectious (see Salje et al. 2020a: 10). They then move to I where they become infectious, and can contaminate other agents over the course of 4 days (see Salje et al. 2020a: 10). Infected agents move to R with probability following a normal distribution with average 0.993 (and possible range at the agent-level between 0.990 and 0.996) (see again Salje et al. 2020b) provided they spent a number of days in I at least equal to a given recovery time. The recovery time follows a Poisson distribution centered on 2 weeks (with possible range at the agent-level between 1 and 6 weeks) (for these values, see empirical estimates in World Health Organization 2020: 14).
We combine this basic compartment structure with network topology such that the agents an infectious-infected agent can infect are determined by the network of close-range contact (see Barrat et al. 2008: ch. 9). During each day, an infectious-infected agent can only transmit the disease to its direct contacts. The agent-to-agent per-day-per-contact transmission probability is assumed to be normally distributed around 0.0300 or 0.0186 (and possible range at the agent-level between 0.0005 and 0.12). This parameterization at the dyadic level generates at the aggregate level an average basic reproduction rate R0 equal to 2.9 and 1.785 respectively. The value 2.9 corresponds to the empirically estimated R0 in France before lockdown (see Salje et al. 2020b): we adopt it to assess whether the model may have generated an epidemic in the absence of any intervention that is in line with existing counterfactual studies (figure 4).
The 1.785 value corresponds to the average between pre-and post-lockdown estimates (the latter being 0.7, see Salje et al. 2020b). We switch to 1.785 to study the impact of targeted interventions in a scenario after partial lifting of the French lockdown on May 11. It represents a reduction in R0 down from the original pre-lockdown level of 2.9 reflecting post-lockdown continuation of general policy measures such as social distancing, group size restrictions and sanitary improvements. At R0 = 1.785, further person-specific interventions are needed to bring down the effective R0 and mitigate a second wave. We study how the targeting of these interventions impacts their effectiveness in doing so.
All the simulations we performed start with five (randomly chosen) initially infectious infected agents. Given the size of our model population, this is the lowest number of seeds that prevents excessive variability across simulation trials. Proportionally to the size of the French population, this number at 160000 roughly approximates, for our basic testing scenario (i.e. R0=2.9), estimates of infected cases around March 7 (see Salje et al. 2020b: figure 3E); on the other hand, for our intervention scenario (i.e. R0=1.785), this number of initial seeds would roughly correspond to targeted interventions commencing a week after partial lockdown release, with R0=1.785 having allowed infections to climb back up by a factor of ten from current levels at about 4000 new cases per day (see Salje et al. 2020b: figure 3G). In simulations on larger networks, which take much longer to run, in which we implemented the same degree distribution and used the same number of seeds (smaller fraction), we find that peaks naturally occur later, while the interventions we present next show qualitatively the same relative effects.

Interventions
We consider a scenario in which France progressively exits the strict March-17/May-11-2020 lockdown and has a fixed daily (medical / technological / financial / ethical) capacity to immunize a portion of susceptible/infected individuals. In the absence of a vaccine, "immunization" describes an effort equivalent to the medical act of 100% effective vaccination. This could be any combination of preventive interventions, such as medical testing, quarantining-if-positive, protective measures in high-risk professions and targeted informational campaigns. Accordingly, we assume that the French government has a daily budget b with which it can immunize b agents, preventing future infection of these and / or spread from these targeted agents to other agents. On day 1, spread from b agents is permanently prevented, on day 2 spread from an additional b agents is permanently prevented, and so on. This is implemented as follows: Each iteration, representing one day, b S, E or I agents are selected and moved to R.
We study four budgets for each intervention: b = 1, 3, 5, and 10. We consider three methods for selecting agents for intervention. The first method, "NO-TARGET", simply randomly samples b agents for immunization each day, and is intended as a benchmark against which to contrast the other two methods.
The second method, "CONTACT-TARGET", follows the strategy described in Cohen & Havlin (2010), whereby each day b random agents are sampled who each select one random contact (without replacement) for immunization. Because of the friendship paradox (Feld 1991), these targets have above-average expected degree (Galvani et al. 2005;Christakis & Fowler 2010;Kitsak et al. 2010). This is so because high-degree nodes are by definition overrepresented among other nodes' contacts (Feld 1991). The CONTACT-TARGET strategy is implementable in practice as a government could in fact randomly sample from the known population and have sampled individuals suggest their contacts. The third method, "HUB-TARGET", assumes that agents' numbers of contacts are perfectly observed. During each iteration, nodes are targeted in strictly decreasing order of their network degree, starting with the b largest hubs.   shows results for the NO-TARGET procedure, whereby each day b randomly selected susceptible, exposed or infected agents are immunized. The NO-TARGET procedure's maximally achievable impact, using the most generous budget considered, b = 10, corresponding to 10% of the population being immunized during the first 20 days, leaves the peak at 69% (Table 3: 326 / 470) of what it would have been without any intervention. Also, the peak is not delayed.

Results
Panels B show results for the CONTACT-TARGET procedure, which assumes that no global information on connectivity is available. Lacking this information, it attempts to find high-degree nodes by drawing a random sample of agents with unknown degree and selecting a random neighbor of each sampled agent for immunization. This procedure is much more effective than the NO-TARGET intervention, as the figure shows. A budget of b = 3 CONTACT-TARGET immunizations produces an impact comparable to a NO-TARGET immunization regime with b = 10 daily immunizations. At b = 10, the CONTACT-TARGET intervention achieves a reduction down to only 5% of the peak in the no-intervention scenario ( The CONTACT-TARGET method would be more effective if randomly chosen agents would be able to select a random neighbor for immunization among relatively high degrees at a higher chance than network structure per se allows. Survey data suggest that targeting of certain professions may also effectively identify high-degree agents (see appendix A1). To evaluate the maximally achievable impact of any degree-based intervention, panels C of figure 5 show the impact of the HUB-TARGET policy, whereby each day the b non-immune agents with highest degree are immunized. Even a single agent per day (b = 1) reduces the peak down to only 25% (Table 2: 118 / 470) and delays it by 17 days. This peak reduction exceeds what NO-TARGET immunization achieves with ten agents per day (b = 10). Three agents per day accelerates the peak again, but this happens as peak height is reduced to only 4%. With five agents (5% of the population after 20 days) virus spread is effectively halted. In other words, if one fully protects a small vulnerable fraction of the population, one can prevent nearly all infections that would have otherwise occurred.

Implications for contemporary policy
According to estimates available on 11 May 2020, when lockdown progressively started to be eased in France, only between 2.8% and 7.2% of the French population had acquired immunity against COVID-19 (Salje et al. 2020b). This would mean that the vast majority of the population is still susceptible. As a consequence, in absence of a vaccine, many observers consider the only alternative to be massive testing, possibly coupled with digital contact tracing (Sustained Suppression, Nature Biomedical Engineering, 2020).

Contemporary French post-lockdown policy is indeed based on this strategy, and policy in several
other European countries is similar. French government and health authorities ask persons with symptoms to call their city doctor, who should prescribe a free-of-charge test. Should the person be positive, the patient is asked to describe all her/his close contacts during the last seven days; each is also asked to be reached by phone, interviewed, and, if established that contact with the focal individual took place, this contact-case too is tested and quarantined if positive (Info Coronavirus COVID-19, 2020). The government early May 2020 claimed to be able to perform up to 10,000 tests per day (which amounts to 0.015% of the French population). This advertised capacity is widely questioned (France Info 2020). Even if this capacity were available, it still remains below the agreed effective fraction of the population to be daily tested, i.e. at least 2% (Humanity tested, Nature Biomedical Engineering, 2020). Moreover, as acknowledged by experts advising the French government (Conseil Scientifique Covid-19, 2020: 29), this strategy requires unrealistic levels of human resources, in particular in case of citizen resistance to and suspicion of contact tracing, as is the case in France (Casilli et al. 2020). Last but not least, the proposed strategy focuses on self-reported symptoms, which exclude asymptomatic cases. Asymptomatic cases are thought to be a non-negligible fraction of COVID-19 infected persons, especially among children (Heneghan et al. 2020).
Our results suggest a third way in addition to contemporary resource-intensive testing and contact tracing strategies and future mass vaccination. This alternative strategy is based on four pillars: First, interventions should concern both susceptible and infected persons; Second, in both cases, given current practical limitations, search over randomly selected portions of the population can fruitfully replace exhaustive search; Third, sampling should be performed in such a way that high-contact persons are more likely selected for intervention; Fourth, high-contact persons should be "treated" with priority. The rationale behind the third and fourth pillar is that high-contact individuals, if susceptible, are simultaneously at higher risk of being contaminated because of their high social exposure, and, when they are infected, also transmit the virus to a larger fraction of the population.
Our model results suggest this strategy can be effective. When a virus spreads through a network that includes individuals with dramatically more close-range contacts than the average (as observed in France), the CONTACT-TARGET method (which attempts to find high-degree nodes by drawing a random sample of agents with unknown degree and selecting one random neighbor of each sampled agent for immunization) can reduce the concurrent number of infected agents by a factor of 1.3, 2.4, 4.4 or 18.8 depending on whether one samples 1, 3, 5 or 10 agents per iteration (our budget "b"). If simulated peak heights are transformed into ICU admissions proportionally to French population (see appendix, table A4, and computational details therein), this would amount to reducing ICU admissions from 102826 without intervention to 79197, 42661, 23191 or 5469 respectively (5000 ICU beds being normal French national capacity). If b-values are also expressed proportionally to the French population, our CONTACT-TARGET method leads to the largest ICU-admission reduction by targeting approximately 320000 individuals per day. In contrast, for the same budget, the NO-TARGET method (which searches agents for immunization at random within the population) can generate at best a peak reduction of a factor of 1.4. Thus, keeping budget constant, systematic targeting of high-contact individuals is much more effective than random search of average-contact persons.
The sampling burden could be further reduced if one efficiently exploits the empirical observation that high-contact individuals tend to be concentrated within certain occupations (see appendix A1).
Following this logic, the sampling person may be asked preferentially to report her/his social contacts within a given list of highly socially exposed professions. Under this condition, our CONTACT-TARGET method would get closer to the HUB-TARGET one. With the latter method a sample of approximately 95000 or 160000 individuals per day would suffice to reduce the peak, and corresponding ICU admissions, under the level of normal French hospital capacity (i.e. 5000 ICU admissions).
Our simulations are silent about the specific content of the actions to be performed on each finally selected individual. We only provide a method to maximize the efficiency of that selection. However, we do believe that, in the absence of a vaccine, and given the relative scarcity of tests, the skeleton of a sustainable recipe could be as follows: a) if the selected high-contact person is symptomatic, directly quarantine; b) if the selected high-contact person is not symptomatic, test with probability equal to the daily number of available tests divided by the number of daily sampled people. In both cases, the selected highcontact person should receive a targeted informational message explaining why, given her/his social exposure, it is more crucial for her/him than for the "average" person to adopt basic protective gestures during all his social and professional interactions. The feasibility of this strategy thus comes from the fact that it relies more on actions, i.e. targeted messages, that can be performed at distance and does not require mobilizing huge human resources across a country. This is in line with recent calls for approaching COVID-19 in terms of massive, but targeted, preventive interventions based on pedagogic actions rather than on actions constraining individuals' freedom, which have proven not to be effective in the long-run for other aggressive infectious diseases (Marcus 2020).

Discussion
As countries exit the Covid-19 lockdown many have limited capacity to prevent flare-ups of the coronavirus. With medical, technological, and financial resources to prevent infection of only a fraction of its population, which individuals should countries target for testing and tracking? Together, our results suggest that targeting individuals characterized by high frequencies of short-range contacts dramatically improves the effectiveness of interventions. An additional known advantage of targeting hubs with medical testing specifically is that they serve as an early-warning device that can detect impending or unfolding outbreaks (Christakis & Fowler 2010;Kitsak et al. 2010).
This conclusion is reached by moving away from the standard compartmental models that rely on random mixing assumptions toward a network-based modeling framework that can accommodate personto-person differences in infection risks stemming from differential connectedness. The framework allows us to model rather than average out the high variability of close-contact frequencies across individuals observed in contact survey data. Simulation results show that consideration of realistic close-contact distributions with high skew strongly impacts the expected impact of targeted versus general interventions, in favor of the former.
A key limitation of our approach is that we have not considered the duration of close-range contact.
A negative relationship may exist between the number of close-range contacts and the per-contact transmission probability if the per-contact duration decreases in the number of contacts. In this case our results could overstate the importance of high-contact individuals and the effectiveness of the targeted intervention strategies we have proposed. Too little is still known about Covid-19 transmission through droplet "clouds" (see Bourouiba 2020;Mittal 2020) to say anything definite in this direction. Nonetheless, limited evidence on super-spreading events involving large crowds interacting in close range (for an overview, see Kay 2020) suggest that the risk posed by large numbers of interactions with many people is not necessarily mitigated by the brevity of such contacts.
How could public policy effectively target high-contact individuals? We propose two concrete methods. First, one could test and track workers in professions characterized by high frequencies of contact.
Survey data show that some professions involve ten times as much close-range contact than others, with elementary school teachers and cashiers topping the list. The occupational categories used in the survey follow a common international standard used by the US Bureau of Labor Statistics. Legislation could straight-forwardly be set on its basis. A second method is entirely agnostic of who the high-degree individuals are and targets random acquaintances of random individuals, who statistically have high expected degree. In our simulations this second method shows to be effective yet at the same time is conservative in assuming no knowledge of degree or use thereof. If individuals nominated contacts they knew to have many other contacts, the difference targeted intervention could make would be even greater.

A1. Segregation of network degree by profession
One potential method for finding individuals with high contact frequencies is to target professions characterized by high contact. The professional categories in the COMES-F data (Figure 3 in main text) are too coarse to evaluate the effectiveness of such a method. We therefore exploit a recent survey that has a somewhat less thorough measurement of close-range contact (Belot et al. 2020) but fine-grained professional categories. The Belot et al. (2020) Table A1. Close contact by profession in Belot et al. data (2020) The distribution of close contact frequency in the Belot et al. (2020) data is displayed in Figure A1. The distribution is severely right-skewed, as we also observed for the French survey data. This provides confidence that the existence of hubs is not a measurement artifact but a robust feature of contact networks: Using different methods for measuring contact the same distributional characteristic is obtained. Table A1 shows the mean and median number of close-range contacts by profession, in descending order of mean contact frequency, combining short-and long-duration contacts, excluding zero answers and professions with 15 or fewer cases. Table A1 has face validity, topped by professions that clearly involve close contact with many individuals --elementary school teachers, cashiers --and at the very bottom individuals who mostly work from home --computer programmers. Some professions have an order of magnitude greater mean close-range contact than others. The spread is substantial especially when considering the ambiguity in the possible interpretation of the phrase "social contact" used in the questionnaire, translated into different languages, and the difficult task of estimating such numbers without use of a contact diary, which will produce noise that suppresses measured occupational differences. These results suggest that targeting select professions may be an effective strategy for finding hubs in contact networks. Table 1 in the main text shows a number of topological features of the empirical network that we generated to match the empirical distribution of close contacts. Here we discuss the topological features of the empirical network in greater detail and compare them with those observed for social networks measured elsewhere. The first is degree assortativity. Social networks tend to be characterized by positive degree assortativity, i.e. nodes with similar degree are more likely to be connected (see Newman 2003a). However, in presence of broad degree distributions, the network can turn to be disassortative. This results from the fact that, while high-degree nodes still tend to be connected to one another, thus pushing assortativity up, low-degree nodes are preferentially attached to high-degree nodes, thus pushing assortativity down.

A2. Topology of the empirical network
Depending on the specific distributions of high-and low-degree nodes these two effects will be more or less balanced (see Johnson et al. 2010). In our case, the simulated empirical network is slightly disassortative, which suggests that the highly connected individuals are not only connected amongst themselves but they also widely connect to individuals with less close-range contacts. The corresponding random network has assortativity close to zero. Degree disassortativity renders targeting of high-degree nodes more effective: "(...) those vertices are strewn far apart across the network, so that attacking them attacks all parts of the network at once" (Newman 2003a: 10).
The clustering coefficient of the empirical network is 0.12, which is in line with values observed for many social networks (see Newman 2003b: . It is about 7 times higher than the corresponding value for our benchmark random network. Analysis of the entire distribution of nodal clustering coefficients (available upon request) suggests systematic variation across nodes with different degrees. In particular, low-degree nodes tend to have higher clustering than high-degree nodes.
On the overall simulated empirical network, this produces a negative correlation of -0.26 between nodal degree and clustering coefficient (meaning that the higher the degree of a node the lower the fraction of connections among its own neighbors). This means that, in addition to a broad degree distribution, our simulated empirical network contains hubs that span across the network rather than clustering together. This is a feature shared with some previously studied networks with high-degree nodes (Barabási 2014: 232-237). It is also a topological feature that makes hub-centered interventions especially effective: attacking the hubs means interrupting (or slowing down) communication among the modules (ibid: 236).
By contrast, the correlation between nodal degree and clustering in the random network we study tends toward zero.
Finally, the network we generated based on the empirically observed broad degree distribution of close-range contacts in France exhibits the typical high reachability of small-world topologies: its average path length and diameter are comparable to that of an Erdős-Rényi network with the same size and degree, and is consistent with what is usually observed in pure scale-free models (see Albert and Barabasi 2002:74).
Highly connected individuals are effective in bringing many parts of the network close.
Thus, overall, the network that we consider as the possible relational infrastructure of the virus spread in France is anchored to the French long-tailed distribution of close-range contacts and, at the same time, exhibits reasonable topological features in terms of degree assortative, community structure, and reachability.  A comparison of panels A between figures A2 and 5 shows that NO-TARGET interventions are less effective in empirical networks with high degree skew than in random networks with low degree variance.

A3. Results for the random network
This suggests that models that do not account for empirical network structure may overestimate the expected impact of interventions. Comparing panels B and C across figures we find that HUB-TARGET and CONTACT-TARGET interventions are much more effective in the empirical network than in random networks, where the to-be-immunized agents have lower network degree.  Table A4. ICU admissions at the national scale corresponding to Peak height (maximum # concurrently infected agents) generated by the model (see table 3) under three interventions (rows) and four budgets (column) on the empirical network. Shown are median, 5% and 95% percentiles across 100 iterations.