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Abstract

The COVID-19 pandemic highlights the importance of effective non-pharmaceutical public health interventions. While social distancing and isolation has been introduced widely, more moderate contact reduction policies could become desirable owing to adverse social, psychological, and economic consequences of a complete or near-complete lockdown. Adopting a novel social network approach, we evaluate the effectiveness of three targeted distancing strategies designed to 'keep the curve flat' and aid compliance in a post-lockdown world. We simulate stochastic infection curves that incorporate core elements from infection models, ideal-type social network models, and statistical relational event models. Our models demonstrate that while social distancing measures clearly do flatten the curve, strategic reduction of contact can strongly increase their efficiency, introducing the possibility of allowing some social contact while keeping risks low. Limiting interaction to a few repeated contacts emerges as the most effective strategy. Maintaining similarity across contacts and the strengthening of communities via triadic strategies are also highly effective. This approach provides empirical evidence which adds nuanced policy advice for effective social distancing that can mitigate adverse consequences of social isolation.
Social network-based distancing strategies to flatten the
COVID-19 curve in a post-lockdown world
Per Block, Marion Hoffman, Isabel J. Raabe, Jennifer Beam Dowd
Charles Rahal,§, Ridhi Kashyap,§,, Melinda C. Mills,§
Leverhulme Centre for Demographic Science, Department of Sociology, University of Oxford
Department of Humanities, Social and Political Sciences, ETH Zurich
Institute of Sociology, University of Zurich
§Nuffield College, University of Oxford, Oxford, UK
School of Anthropology and Museum Ethnography, University of Oxford, Oxford, UK
25th May, 2020
Abstract: Social distancing and isolation have been introduced widely to counter the COVID-
19 pandemic. However, more moderate contact reduction policies become desirable owing to ad-
verse social, psychological, and economic consequences of a complete or near-complete lockdown.
Adopting a social network approach, we evaluate the effectiveness of three targeted distancing
strategies designed to ‘keep the curve flat’ and aid compliance in a post-lockdown world. These
are limiting interaction to a few repeated contacts, seeking similarity across contacts, and strength-
ening communities via triadic strategies. We simulate stochastic infection curves that incorporate
core elements from infection models, ideal-type social network models, and statistical relational
event models. We demonstrate that strategic reduction of contact can strongly increase the effi-
ciency of social distancing measures, introducing the possibility of allowing some social contact
while keeping risks low. This approach provides nuanced insights to policy makers for effective
social distancing that can mitigate negative consequences of social isolation.
Keywords:COVID-19, social networks, stochastic infection curves, statistical relational events
1. INTRODUCTION
The non-pharmaceutical intervention of ‘social distancing’ is a central policy to reduce the spread of
COVID-19, largely by maintaining physical distance and reducing social interactions (Glass et al.,
2006). The aim is to slow transmission and the growth rate of infections to avoid overburdening
health-care systems, widely known as ‘flattening the curve’ (Roberts,2020). Social distancing
includes bans on public events, the closure of schools, universities and non-essential workplaces,
limiting public transportation, travel and movement restrictions, and urging citizens to limit social
interactions.
The majority of existing research on mitigating influenza pandemics focus on the effectiveness
of different individual measures, such as travel restrictions, school closures, or vaccines (Ferguson
et al.,2006;Germann et al.,2006). Few have simultaneously considered interventions and the
structure of social networks. When social networks are examined, it is generally in relation to
vaccination (Ventresca and Aleman,2013), contact tracing, or analysing the spread of the virus
For correspondence: Per Block and Melinda C. Mills, Leverhulme Centre for Demographic Science, Department of
Sociology, University of Oxford, OX1 1JD, United Kingdom. Tel: 01865 286170. Email: per.block@sociology.ox.ac.uk
and melinda.mills@nuffield.ox.ac.uk. The replication files for this paper including customised functions in the statis-
tics environment R and an example script are available on Zenodo, a general-purpose open-access repository devel-
oped under the European OpenAIRE program and operated by CERN (https://zenodo.org/record/3782465).
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(Sun et al.,2020;Wu and McGoogan,2020). We outline key behavioural strategies for selective
contact reduction that every individual and organisation can adopt to maximise the benefits of
limiting contact and engaging in strategic social distancing. Applying insights from social and
statistical network science, we demonstrate how changing network configurations of individuals’
contact choices and organisational routines can alter the rate and spread of the virus, by providing
guidelines to differentiate between ‘high-impact’ and ‘low-impact’ contacts for disease spread. This
can contribute to balancing public health concerns and socio-economic needs for interpersonal
interaction. We introduce and assess three strategies: contact with similar people, strengthening
contact in communities, and repeatedly interacting with the same people.
Conclusions regarding the effectiveness of non-pharmaceutical public health interventions have
often been made on the basis of on ‘expert recommendations’ rather than scientific evidence (Bell,
2006). During previous outbreaks (e.g. SARS-CoV), social distancing measures such as workplace
closures, limiting public gatherings, and travel restrictions were implemented. Cancelling public
gatherings and long-distance travel restrictions appears to decrease transmission and morbidity
rates (Aledort et al.,2007). There is mixed evidence regarding the effectiveness of school closures
on respiratory infections, possibly because of the timing of school closures, or since this affects
only on school-aged children (Jackson et al.,2013).
There has been considerably less research on the effectiveness of other types of social distancing
measures, such as strategies based on individual’s knowledge of their social surrounding. Existing
research has demonstrated that interventions are only effective and feasible when the public deems
them acceptable (Aledort et al.,2007). Our approach recognises the social, psychological, and
economic cost of – and potential compliance fatigue with – complete isolation (Morse et al.,2006).
Fully quarantining non-infected, psychologically vulnerable individuals over prolonged periods can
have severe mental health consequences. Many facets of economic and social life require some
amount of person-to-person contact. Compliance with recommendations to strategically reduce
contact is more favourable than compliance with complete isolation and, thus, can keep the curve
flat in the long run. We therefore propose a novel approach that assesses the effectiveness of
network adaptations that rely on less confinement and allow some degree of social contact while
still ‘flattening the curve’.
Flattening the (infection) curve represents a decrease in the number of infected individuals at
the height of the epidemic, with the incidence of cases distributed over a longer time horizon
(Roberts,2020). This is largely achieved by reducing the reproduction number (R), which is how
many individuals are infected by each carrier. Social distancing policies are implicitly designed to
achieve this by limiting the amount of social contact between individuals. By introducing a social
network approach, we propose that a decrease in R can simultaneously be achieved by managing
the network structure of interpersonal contact. From a social network perspective, the shape of the
infection curve is closely related to the concept of network distance or path lengths (Wasserman
et al.,1994), which indicates the number of network steps needed to connect two nodes. Popularised
examples of network distance include the ‘six degrees of separation’ phenomenon (Milgram,1967),
which posits that any two people are connected through at most five acquaintances.
The relation between infection curves and network distance can be illustrated with a simple
network infection model (Figure 1). Panels A and C depict two networks with different path
lengths, each with one hypothetically infected COVID-19 seed node (purple square). At each time
step, the disease spreads from infected nodes to every node to which they are connected; thus, in
the first step the disease spreads from the seed node to its direct neighbours. In the second step, it
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Figure 1: Two example networks A and C. Both networks have the same number of nodes (individuals)
and ties (social interactions) but different structures which imply different infection curves (B and D). Bold
ties highlight the shortest infection path from the infection source to the last infected individual in the
respective networks. Network node colour indicates at which step a node is infected and maps onto colours
of histogram bars.
spreads to their neighbours, who are at network distance 2 from the seed node, and so forth. Over
time, the virus moves along network ties until all nodes are infected. The example shows that the
network distance of a node from the infection source (indicated by node colour in Figure 1A and
C) is identical to the number of time-steps until the virus reaches it. The distribution of network
distances to the source thus directly maps onto the curve of new infections (Figure 1B and D).
In our example, both networks have the same number of nodes (individuals) and edges (inter-
actions); however, the network depicted in panel C has a much flatter infection curve than the
network in panel A, even though all nodes are eventually infected in both cases. This is because the
latter network has longer path lengths than the former one – or in other words – more network dis-
tance between the individuals due to a differing structure of interaction, despite the same absolute
contact prevalence. Thus, when adopting a network perspective, flattening the curve is equivalent
to increasing the path length from an infected individual to all others, which can be achieved
by restructuring contact (besides the generally proposed reduction of contact). Consequently, one
aim of social distancing should be increasing the average network distance between individuals by
smartly manipulating the structure of interactions. Our illustration shows a viable path to keep the
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COVID-19 curve flat while allowing some social interaction: we must devise interaction strategies
that make real-life networks look more like network C, and less like network A.
We propose a series of strategies for how individuals can make local decisions to achieve this
goal. Understanding which types of strategies of targeted contact reduction and social distancing
are more efficient in increasing path lengths and flattening the curve can inform how to shift from
short-term (complete lockdown) to long-term management of COVID-19 contagion processes. The
contact reduction strategies we propose are based on insights of how items flow through networks,
such as diseases, memes, information, or ideas (Watts et al.,2006;Podolny,2001;Borgatti,2005;
Centola,2010). Such spread is generally hampered when networks consist of densely connected
groups with few connections in-between, such as individuals who live in isolated villages scattered
over sparse rural areas (Watts,1999). In contrast, contacts that bridge large distances are related to
short paths and rapid spread. When commuters travel between these isolated villages, for instance,
network distances decrease substantially (Milgram,1967;Centola,2010). Using this knowledge,
we can avoid rapid contagion by encouraging social distancing strategies that increase clustering
and reduce network short-cuts to reap the largest benefit of reducing social contact and limiting
disease spread to a minimum. We propose three strategies aimed at increasing network clustering
and eliminating short-cuts.
While more realistic examples of the proposed strategies are simulated in the next section, we
first outline the underlying principles of the model in Figure 2. Panel A depicts a network in which
densely connected communities are bridged by random, long-range ties. This type of network is
commonly known as a ‘small world network’ (Centola,2010). It is widely used in simulations, as
it represents core features of real-world contact networks, in particular social clustering combined
with short network distances, making it particularly useful for our illustration (Milgram,1967).
Within clusters, individuals are similar to each other, indicated by their node colour, and live in
the same neighbourhood, indicated by node location. The further away two clusters are in the
figure, the further they live from each other and the more dissimilar their members. Panels A to
D illustrate the successive, targeted contact reduction strategies, while the bar-graph depicts the
distribution of distances of all individuals from one of the two highlighted infection sources.
Strategy 1: ‘Seek similarity’ strategy: Reduce geographic and socio-demographic difference to
contact partners (A to B in Fig. 2). In the first strategy, individuals choose their contact partners
based on their individual characteristics. Generally, individuals tend to have contact others who
share common attributes, such as those in the same neighbourhood (geographical), or of similar in-
come or socio-demographic characteristics such as age (Feld,1981;Rivera et al.,2010;McPherson
et al.,2001). The tendency to interact with similar others is called ‘homophily’ in the sociolog-
ical network literature (Rivera et al.,2010) and is a ubiquitous and well-established feature of
social networks (thus, we use ‘seek similarity’ strategy and ‘homophily’ strategy interchangeably).
Because we are mostly connected to similar others, contact with dissimilar individuals tends to
bridge to more distant communities. Restricting one’s contact to those most similar helps limit
network bridges that substantially reduce network path lengths. This entails choosing to inter-
act with those geographically proximate (e.g., living in the same neighbourhood), or individuals
with similar characteristics (e.g., age). Panel B in Figure 2shows the network structure after the
implementation of this strategy of tie reduction. The associated bar-graph illustrates that follow-
ing this network-based intervention, a substantial number of nodes are at a larger distance from
the infection source. This strategy will be successful when the characteristic or variable which
determines the communities can take on a variety of different (categorical or continuous) values
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A B C
DA B C D
Proportion,of,nodes,,,
with,distance
disconnected,(safe)
4+
2,or,3
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Figure 2: Example networks that result from the successive tie reduction strategies. Node
colour represents an individual characteristic, where similarity in node colour represents similarity in this
characteristic. Node placement represents geographic location of residence. A: initial small world network;
B: removing ties to dissimilar others that live far away; C: removing non-embedded ties that are not part
of triads or 4-cycles; D: repeating rather than extending contact. Bar graphs show network distances from
the infection sources, highlighted in yellow, for the different scenarios.
for different individuals, thereby promoting the formation of small communities. A broader split,
such as along gender or ethnic lines does not promise measurable success but will instead likely
exacerbate the negative consequences of distancing measures. This strategy is supported by epi-
demiological modelling which suggests that co-residence and mixing of individuals from different
ages strongly increases the spread of infectious disease, such as COVID-19 (Pellis et al.,2020).
Providing a concrete example, if people only interact with others in a 3-block radius (increase geo-
graphic similarity), more than 30 transmission events would be necessary for a virus to travel 100
blocks. Workplaces where many individuals come together could, for instance, implement routines
to decrease contact between groups from different geographic areas or age-groups.
Strategy 2: ‘Strengthen triadic communities’ clustering strategy: Increase triadic clustering
among contact partners (B to C in Fig. 2). For the second strategy, individuals must consider with
whom their contact partners usually interact. A common feature of contact networks is ‘triadic
closure’, referring to the fact that contact partners of an individual tend to be connected themselves
(Feld,1981;Granovetter,1973;Goodreau et al.,2009). Tie embedding in triads is a particularly
useful topology for containing epidemic outbreaks. Consider a closed triad of individuals i,j, and
h. When iinfects jand h, the connection between jand hdoes not contribute to further disease
spread: it is a ‘redundant’ contact (Burt,1995). When comparing networks with an identical
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number of connections, networks with more redundant ties tend to have longer path lengths.
Accordingly, when removing contact to others, one should prioritize removing ties not embedded
in triads, since these ties generally decrease path lengths. In practice, this means that physical
contact should be curtailed with people who are not also connected to one’s usual other social
contacts. Panel C in Figure 2illustrates the structure if ties that are not part of closed triads or
4-cycles are removed. In this ideal-type example, this intervention not only further reduces the
network distance of many nodes from the infection sources, but also creates isolated communities
or that cannot be infected by the virus.
Strategy 3: ‘Repeat contact and build micro-communities’ strategy: Repeated contact to same
others, rather than changing interaction partners (C to D in Fig. 2). For the third strategy, individ-
uals need to consider who they want to regularly interact with and, over time, restrict interaction
to those people; this reduces the number of contact partners rather than number of interactions,
which is particularly important when contact is necessary for psychological well-being. This strat-
egy of limiting contact to very few others with repeated interactions is in the spirit of a social
contract with others to create micro-communities to only interact within the same group delineated
by common agreement. Although this requires coordination, micro-communities would be difficult
for a virus to penetrate, or – importantly – if the infection is contracted by one contact, for the
virus to spread further. Another implication of this strategy includes the repetition of interaction
with others that overlap across more than one contact group. For example, meeting co-workers
outside of work for socializing will have less of an impact on the virus spread relative to a sepa-
rate group of friends, since a potential infection path already exists. Having tight and consistent
networks of medical or community-based carers for those more vulnerable to COVID-19 (elderly,
pre-existing conditions) limits the transmission chain. Organisations can leverage this strategy by
structuring staggered and grouped shifts so that individuals have repeated physical contact with
a limited group rather than dispersing throughout an organisation. Panel D in Figure 2illustrates
the resulting network structure.
Strategy 2 and 3 are similar in that they build on pre-existing network structures. However,
their difference lies in the determinants of individual interaction. Strategy 2 relies on a stable and
established network structure of durable relations: who are members of my usual ‘groups’ (e.g.,
friends, family, co-workers) and which pairs of individuals among my usual contacts interacts with
each other, too? Strategy 3 relies on a strategic decision to form most convenient and effective
“interaction bubbles” and repeat contact to them over time. In this sense, strategy 2 is easier to
implement, since individuals are able to shape their contacts themselves, while strategy 3 requires
coordinated action of everyone involved in a given “bubble”. Until now, we have illustrated our
strategies with an intuitive but stylized model of epidemic spread. We now demonstrate how our
three contact strategies impact infection curves using more formal stochastic infection models that
incorporate core elements from infection models, ideal-type network models and statistical rela-
tional event models. These strategies are compared to a baseline (null) model that represents how
the COVID-19 infection would spread if there was unrestricted contact (i.e., no social distancing).
First, our model draws from classical disease modelling (Kermack and McKendrick,1927;An-
derson and May,1992), in which individuals (actors) can be in four states: susceptible, exposed
(infected but not yet infectious), infectious, and recovered (no longer susceptible to infection).
Most actors begin in the susceptible state, while qrandom actors are in the infectious state (one
per thousand in our simulations). This can represent, for example, the post-lockdown scenario in
which only a few cases of COVID-19 remain in the population; however, variation of qmight also
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be used to determine the levels at which a lock-down can be eased. During the simulation, suscep-
tible actors can transition to the exposed state by having contact with infectious others (contact
partners will be called ‘alters’ from here on). Whether contact between a susceptible actor and
an infectious alter results in contagion is determined probabilistically. A designated time after be-
coming exposed, actors become infectious themselves, and later move to the recovered state after
another fixed amount of time.
Second, as in many previous modelling efforts of the dynamics of epidemics such as influenza,
we do not assume homogeneous contact probabilities in an affected population but rather impose
a network structure that limits contact opportunities between actors (Newman,2002;Halloran
et al.,2008;Salath´e et al.,2010). This network represents the typical contact people had in a pre-
COVID-19 world. The networks we generate stochastically for our model follow fairly standard
ideal-type network generating approaches. Representing place of residence, actors are assumed
to have a geographic location, determined by coordinates in a two-dimensional space. They are
members of groups, such as households, institutions like schools or workplaces, and have individual
attributes, such as age, education, or income. Network ties are generated so that actors have some
connections to geographically close alters, some ties to members of the same groups (representing
e.g., co-workers), some ties to alters with similar attributes (e.g., similar age), and, finally, some
ties to random alters in the population. The generated networks represent the structure of alters
that an actor can possibly interact with. They represent the members of their so-called ‘social
circles’ (Watts,1999;Feld,1981;Block,2018) with whom they interact in their normal, pre-
COVID life (including family, friends, schoolmates or co-workers). The exact algorithms which
define the networks are described in the Methods section.
In the third component of the model, actors in the network interact at discrete times with alters
with which they have a connection in the underlying network, or in other words, someone they
meet from their usual social contacts. This represents the actual contact people have in their lives
during which the disease can be transmitted from infectious actors to susceptible alters. Notably, in
contrast to other modelling approaches, we do not assume that actors interact with alters in their
personal network with uniform probability (i.e. at random), but, rather, that they are purposeful
actors who make strategic choices about interactions. These strategic choices are at the core of our
advice for policy interventions, where individuals can strategically increase the efficiency of social
distancing. In our model, all choices are stochastic; strategies increase the likelihood of interacting
with specific alters but are not deterministic. The exact formulation of with whom to interact
follows a multinomial logit model to choose among possible interaction partners, given by the
network structure. This type of model has previously been used in network evolution (Snijders,
2001) and relational event models (Butts,2008;Stadtfeld and Block,2017).
Our simulations explore the three interaction strategies we propose. First, in our ‘seek similarity’
strategy, actors choose to interact predominantly with others that are similar to themselves based
on one or several specified attributes used at the network generation stage. Second, actors can adopt
our ‘strengthen triadic community’ or triadic strategy and choose to mostly interact with alters
that have common connections in the underlying network. Third, adopting our ‘repeat contact’
strategy, actors can base their choices on whom they have interacted with in their previous contacts,
both as sender and receiver of an interaction. In each case, a separate statistical parameter in the
multinomial model determines the probabilities of interaction partners based on the: (i) similarity
of alters, (ii) number of common contacts the actor and alter have; and, (iii) repeat interaction
with one of the last jcontact partners (see Methods). In our analyses, these three strategies are
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compared to a baseline case that mirrors the simple reduction of contact in which individuals have
the same amount of interactions but choose randomly amongst their network contacts (a na¨ıve
contact reduction strategy) and a null model that represents unbridled contact without any social
distancing. To make the comparison of interaction strategies independent of the arbitrary size
of statistical parameters, we empirically calibrate parameters so that the average entropy in the
probability distribution that represents the likelihood of different interaction choices is identical
for all strategies, as documented in our Methods section (Snijders,2004).
Following an initial analysis that represents a benchmark scenario of our disease model, we
present a series of variations in modelling parameters that explore alternative scenarios and ensure
our main conclusions are independent of user-defined parameters and arbitrary modelling choices.
Variations are fully described in the methods section and include: (i) different operationalisations
of homophily; (ii) the effect of employing mixed strategies; (iii) number of actors in the simulation;
(iv) varying the underlying network structure in the simulations; (v) length of the interval in which
actors are exposed relative to the time they are infectious; and (vi) the infectiousness of the virus.
2. RESULTS
The average outcome of the benchmark scenario is presented in Figure 3. The x-axis represents
time as measured in simulation steps per actor and the y-axis the number of individuals infected
at this time step out of a total population of 2,000. Curves are averaged over 40 simulation runs.
The first scenario in blue shows a null or control interaction model in which there is no social
distancing and actors interact at random. The next four strategies all employ a 50% contact
reduction relative to the null model and compare different contact reduction strategies. The black
line represents na¨ıve social distancing in which actors reduce contact in a random fashion. The
golden line represents the infection curve when actors employ our first ‘seek similarity’ strategy.
The green line models our second triadic strategy of ‘strengthening communities’ and represents
the associated infection curve. Finally, the dark red line shows how infections develop when actors
employ our third strategy of ‘repeat contact’.
All three of our strategies substantially slow the spread of the virus compared to either no
intervention or simple, un-strategic social distancing. The most effective is the strategic reduction
of interaction with repeated contacts. In comparison to the random contact reduction strategy, the
average infection curve delays the peak of infections by 37%, decreases the height of the peak by
60%, and results in 30% fewer infected individuals at the end of the simulation. This is marginally
more efficient than the triadic strategy and the homophily strategy, in this order (delay of peak
18% and 34%, decrease in peak height of 44% and 49%, and reduction of infected individuals by
2% and 19%, for homophily and triadic strategies, respectively). Note that these metrics cannot
be interpreted as general estimates of the efficiency of these strategies in real-world networks.
Summarizing the sensitivity and robustness analyses carried out, strategic contact reduction has a
substantive effect on flattening the curve compared to simple social distancing consistently across
all scenarios. However, interesting variations occur as discussed below. Full average infection curves
and results description for all model variations are presented in the Supplementary Information.
Different operationalisations of homophily
In the benchmark model, the ‘seek similarity’ strategy was employed on one demographic attribute.
However, in real-world social networks, individuals are homophilous on multiple attributes (Block
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500
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Steps of interaction (per person)
Null model: full contact
50% Contact: random reduction
50% Contact: community strategy
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Number infected (out of 2000)
Figure 3: Average infection curves. Curves compare 4 contact reduction strategies to the null model
of no social distancing. Underlying network structure includes 2000 actors and the benchmark network
characteristics described in the main text.
and Grund,2014). Furthermore, the benchmark model only uses demographic homophily, while we
previously also discuss the importance of geographic homophily. In a variation of the homophily
strategy, we show that using geographic homophily for contact reduction is highly efficient, much
more than homophily based on demographic attributes (Figure S1b.). Geographic homophily ef-
fectively eliminates contacts to distant others in the network. In a further analysis, we compare
the benefits of using one dimension of demographic homophily or a composite of two dimensions
that structure the network. This explores whether we should focus on interacting with persons
similar in one dedicated dimension or seek out others who are as similar as possible in multiple
dimensions. Encouragingly, the focus on one strategic dimension of homophily provides similar
outcomes to reducing overall demographic distance, meaning that homophily should be encour-
aged on the dimension that has the least adverse consequences for societal cohesion. Infection
curves are presented in Figure S1c.-d.
Employing mixed strategies
Since most individuals in a post-lockdown world need to interact across multiple social circles
(e.g., workplace, extended family), employing only one strategy might not be practical. A mix of
different strategies could therefore be more realistic for everyday use. We tested how four possible
combinations of mixing strategies (three two-way combinations and one three-way combination)
compare to the single strategies of seeking similarity and strengthening communities. We find that
the combined strategies are comparably as effective as single strategies (Figure S2) and can be
recommended as alternatives if single strategies are not practicable in some contexts. Importantly,
each combination performs better in limiting infection spread compared to the na¨ıve contact
reduction strategy.
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Varying the number of actors in the simulation
The computational complexity of our simulation prohibits assessing disease dynamics in very large
networks (e.g. 100k+ actors), even on large distributed systems. Nevertheless, we can compare
simulations using the same local network topology as the benchmark model on networks of 500,
1000, 2000, and 4000 actors. Reassuringly, we find no variation of the relative effectiveness of the
different interaction strategies by network size (see Figure S3). While this does not fully allow
extrapolation to very large networks, it provides initial support that disease spread under the
model could be similar within differently sized sub-regions of larger, real-world networks.
Varying the underlying network structure
The generating process of the ideal-type network that provides the opportunity structure among
individuals with whom they can interact contains multiple degrees of freedom. These include the
average number of contacts and the importance of different foci (geography, groups, and attributes)
in structuring contact. We provide infection curves for multiple scenarios in the Supplementary
Information (Figure S4 and S5), showing that our strategies work mostly independent of the
underlying structure. A first noteworthy finding from these simulations is that in networks with
fewer connection opportunities, all strategies have much larger benefits compared to networks with
more connection opportunities (panels C and D in Figure S4). In fact, the triadic strategy does
not seem to work anymore in the scenarios with very high average connectivity in the underlying
network – most likely because of a large number of closed triangles. This shows that in communities
that have lower connectivity, spread can be contained even better. As a second finding, we see
that in the case where the underlying network is not structured by homophily, the homophily
strategy does not work (panel C in Figure S5), illustrating how the strategy relies on predetermined
structural network features.
Variation in infectiousness and the length of the exposed period
Average infection curves under conditions of differences in infectiousness of the virus, and varia-
tions of the time individuals are in the state “exposed” relative to the time of being in the state
“infectious” do not influence the relative effectiveness of the different strategies and are presented
in Figures S6 and S7 respectively.
3. DISCUSSION AND CONCLUSION
In the absence of a vaccine against COVID-19, governments and organisations face economic and
social pressures to gradually and safely open up societies but lack scientific evidence on how to best
do so. We provide clear social network-based strategies to empower individuals and organisations
to adopt safer contact patterns across multiple domains by enabling individuals to differentiate
between ‘high-impact’ and ‘low-impact’ contacts. The result may also be higher compliance since
actors will hold the power to strategically adjust their interactions without being requested to fully
isolate. Instead of blanket self-isolation policies, the emphasis on similar, community-based, and
repetitive contacts is both easy to understand and implement thus making distancing measures
more palatable over longer periods of time.
How can this be applied to real-world settings? When a firm lock-down is no longer mandated
or recommended, it is likely that individuals will want or need to interact in different social
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circles, e.g. at the workplace and with the wider family. Consequently, the simple one-at-a-time
strategic recommendations we analysed in most simulations might be impossible to follow strictly
by some. Our sensitivity analysis using mixed strategies addresses this concern. For example, does
mixing the three strategies still provide benefits or do they counteract one another in their effect?
Reassuringly, a mix of strategies still provided comparable benefits to single strategies, compared
to na¨ıve contact reduction. Further modelling is needed to assess the implications in a variety of
contexts. However, when approaching this issue from a policy perspective, designing steps to ease
lockdown can be done with potential behavioural recommendations in mind: if network structures
and demographic characteristics of individuals in particular regions suggest that the use of one
strategy will yield the best results, decisions on which contact opportunities to allow – such as
opening schools or local shops – might be taken so that this strategy can be adhered to most easily.
A second discussion point concerns the potential unintended consequences of recommending our
triadic and homophilous strategies. Advocating the creation of small communities and contact to
mostly similar others can potentially result in the long-term reduction of intergroup contact and
an associated rise in inequality (DiMaggio and Garip,2012). In our simulations we explored this
concern by comparing the scenarios when homophilous ties in the underlying network are formed
following similarity in multiple dimensions, e.g. age and income. Our test of whether minimising
the overall difference in attributes of contacts versus only reducing homophily on one dimension
suggests that choosing one salient attribute can already go a long way. Thus, policymakers can make
smart choices in deciding which attribute people should pay attention to, keeping the potential
social consequences in mind. Nevertheless, understanding the long-term social consequences of
which types of public spaces are opened and, accordingly, which types of interaction are allowed
should be a major policy concern.
A number of concrete policy guidelines can be deduced from our network-based strategies. For
hospital or essential workers, risk is minimized in sustained shifts with similar composition of
employees (i.e., repeating contact) and, to distribute people into shifts based on, for example, resi-
dential proximity where possible (i.e., homophily). In workplaces and schools, staggering shifts and
lessons with different start, end and break-times by discrete organisational units and classrooms
will keep contact in small groups and reduce contact between them. When providing private or
home care to the elderly or vulnerable, the same person should visit rather than rotating or taking
turns, but that person should be the one with fewest bridging ties to other groups and who lives
the closest (geographically). Repeated social meetings of individuals of similar ages that live alone
carry a comparatively low risk. However, in a household of five, when each person interacts with
disparate sets of friends, many short cuts are being formed that are potentially connected to a
very high risk of spreading the disease.
Simple behavioural rules can go a long way in ‘keeping the curve flat’. As the pressure grows
throughout a pandemic to ease stringent lockdown measures increases to relieve social, psycho-
logical, and economic burdens, our approach provides insights to individuals, governments and
organisations about three simple strategies: interacting with similar types of people, strengthening
interaction within communities, and repeating interaction with the same people.
12
4. METHODS
Generation of stylised networks
The stylised binary networks xthat represent interaction opportunities of individuals are generated
as the composite of four sub-processes. Jointly, the sub-processes create networks that have realistic
values of local clustering, path-lengths, and homophily. All ties in the network are defined as
undirected. The number of actors in the network is denoted by n.
The first sub-process represents tie formation based on geographic proximity (Hamill and Gilbert,
2009). First, all actors in the network are randomly placed into a two-dimensional square. Second,
each actor draws the number of contacts which it forms in this subprocess dgeo,i from a uniform
distribution between dgeo,min and dgeo,max; for example, if dgeo,min = 10 and dgeo,max=20, every
actor forms a random number of ties between 10 and 20 in this sub-process. Third, the user-defined
density in geographic tie-formation ggeo defines the geographic proximity of contacts drawn, so that
actor irandomly forms dgeo,i ties among those dgeo,i
ggeo that are closed in Euclidean distance from
actor i. For example, if actor iis posed to form dgeo,i =12 ties and ggeo =0.5, the actor randomly
chooses 12 out of the 24 closest alters to form a tie to. Across all simulated networks we set
ggeo=0.3. Fourth, unilateral choices (where only iselected jbut not vice versa) are symmetrised
so that a non-directed connection exists between the actors.
The second sub-process represents tie formation in organizational foci, e.g. workplaces (ebert-
Dufresne and Althouse,2015). First, each actor is randomly assigned to a group so that all groups
have on average mmembers. Second, each actor forms ties at random to other members within the
same groups with a probability of ggroups. For example, when m=10 and ggroups=0.5, a tie from
each actor to every alter in the same group is formed with a probability of 50%. Third, unilateral
ties are symmetrised as above.
The third sub-process represents tie-formation based on homophily, for example similarity in
age or income (Pellis et al.,2020). First, each actor is assigned an individual attribute aibetween
0 and 100 with uniform probability (the scale of aicancels later in the model). Second, for each
actor, the normalized similarity simi,j to all alters jis calculated, which is one minus the absolute
difference between aiand ajfor actor j, divided by 100 (the range of the variable), so that simi,j =1
in case iand jhave the identical value of aand simi,j=0 if they are at opposite ends of the scale.
Third, each actor draws the number of contacts it forms in this subprocess dhomo,i from a uniform
distribution between dhomo,min and dhomo,max. Fourth, each actor creates dhomo,i ties to alters j
in the networks with a probability that is proportional to (simi,j )w, where higher values of wmean
that individuals prefer more similar others. Across all reported simulations, we set w=2. Fifth,
unilateral ties are symmetrised as above.
The fourth sub-process represents haphazard ties that are not captured by any of the above
processes. Here simply zties per actor are created with respect to randomly chosen alters.
Definition of simulation model
Let the binary network xrepresent the underlying social ties between nindividuals, labeled from 1
to n. Each node iis characterized by a set of attributes ak
i(such as age or location). Our model aims
to reproduce the process of individuals interacting with some of their social connections. Similar
to the classic SIR model (Kermack and McKendrick,1927) and its SEIR extension (Anderson and
May,1992), we assume that individuals can be in four different states: either susceptible to the
disease, exposed (infected but not yet infectious), infectious, or recovered. Infection occurs through
13
social interactions, which are modeled in a similar fashion to the Dynamic Actor-Oriented Model
(Salath´e et al.,2010) developed for relational events. More specifically, our model is comprised of
the following steps:
1 At each step of the process, one individual is picked at random and initiates an interaction
with probability πcontact.
2 An actor initiating an interaction can only pick one interaction partner. Only potential part-
ners as defined by the network xcan be chosen. The decision to interact is unilateral and
depends on characteristics of the two persons through a probability model p.
3 An infectious individual infects a healthy person when they interact, who then becomes
exposed. This contagion occurs with the probability πinf ection.
4 After a fixed number of steps (Texposure), an exposed individual becomes infectious.
5 After becoming infectious, recovery occurs within Trecovery steps. Once recovered, individuals
can no longer be infected.
6 The process ends once there is no longer anyone exposed or infectious.
The steps of the model are illustrated in Figure 4. One can note that the mechanics of the
infection align with previously proposed agent-based versions of the SIR and SEIR models (Chowell
et al.,2016;Siettos and Russo,2013). Together, the probabilities πcontact and πinf ection play a
similar role as the classic infectivity rate (β) in SIR models. The rate models the average number of
contacts per person (modelled here through πcontact ) and the likelihood of infection (represented
by πinfection ), however the equivalence is not direct due to the added step of the interaction
probability (p). The exposure and recovery times replace the classic exposure and recovery rates
(often traditionally denoted as σand y) in a straightforward manner. Let us turn to the definition
of the probability model p. Let Nibe the set of potential contacts, or alters, jof a given individual i
in the network x. We define for each step tof the process, Li(j, t) as the number of prior interactions
between iand an alter j, within the last Kinteractions of i. In our simulations, the number Kwas
arbitrarily set to 2 but can be easily adjusted in the replication files. For each alter jN, the value
s(i, j) represents the statistic driving the strategical choice of ito pick j. Specifically, we define
three different ways depending on whether the homophily, the triadic, or the repetition strategy
is chosen (however, arbitrary other statistics can be defined). The statistic shomophily accounts for
the level of similarity between iand jgiven a set of attributes, striadic corresponds to the number
of alters they share, and srepetition is the count of previous interactions within the last Kcontacts
of i. In practice, these statistics are calculated as:
shomophily(i, j)=1qPk(ak
iak
j)2
max
i,j (qPk(ak
iak
j)2)min
i,j (qPk(ak
iak
j)2)
(4.1)
striadic(i, j ) =
n
X
k=1
xi,kxj,k (4.2)
srepetition(i, j ) = Li(j, t) (4.3)
The probability for ito pick jis defined as a multinomial choice probability (McFadden et al.,
1973), following the logic of previous relational event (Block,2018) and stochastic network mod-
els (Halloran et al.,2008). The intuition behind this distribution is that each potential partner
in Niis assigned an objective function value, and choosing a partner is based on these values.
14
An individual iis
randomly chosen
Yes
No
Does icontact
someone?
Individual icontacts a neighbour
with probability
A neighbour jis chosen with
probability
Yes
No
Is iinfectious
and jsuceptible?
Yes
No
Is j infectious
and i suceptible?
Individual jgets exposed with
probability
Individual i gets exposed with
probability
Time step
All individuals are
susceptible
q seed nodes
become infectious
Yes
No
Is anyone expo-
sed or infectious?
Simulation
ends
Individuals who
passed
recover
Individuals who
passed
become infectious
Time step
Figure 4:Figure 4. Flowchart of the simulation model. Squares indicate updating steps to individuals or the
entire system. Diamond shapes represent decisions that determine the subsequent step in the simulation. In
the iterative part of the model, a random individual iis chosen, to initiate an interactions with probability
πcontact . In case an interaction is initiated, a contact partner jis chosen with probability p(ij) following
a multinomial choice model. If either interaction partner is infectious and the other is susceptible, contagion
occurs with probability πinfection . Subsequently, among all individuals in the simulation, those that are in
the exposed state for more than Texposure transition to infectious state and those that are in the infectious
state for more than Tinfection recover. These recursive steps are repeated until all individual are either in
the susceptible or recovered state. The colors red, green, and yellow relate closely to the steps in the SEIR
model, where red squares govern the transition from susceptible to exposed, the yellow square governs
the transitions from exposed to infectious, and the green square governs the transition from infectious
to recovered. The purple square represents the step at which individuals strategically chose interaction
partners to limit disease spread.
Mathematically, the objective function is an exponentiated linear function of the statistic s(i,j),
weighted by a parameter α. We further assume that individuals can reduce a certain percentage
of their interactions. Considering the probability (πcontact) of initiating an interaction in the first
place, the relevant probability distribution becomes:
p(ij|πcontact, α) = πcontactexp(α×s(i, j ))
PjNiexp(α×s(i, j)) (4.4)
These probabilities can be loosely interpreted in terms of log-odd ratios, similarly to logit mod-
els. Given two potential partners j1and j2for which the statistic sincreases of one unit (i.e.
s(i,j2)=s(i,j1)+1), the following log ratio simplifies to:
logp(ij2|πcontact, β )
p(ij1|πcontact, β )=α(4.5)
For example, if we use s=srepetition and αrepetition = log(2), the probability of picking one alter
present in the last contacts of iis twice as high as picking another alter who is not.
15
Calibration of model parameters
The strategy of picking a neighbor at random corresponds to the model without any statistic s, re-
ducing the probability distribution to a uniform one. For the three other strategies, the parameters
αhomophily,αtriadic, and αrepetition are adjusted to keep the models comparable. To this end, we
use the measure of explained variation for dynamic network models devised by Snijders (Snijders,
2004). This measure builds upon the Shannon entropy and can be applied to our model to assess
the degree of certainty in the choices individuals make. For a given individual iat a step t, this
measure is defined as:
rH(i, t|πcontact, α) = 1 + PjNip(ij|πcontact , α)×log2(p(ij|πcontact, α))
log2(|Ni|)(4.6)
Intuitively, this measure equals 0 in the case of the random strategy where the probability of
picking any alter is identical. It increases whenever some outcomes are favored over others and
equals 1 if one outcome has all of the probability mass. Since the model assumes all individuals are
equally likely to initiate interactions, we can average this measure over all actors. Moreover, in the
case of the repetition strategy, the measure is time dependent. In that case, we use its expected
value over the whole process. We finally use the following aggregated measure in order to evaluate
the certainty of outcomes of a specific strategy:
RH(πcontact, α) = 1
n
n
X
i=1
E[rH(i, t)] (4.7)
For this article, we first fix the parameter αrepetition at a value of 2.5, and calculate an estimated
value ˆ
RH(πcontact,αrepetition ) of this measure. This experience-based parameter choice results in
an associated RHvalue between 0.3 and 0.5 in the different scenario, which is realistic in terms
of size (see definition above). To compare this model to others, we then define the parameters
αhomophily and αtriadic that verify:
ˆ
RH(πcontact, αrepetition ) = RH(πcontact, αhomophily) = RH(πcontact , αtriadic ) (4.8)
using a standard optimisation algorithm. The average parameters across simulations for the
different network scenarios are αtriadic=0.75 and αhomophily=17.6. While that latter parameter
appears large, note that the associated statistic shomophily ranges from 0 to 1, with most realised
values close to 1.
Parametrisation of the different simulations
Unless otherwise noted, all simulations use πcontact=0.5 except for the null model, which uses
πcontact=1. In all simulations except the ones that vary the infectiousness, πinfection =0.8. Unless
otherwise noted, Texposure = 1nand Tinf ection = 4n. Given the substantial computational burden
involved in conducting the simulations, 48 repetitions were run for networks with n1000, with
40 for larger networks. Experiments varying Texposure and πinfection used 24 repetitions.
For the experiments that vary the structure of the underlying network and the network size,
the parameters that guide the stochastic network creation are presented in Table S1. Descriptive
statistics of these networks are presented in Table S2. The underlying networks that are used in
the other variation experiments are generated according to the parameters denoted ‘1: Baseline’ in
Table S1-S2. The four experiments that vary the time individuals are in the ‘exposed’ state before
16
becoming ‘infectious’ use values for Texposure of 0, 1n, 2n, 3nand 4n. The four experiments that
vary the infectiousness of the disease use values for πinf ection of 0.55, 0.65, 0.8, and 0.95.
The experiment that used geography as the basis of the homophily strategy was created ac-
cording to the ‘1: Baseline’ parameters but used the Euclidean distance in geographic placement
as the basis for choosing interaction partners in the homophily strategy. The two experiments on
multidimensional homophily used underlying networks created following the ‘1: Baseline’ param-
eters, with the exception that instead of one homophilous attribute, two attributes were defined
and the number of ties created according to the homophily parameter was split evenly between
the two dimensions. The homophily strategy used for the simulated infection curves in the two
scenarios differs in the sense that in the first, individuals interact according to minimising the
absolute difference in both attributes. In the second scenario, only the first attribute was used as
the basis of the homophily strategy and the second attribute was ignored.
For the experiments using mixed strategies, the probability of partner choice p(ij) can depend
on a vector of statistics and parameters (Stadtfeld and Block,2017). The entropy based on a set
parameter vector was used to calibrate the parameter for the homophily and triadic closure strategy
as comparison cases. Parameter choices rely on experimentation to result in similar entropy values
as when using single strategies. For the mixed strategy of repetition and homophily, the parameters
were set to αhomophily=0.7 and αrepetition =1.6. For the mixed strategy of repetition and triadic
closure, the parameters were set to αtriadic=0.35 and αrepetition =1.6. For the mixed strategy of
homophily and triadic closure, the parameters were set to αhomophily=6 and αtriadic =0.35. For
the mixed strategy incorporating all three, parameters were set to αhomophily =4, αtriadic=0.3, and
αrepetition=1.2.
The simulated average infection curves for all experiments can be found in Figures S1-S7. De-
scriptive results for the simulations in terms of delay of peak, height of peak and total number
infected at the end of the simulation are presented in Table S3. Note that the descriptive statistics
in this table present the averages of characteristics of the repetitions of the simulated infection
curves, which are not the same as the characteristics of the average infection curves as presented
in the supplementary figures.
Acknowledgements: PB, JBM, CR, RK and MCM are supported by The Leverhulme Trust,
Leverhulme Centre for Demographic Science. CR is supported by the British Academy. MCM is
supported by ERC Advanced Grant CHRONO (835079). The authors would like to thank Mark
Verhagen, Valentina Rotondi, and Liliana Andriano for feedback.
Author contributions: PB, MH, and IJR conceptualised the study; PB and MH contributed
methodology, implementation and performed analyses; PB, MH, IJR, and MCM wrote initial
manuscript and provided visualisation; all authors (PB, MH, IJR, JBD, CR, RK, and MCM) dis-
cussed research design and reviewed and edited manuscript.
Competing interests: The authors declare no competing interests.
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19
SUPPLEMENTARY INFORMATION
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person) Steps of interaction (per person)
Number infected (out of 1000) Number infected (out of 1000)
Number infected (out of 1000) Number infected (out of 1000)
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
60 0
0 10 2 0 30 4 0
0
20 0
40 0
60 0
0 10 2 0 30 4 0
0
20 0
40 0
60 0
0 10 2 0 30 4 0
c.
a.
d.
b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Repetition
strategy
Figure S1:Curves compare 4 contact reduction strategies to the null model of no social distancing, as
described in the main text. (A) Reference model with standard operationalisation of homophily; (B) model
with homophily based on geographic proximity; (C) underlying network model with homophily based on
two dimensions, interaction strategy minimises the overall difference along both attributes; (D) underlying
network model with homophily based on two dimensions, interaction strategy minimises the difference
only on the first attribute.
20
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person) Steps of interaction (per person)
Number infected (out of 1000) Number infected (out of 1000)
Number infected (out of 1000) Number infected (out of 1000)
0
20 0
40 0
60 0
800
0 10 2 0 30 4 0
0
20 0
40 0
60 0
80 0
0 10 2 0 30 4 0
0
20 0
40 0
60 0
80 0
0 10 2 0 30 4 0
0
20 0
40 0
60 0
80 0
0 10 2 0 30 4 0
c.
a.
d.
b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Mixed
strategy
Figure S2:Curves compare 4 contact reduction strategies to the null model of no social distancing, as
described in the main text. (A) Mixed strategy of repetition and triadic closure; (B) mixed strategy of
repetition and homophily; (C) mixed strategy of repetition, homophily, and triadic closure; (D) mixed
strategy of homophily and triadic closure.
21
0
10 0
20 0
30 0
0 20 4 0 60
0
20 0
40 0
60 0
0 20 4 0 60
0
50 0
10 00
15 00
0 20 4 0 60
0
10 00
20 00
30 00
0 20 4 0 60
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person) Steps of interaction (per person)
Number infected (out of 2000) Number infected (out of 500)
Number infected (out of 4000) Number infected (out of 1000)
c.
a.
d.
b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Repetition
strategy
Figure S3:Curves compare 4 contact reduction strategies to the null model of no social distancing, as
described in the main text. (A) 500 actors; (B) 1000 actors; (C) 2000 actors; (D) 4000 actors.
22
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person) Steps of interaction (per person)
Number infected (out of 1000) Number infected (out of 1000)
Number infected (out of 1000) Number infected (out of 1000)
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
60 0
80 0
0 20 4 0 60
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
60 0
0 20 4 0 60
c.
a.
d.
b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Repetition
strategy
Figure S4:Curves compare 4 contact reduction strategies to the null model of no social distancing, as
described in the main text. Names refer to the parametrisation given in Table S1. (A) baseline scenario;
(B) random network; (C) higher degree; (D) lower degree.
23
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person)
Number infected (out of 1000) Number infected (out of 1000)
Number infected (out of 1000)
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
0 20 4 0 60
c.
a. b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Repetition
strategy
Figure S5:Curves compare 4 contact reduction strategies to the null model of no social distancing, as
described in the main text. Names refer to the parametrisation given in Table S1. (A) no groups; (B) no
geography; (C) small world-ish.
24
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person) Steps of interaction (per person)
Number infected (out of 1000) Number infected (out of 1000)
Number infected (out of 1000) Number infected (out of 1000)
0
20 0
400
0 20 4 0 60
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
60 0
0 20 4 0 60
0
20 0
40 0
60 0
80 0
0 20 4 0 60
c.
a.
d.
b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Repetition
strategy
Figure S6:Curves compare 4 contact reduction strategies to the null model of no social distancing, as de-
scribed in the main text. (A) πinf ection =0.55; (B) πinf ection=0.65; (C) πinf ection =0.8; (D)πinf ection =0.95.
25
Steps of interaction (per person) Steps of interaction (per person)
Steps of interaction (per person) Steps of interaction (per person)
Number infected (out of 1000) Number infected (out of 1000)
Number infected (out of 1000) Number infected (out of 1000)
Steps of interaction (per person)
Number infected (out of 1000)
e.
0
200
400
600
800
0 25 50 75 100
0
200
400
600
0 25 50 75 100
0
200
400
600
0 25 50 75 100
0
200
400
600
0 25 50 75 100
0
200
400
600
0 25 50 75 100
c.
a.
d.
b.
Null
model
Random
reduction
Community
strategy
Similarity
strategy
Repetition
strategy
Figure S7:Curves compare 4 contact reduction strategies to the null model of no social distancing, as
described in the main text. (A) Texposed=0; (B) Texposed =1n; (C) Texposed =2n; (D) Texposed=3n; (E)
Texposed=4n.
26
Scenario nActors d geo min d geo max g geo m groups g groups d hom min d hom max w hom. z random
1: Baseline scenario 1000 4 12 0.3 8 0.9 4 12 2 0.5
2: Higher degree 1000 8 24 0.3 16 0.9 8 24 2 1
3: Lower degree 1000 2 6 0.3 4 0.9 2 6 2 0.25
4: No groups 1000 10 30 0.3 0 NA 5 15 2 0.5
5: No geography 1000 0 0 NA 20 0.9 5 15 2 0.5
6: Random net. 1000 0 0 NA 0 NA 0 0 NA 32
7: Geography 1000 15 45 0.3 0 NA 0 0 NA 0.5
8: 500 Actors 500 4 12 0.3 8 0.9 4 12 2 0.5
9: 1000 Actors 1000 4 12 0.3 8 0.9 4 12 2 0.5
10: 2000 Actors 2000 4 12 0.3 8 0.9 4 12 2 0.5
11: 4000 Actors 4000 4 12 0.3 8 0.9 4 12 2 0.5
Table S1:Parameters used in the stochastic generation of underlying networks. Full description of procedure is described in the Methods section in the main text.
27
Scenario n deg. clus. av. path dia. hom.
1: Baseline scenario 1000 38.4 0.11 2.23 3 1.08
2: Higher degree 1000 75.9 0.14 1.93 3 1.08
3: Lower degree 1000 19.4 0.09 2.69 4 1.08
4: No groups 1000 55.4 0.16 2.07 3 1.07
5: No geography 1000 40.2 0.26 2.24 3 1.09
6: Random net. 1000 62 0.06 1.96 3 1
7: Small world-ish 1000 53.9 0.3 2.57 4 1
8: 500 Actors 500 38.11 0.14 2 3 1.08
9: 1000 Actors 1000 38.4 0.11 2.23 3 1.08
10: 2000 Actors 2000 38.72 0.09 2.49 3.4 1.08
11: 4000 Actors 4000 38.85 0.08 2.7 4 1.08
Table S2: Characteristics of the networks created under different scenarios. Descriptive statis-
tics are averaged over 40-48 simulations. Notes: n: number of actors; deg.: average degree / number of
connections per actor; clus.: clustering coefficient / proportion of closed triads over possibly closed triads;
av. Path: average network distance between pairs of nodes; dia.: diameter / maximum distance in between
nodes in the network; hom.: average similarity of interaction partners divided by average similarity among
all actors.
28
Variation: network structure
Scenario Strategy Delay Peak Inf.
1: Baseline scenario Random 1.30 0.50 75%
Triads 1.66 0.35 73%
Homophily 1.50 0.23 71%
Repetition 1.59 0.24 57%
2: Higher degree Random 1.25 0.48 71%
Triads 1.49 0.48 79%
Homophily 1.52 0.44 80%
Repetition 1.24 0.31 56%
3: Lower degree Random 1.06 0.40 62%
Triads 1.85 0.14 51%
Homophily 1.88 0.12 52%
Repetition 2.00 0.06 24%
4: No groups Random 1.34 0.52 77%
Triads 1.68 0.29 72%
Homophily 1.46 0.38 77%
Repetition 1.24 0.27 54%
5: No geography Random 1.12 0.41 63%
Triads 0.71 0.01 2%
Homophily 1.38 0.22 65%
Repetition 0.99 0.14 32%
6: Random net. Random 1.46 0.59 90%
Triads 1.38 0.45 72%
Homophily 1.39 0.41 73%
Repetition 1.46 0.32 63%
7: Small world-ish Random 1.11 0.43 65%
Triads 1.61 0.25 66%
Homophily 1.44 0.45 74%
Repetition 1.30 0.29 56%
Variation: network size
8: 500 Actors Random 1.59 0.62 94%
Triads 1.83 0.41 81%
Homophily 2.09 0.33 93%
Repetition 1.87 0.34 75%
9: 1000 Actors Random 1.30 0.50 75%
Triads 1.66 0.35 73%
Homophily 1.50 0.23 71%
Repetition 1.59 0.24 57%
10: 2000 Actors Random 1.30 0.49 74%
Triads 1.49 0.28 60%
Homophily 1.60 0.26 73%
Repetition 1.47 0.21 52%
11: 4000 Actors Random 1.63 0.63 95%
Triads 2.20 0.42 93%
Homophily 1.90 0.36 94%
Repetition 2.08 0.29 71%
Variation: infectiousness
12: pinfect =0.55 Random 1.11 0.32 57%
Triads 0.96 0.10 29%
Homophily 1.25 0.16 49%
Repetition 1.25 0.07 22%
13: pinfect =0.65 Random 1.29 0.42 68%
Triads 1.34 0.19 47%
Homophily 1.21 0.16 48%
Repetition 1.65 0.17 46%
14: pinfect =0.8 Random 1.30 0.50 75%
Triads 1.66 0.35 73%
Homophily 1.50 0.23 71%
Repetition 1.59 0.24 57%
15: pinfect =0.95 Random 1.32 0.57 80%
Triads 1.44 0.33 62%
Homophily 1.33 0.24 63%
Repetition 1.48 0.27 57%
Variation: Texposure
Scenario Strategy Delay Peak Inf.
16: Texposure = 0 Random 1.44 0.50 74%
Triads 1.69 0.33 65%
Homophily 1.48 0.27 63%
Repetition 1.52 0.21 46%
17: Texposure = 1 Random 1.30 0.50 75%
Triads 1.66 0.35 73%
Homophily 1.50 0.23 71%
Repetition 1.59 0.24 57%
18: Texposure = 2 Random 1.40 0.54 84%
Triads 1.20 0.26 56%
Homophily 1.54 0.26 74%
Repetition 1.24 0.19 47%
19: Texposure = 3 Random 1.22 0.47 73%
Triads 1.58 0.32 68%
Homophily 1.24 0.21 62%
Repetition 1.49 0.23 57%
20: Texposure = 4 Random 1.11 0.43 68%
Triads 1.85 0.34 78%
Homophily 1.50 0.23 73%
Repetition 1.11 0.19 44%
Variation: mixed strategies
21: Homo. + Triad. Random 1.52 0.69 97%
Triads 1.86 0.55 96%
Homophily 1.73 0.49 97%
Mixed strat. 1.75 0.58 96%
22: Homo. + Rep. Random 1.39 0.60 84%
Triads 1.51 0.40 70%
Homophily 1.60 0.37 82%
Mixed strat. 1.52 0.36 68%
23: Rep. + Triad Random 1.53 0.63 89%
Triads 1.47 0.34 64%
Homophily 1.67 0.31 80%
Mixed strat. 1.92 0.35 74%
24: Ho. + Re. + Tr. Random 1.53 0.63 88%
Triads 1.51 0.39 70%
Homophily 1.43 0.32 74%
Mixed strat. 1.39 0.34 64%
Variation: operationalisation homophily
25: Normal homo. Random 1.30 0.50 75%
Triads 1.66 0.35 73%
Homophily 1.50 0.23 71%
Repetition 1.59 0.24 57%
26: Geogr. homo. Random 1.20 0.46 69%
Triads 1.40 0.30 62%
Homophily 2.19 0.11 50%
Repetition 1.55 0.23 54%
27: 2-Dim. abs. diff. Random 1.31 0.50 75%
Triads 1.47 0.25 61%
Homophily 1.00 0.31 53%
Repetition 1.22 0.26 52%
28: 2-Dim. only 1st Random 1.32 0.52 78%
Triads 1.63 0.27 67%
Homophily 1.33 0.36 68%
Repetition 1.19 0.25 50%
Table S3: Characteristics of average infection curves for
different strategies. All entries denoting averaged results of
simulations are relative to the null model of no contact reduction
(blue line in Fig. 3). Delay: delay of the peak of the infection
curve compared to the null model; Peak: height of the peak of
the infection curve compared to the null model; Inf.: proportion
of the population infected compared to the null model.
Table S3
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