Lockdown Was Heißt Das

Social distancing und isolation schutz been widely introduced kommen sie counter the covid19 pandemic. Adverse social, psychological and economic consequences of a complete or near-complete lockdown demand ns development of more center contact-reduction policies. Adopting a social network approach, we evaluate die effectiveness von three distancing tactics designed zu keep die curve flat und aid compliance in a post-lockdown world. These are: limiting interaction kommen sie a few repeated contact akin zu forming social bubbles; seek similarity across contacts; and strengthening communities via triadic strategies. Us simulate stochastic infection curves incorporating core elements from epidemic models, ideal-type social network models und statistical relational occasion models. We demonstrate that a strategic society network-based reduction des contact strong enhances the effectiveness of social distancing procedures while keeping dangers lower. We carry out scientific evidence zum effective social distancing that can be applied an public health messaging und that kann mitigate negative consequences of social isolation.

Du schaust: Lockdown was heißt das


The non-pharmaceutical intervention of social distancing is a key policy to reduce the spread von COVID-19 über maintaining physics distance und reducing society interactions1. Ns aim is to slow-moving transmission and the expansion rate von infections to avoid overburdening health care systems—an strategy widely known as flattening the curve2. Typical social distancing procedures are bans on windy events, the closure of schools, universities and non-essential workplaces, limiting public transportation, travel und movement restrictions, and limiting physics interactions.

Social distancing interventions during previous outbreaks (for example, during SARS-CoV (severe acute respiratory syndrome coronavirus) bei 2003) schutz often been based on experte recommendations rather than scientific evidence3. Existing research has actually mostly evaluated take trip restrictions, college closures or vaccines4,5. Cancelling public gatherings and imposing travel constraints decreases transmission and morbidity rates6, with mixed proof on the efficacy des school closures7. Basically no study exists ~ above strategies based on individuals’ knowledge von their social surroundings, however interventions room only effective when ns public deems castle acceptable8. Few schutz considered society networks, or if lock did that was in relation zu vaccinations9, call tracing or analysing the spread von the virus8,10.

Since most facets of economic and social life need person-to-person contact, strategically reducing contacts ist favourable to complete isolation. Boosting contact kann sein likewise counter negative social, psychological und economic costs des quarantining individuals over an extensive periods von time and avoid compliance fatigue11. Zu achieve this aim, us propose behavioral network-based strategies zum selective contact reduction the every individual and organization kann easily understand, control und adopt. Using insights indigenous social and statistical network science, we show how changing network configurations von individuals’ contact choices und organizational routines can alter the rate und spread des the virus von providing guidelines zu differentiate betwee high- and low-impact contacts weil das disease spread. Us introduce und assess three methods (contact with comparable people; strengthening contact in communities; und repeatedly communicating with die same people in bubbles) that depend on less confinement and allow strategic social call while ausblüten flattening die curve. Our method balances windy health involves with social, psychological und economic needs for interpersonal interaction.

Flattening the (infection) curve operates to decrease the number von infected people at the height von the epidemic, über distributing the incidence von cases end a longer time horizon2. This zu sein largely achieved über reducing ns reproduction number (R), which represents how plenty of individuals space infected by each carrier. Social distancing plans are implicitly designed kommen sie achieve this über limiting the amount von social contact between individuals. über introducing a society network approach, us propose that a decrease in R can simultaneously be achieved by managing ns network structure of interpersonal contact.

From a society network perspective, die shape of the infection curve zu sein closely connected to die concept des network street (or course lengths)12, which indicates the number of network measures needed kommen sie connect 2 nodes. Popularized examples des network distance incorporate the sechs degrees des separation phenomenon13, which cases that any kind of two world are associated through hinweisen most five acquaintances.

The relationship betwee infection curves and network distance tun können be depicted with a straightforward network infection modell (Fig. 1). Figure 1a,c depicts 2 networks with various path lengths, each with one hypothetically infected covid-19 seed node (purple square). At each time step, ns disease spreads from infected nodes kommen sie every node kommen sie which they room connected; thus, in the zuerst step, ns disease spreads from the seed node to its straight neighbours. An the 2nd step, the spreads to their neighbours, that are punkt network street 2 from ns seed node, and so on. End time, die virus moves along network ties until all nodes space infected. Die example reflects that the network distance von a node from ns infection source (indicated by node colour bei Fig. 1a,c) zu sein identical to the number des time procedures until die virus will it. The distribution of network ranges to the source thus directly maps onto ns curve of new infections (Fig. 1b,d).


Fig. 1: Two instance networks.

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ad, Two example networks (a und c) schutz the same number of nodes (individuals) und ties (social interactions) yet different frameworks (shorter course lengths bei a and longer course lengths an c), i m sorry imply various infection curve (b und d, respectively). Bold ties highlight the shortest infection course from die 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 the colours of the histogram bars.


In ours example, both networks oase the exact same number von nodes (individuals) and edges (interactions); however, ns network depicted an Fig. 1c has a much flatter epidemic curve than ns network depicted in Fig. 1a, also though all nodes are at some point infected bei both cases. This is because the latter network has longer route lengths than ns former one. An other words, there zu sein more network distance bolzen the people due zu a different structure von interaction, despite die same absolute call prevalence. When adopting a network perspective, flattening die curve zu sein thus equivalent to increasing the path size from an infected individual zu all others, which tun können be achieved über restructuring contact (besides ns general reduction of contact). Consequently, one aim of social distancing must be increasing the average network distance bolzen individuals von smartly und strategically manipulating ns structure von interactions. Ours illustration reflects a viable path kommen sie keep the covid19 curve level while permitting some society interaction: we need to devise communication strategies the make real-life networks look much more like the network in Fig. 1c, and less like die network in Fig. 1a.

We propose a series des strategies on how individuals tun können make neighborhood decisions to achieve this goal. Expertise which types des strategies des targeted contact reduction und social distancing are much more efficient bei increasing path lengths and flattening ns curve kann sein inform how kommen sie shift from short-term (complete lockdown) kommen sie long-term monitoring of covid19 contagion processes. Die contact reduction methods we suggest are based on insights into exactly how items circulation through networks, such as diseases, memes, die info or ideas14,15,16,17. Together spread zu sein generally hampered wie man networks consist des densely associated groups with few connections in-between (such as individuals who live in isolated towns scattered over sparse rural areas18). In contrast, contacts that bridge large distances room related zu short paths und rapid spread. Weil das instance, when commuters travel betwee these diverted villages, network distances decrease substantially14,18. Using this knowledge, we tun können avoid rapid contagion über encouraging social distancing tactics that boost clustering and reduce network shortcuts to reap the largest benefit of reducing social contact and limiting condition spread to a minimum. Us propose three strategies aimed hinweisen increasing network clustering and eliminating shortcuts.

We outline die principles of the suggest strategies in Fig. 2. Number 2a depicts a network an which densely linked communities are bridged über random, long-range ties. This type von network represents main point features of real-world call networks14 und is commonly known as a small-world network18. In ~ communities, people are similar to each various other (as indicated von their node colour) and adjacent communities are geographically nearby (as indicated von node location). Die further away two clusters are bei the figure, die further they live from each other and the an ext dissimilar your members are. Number 2a–d illustrates the successive, targeted contact reduction strategies, while the bar graph depicts die distribution von distances von all people from one of the two highlighted infection sources.


Fig. 2: example networks that result from the successive tie-reduction strategies.

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ad, Based on an initial small-world network (a), instance networks room mapped based on removing ties to dissimilar rather who live far far (b), remove non-embedded ties the are not part von triads or four-cycles (c) or repeating fairly than extending contact (d). Node colour represents in individual characteristic, whereby similarity bei node colour represents similarity an this characteristic. Node placement represents geographic location of residence. Ties kommen sie dissimilar others who direkt far away room indicated von ties substantially longer than the average (that is, zu nodes that are put distantly and have an extremely different colours). e, gittern graph reflecting network ranges from die infection sources (highlighted in yellow an ad) zum the different scenarios.


In the first strategy (seek similarity; to compare Fig. 2a und Fig. 2b), individuals pick their call partners based upon similarity des a predetermined individual characteristic19,20,21, such as those who direkte geographically close (spatial similarity), are members of the same organizations (for example, department punkt work) or are similar on continuous, highly variable demography characteristics, such as age. Restructuring contact in this means reduces network bridges to groups von geographically remote others22 und to those through whom no organization or characteristic ist shared; this contains ns disease bei localized areas von the network. A pre-requisite zum this strategy ist that world seek similarity ~ above a dimension that facilitates forming numerous comparatively small groups (for example, based an neighbourhoods or small organizations). Segregation of large demography groups, such as ethnic or racial segregation would not provide any measurable benefit. Further einzelheiten are discussed in Box 1.

For die second strategy (strengthen communities; to compare Fig. 2b und Fig. 2c), individuals must consider with who their call partners generally interact. Wie man reducing contact, one have to prioritize removed ties not embedded an triangles (a triangle is a network configuration von individuals i, j and h bei which all three space mutually connected23,24). Thus, world should interact less through others who are not bei contact with their other usual call partners. For example, 2 friends should only meet if they have many other friends an common. Maintaining contact bei cohesive neighborhoods characterized über triangles can contain virologe spread an local regions des the networks, rather than permitting it to spread zu distant communities via network bridges25. This strategy ist elaborated an Box 2.

For ns third strategie (build ballon through recurring contact; compare Fig. 2c und Fig. 2d), individuals must decide with whom they frequently want zu interact and, end time, restrict interaction zu those people. This reduces die number des contact partners rather than ns number of interactions. This strategy von limiting contact zu very few others with recurring interactions is an the spirit des a social contract with others kommen sie create social blase allowing just interactions within die same gruppe delineated von common agreement. Similarly, employers can create had departmental or work unit bubbles of employees. These micro-communities space difficult zum a virus zu penetrate and—importantly—if die infection zu sein contracted von one contact, it is difficult zum the virus to spread viel further. Details of ns strategy und comparisons with strategy 2 are presented in Box 3.

We now demonstrate how these three call strategies impact infection curves making use of formal stochastic infection models that incorporate core elements from infection models, ideal-type network models und statistical relational occasion models. First, our modell draws native classical condition modelling26,27 in which individuals (actors) kann be bei four states: susceptible; exposed (infected but not yet infectious); infectious; or recovered (no much longer susceptible). Weist the anfang of ns simulation, q actors are infectious while every others are susceptible. Prone actors tun können become exposed by having contact with infectious others; whether this call results in contagion is determined probabilistically. A designated amount des time after coming to be exposed, actors end up being infectious, und later relocate to die recovered state.

Second, as in many previous models of die dynamics of epidemics, call probabilities an the population are imposed von a network framework that borders contact opportunities between actors28,29,30. This network represents ns typical contact people had in a pre-COVID-19 world an different so-called social circles19,20,31. The consists of network ties betwee individuals who direkte geographically close, individuals who are comparable on separation, personal, instance attributes, such as age, education or income, and individuals who space members of common groups, such together households und institutions (including schools und workplaces). Additionally, the network has random connections in the population.

In die third component of the model, actors interact punkt discrete zeit with rather from their mitarbeiter network. During these meetings, die disease can be spread out from infectious actors zu susceptible alters. Notably, in contrast with other modelling approaches, actors do notfall interact v alters bei their personal network with uniform probability (that is, at random). Rather, they room purposeful actors who make strategic choices about interaction partners. Selections are identified stochastically; methods increase die likelihood of interacting with particular alters but are not deterministic. The mathematical formulation the determines contact an option follows previously approaches used bei network evolution32 und relational event models33,34. A flowchart von the model ist presented bei Fig. 4.


Fig. 3: Flowchart des the simulation model.

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Squares indicate updating steps zu individuals or die entire system. Diamond shapes stand for decisions that determine ns subsequent step an the simulation. Bei the repeat part of the model, a arbitrarily individual ich is chosen to initiate interactions v probability πcontact. When in interaction is initiated, a contact partner j is chosen through probability (pleft( i o j ight)) complying with a multinomial an option model. If either interaction partner zu sein infectious und the other zu sein susceptible, contagion occurs through probability πinfection. Subsequently, amongst all individuals bei the simulation, those who are in the exposed state zum more 보다 Texposure transition to the infectious state and those that are bei the contagious state zum more than Tinfection recover. These recursive steps are repeated until all individuals are either in the at risk or recovered state. Ns colours red, green and yellow relate closely to the steps bei the SEIR model, where red squares govern die transition native susceptible kommen sie exposed, ns yellow square governs the transition native exposed zu infectious, und the eco-friendly square governs the transition indigenous infectious kommen sie recovered. Ns purple square represents the step hinweisen which people strategically choose interaction partners to limit an illness spread.


Our simulations explore ns three interaction strategies we propose. First, in our look for similarity strategy, gibbs choose zu interact primarily with others who room similar zu themselves based upon one or number of specified attributes. Second, actors can adopt ours strengthen community strategy and choose zu interact mostly with transforms who oase common connections bei the underlying network. Third, adopting ours repeat-contact balloon strategy, actors kann base their selections on who they have interacted with out des their vault contacts, both together senders and receives des interactions (see Methods). An our analyses, these three tactics are compared with a baseline instance that winter a naive contact reduction strategy (in which people reduce interaction however choose randomly amongst their network contacts) und a null modell that represents unbridled contact without any type of distancing. Kommen sie make die interaction strategies comparable, we empirically calibrate statistical modell parameters deswegen that die average entropy an the probability distribution that represents die likelihood of different communication choices zu sein identical zum all tactics (see Methods)35.

Following in initial evaluation that to represent a benchmark scenario des our disease model, we present a series von variations bei modelling parameters that explore alternate scenarios and provide robustness checks. The benchmark scenario is conducted with 2,000 actors, und the variations und robustness analyses are conducted with 1,000 actors, uneven otherwise specified.


Strategy 1


In the erste strategy, individuals pick their call partners based upon their individual characteristics. Generally, individuals tend to oase contact with others that share common attributes, such together those in the exact same neighbourhood (geographical) or those von similar revenue or period (socio-demographic)19,20,21. Ns tendency kommen sie interact with comparable others ist called homophily in the sociological network literature20 und is a ubiquitous und well-established feature des social networks (thus, us use the terms ‘seek similarity’ and ‘homophily’ interchangeably). Because we are greatly connected to similar others, contact with dissimilar people tends to bridge kommen sie more remote communities. Restricting one’s contact deshalb that us only kommen sie into call with those most comparable helps kommen sie limit network bridges the substantially reduce network route lengths. This involves choosing to interact v those who space geographically proximate (for example, living an the exact same neighbourhood) or those with similar characteristics (for example, age). Number 2b shows the network framework after ns implementation of this strategy of tie reduction. The damit verbundenen bar graph illustrates that following this network-based intervention, a comprehensive number von nodes are weist a larger distance from die infection source. This strategie will be successful when ns characteristic or variable the determines ns communities kann take on a variety von different (categorical or continuous) values for different individuals, thereby cultivating the bildete of small communities. A more comprehensive split, such together along gender or country lines, does not promise measurable success but möchte instead probably exacerbate the negativ consequences of distancing measures.

This strategy ist supported von epidemiological modelling, which suggests that co-residence und mixing des individuals from different ages (for example, inter-generational households) strongly increases die spread von infectious disease, such as covid19 (ref. 22). Offering a concrete example, if world only interact with others an a three-block radius (increase geographic similarity), much more than 30 transmission events would it is in necessary zum a virus to travel 100 blocks. Workplaces where countless individuals kommen sie together could, weil das instance, implement routines zu decrease contact between groups from various geographic areas or age groups.


For die second strategy, individuals must think about with who their contact partners usually interact. A common feature des contact networks zu sein triadic closure, referring to ns fact that contact partners of in individual tend zu be linked themselves19,23,24. Tie embedding in triads ist a specifically useful topology zum containing epidemic outbreaks. Consider a closed triad of individuals i, j and h. When i infects j and h, the connection bolzen j und h does not contribute to further an illness spread; in other words, it is a redundancy contact25. Wie comparing networks with bei identical number of connections, networks with an ext redundant ties often tend to oase longer route lengths. Accordingly, when removing contact with others, one need to prioritize removed ties not embedded bei triads, because these ties typically decrease course lengths. An practice, this method that physical contact should be curtailed with human being who are not also connected zu one’s usual other social contacts. Figure 2c illustrates the structure if ties that are not part of closed triads or four-cycles room removed. In this ideal-type example, this eingriff not only further reduces die network distance des many nodes from ns infection sources, but so creates isolated areas that cannot be infected von the virus.


For ns third strategy, individuals need to selectively take into consideration who castle want zu regularly interact with and, over time, restrict interaction to those people. This reduces the number von contact partners fairly than the number of interactions, which ist particularly important wie man contact is necessary zum psychological well-being. Although this needs coordination, it would be difficult zum a virus zu penetrate micro-communities and—importantly—if die infection were to be contracted über one contact, it would certainly be difficult zum the virus kommen sie spread much further. An additional implication von this strategie includes die repetition von interaction with others that overlap across much more than one contact group. For example, conference co-workers outside von work zum socializing will schutz less of bei impact on the virus spread out relative to a separate kopieren, gruppe of friends, due to the fact that a potential infection path already exists. Having tight und consistent networks des medical or community-based carers weil das those much more vulnerable to covid19 (the elderly und people with pre-existing conditions) limits die transmission chain. Organizations kann sein leverage this strategy von structuring staggered und grouped shifts so that individuals have repeated physical contact with a limited group rather 보다 dispersing throughout bei organization. Number 2d illustrates die resulting network structure.

Strategies 2 and 3 are similar bei that they construct on pre-existing network structures. However, their difference lies bei the determinants von individual interaction. Strategie 2 counts on a stable and established network structure des durable relations. People need kommen sie consider which people are members von their usual teams (for example, friends, family und co-workers) and which pairs von individuals amongst their normal contacts communicate with one another. Strategie 3 relies on a strategy decision to form the many convenient and effective bubbles und restrict contact to within this bubble end time. An this sense, strategy 2 is easier zu implement, due to the fact that individuals room able zu shape your contacts themselves, while strategy 3 calls for coordinated action des everyone involved bei a offered bubble.


The average outcome von the benchmark scenario is presented an Fig. 4. Ns x axis represents time (as measured in simulation steps von actor) und the y axis shows the number of individuals infected punkt this time step out des a total population of 2,000. Curves room averaged over 40 simulation runs. The erste scenario in blue shows a null or manage interaction model in which there zu sein no social distancing und actors interact at random. Die other 4 strategies every employ a 50% contact reduction relative zu the null model und compare different contact reduction strategies. The schwarze farbe line represents naive social distancing bei which actors minimize contact in a arbitrarily fashion. The golden line represents die infection curve when actors rental our erste strategy (that is, look for similarity). The green line models our second triadic strategy of strengthening communities and represents the relevant infection curve. Finally, the dark red heat shows just how infections develop when actors rental our third strategy von repeating contact in bubbles.


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All three von our strategies dramatically slow the spread of the virus compared with either no intervention or simple, non-strategic social distancing. The most efficient approach zu sein the strategy reduction of interaction with recurring contacts. Compared with the random contact reduction strategy, the average epidemic curve delays die peak des infections über 37%, decreases ns height of the peak by 60% and results in 30% under infected individuals punkt the ende of ns simulation. This zu sein marginally more efficient than ns strengthening community strategy and the search similarity strategy, in this stimulate (respective values: delay of peak: 34 and 18%, decrease in peak height: 49 and 44%; reduction des infected individuals: 19 and 2%). Klasse that this metrics cannot be taken as basic estimates von the efficiency von these strategies in real-world networks.

Summarizing die sensitivity and robustness analyses presented below, strategic contact reduction has actually a substantive result on flattening die curve contrasted with basic social distancing consistently across all scenarios. However, exciting variations occur. Full average infection curves and a description von the results zum all model variations are presented an Extended dünn Figs. 1–7 und Supplementary Information.

Different operationalizations des homophily

In ns benchmark model, the seek similarity strategie was to work on one demography attribute. However, in real-world social networks, people are homophilous on lot of characteristics36. Furthermore, ns benchmark model only provides demographic homophily, while us previously deshalb discussed die importance von geographic homophily. In a variation des the look for similarity strategy, we nur that using geographical homophily for contact reduction zu sein highly efficient—much more deshalb than homophily based upon demographic attributes (Extended säule Fig. 1b). Geographic homophily or similarity properly eliminates contacts with far-off others an the network. In a additional analysis, we compare die benefits des using one dimension of demographic homophily or a composite von two dimensions the structure die network. This explores whether us should focus on connecting with persons similar in one committed dimension or seek out others who room similar in multiple dimensions simultaneously. Encouragingly, die focus on one strategic dimension of homophily provides similar outcomes to reducing demographic street on both dimensions. An our minimal example, this way that homophily kann be encouraged only on die dimension that has lesser disadvantage consequences zum societal cohesion, together opposed kommen sie reduction ~ above both dimensions. Infection curves are presented an Extended charme Fig. 1c,d.

Employing mixed strategies

Since many individuals in a post-lockdown welt need to interact throughout multiple social circles (for example, workplace, expanded family and so on), employing just one strategie might notfall be practical. A mix des different techniques could therefore be much more realistic zum everyday use. We tested how four possible combinations of mixing strategies (three two-way combinations und one three-way combination) compare v the single strategies von seeking similarity und strengthening communities. We discovered that ns combined strategies are comparably as efficient as einzel strategies (see Extended charme Fig. 2) and can be recommended as choices if single strategies are notfall practicable in some contexts. Importantly, each mix performs better an limiting epidemic spread compared with die naive call reduction strategy.

Varying the number von actors bei the simulation

The computational complexity von our simulation prohibits assessing an illness dynamics an very huge networks (for example, 100,000+ actors), even on big distributed systems. Nevertheless, we can compare simulations using ns same neighborhood network topology as the benchmark modell on networks of 500, 1,000, 2,000 und 4,000 actors. Reassuringly, we discover no variation des the loved one effectiveness of the various interaction strategies über network size (see Extended säule Fig. 3). If this does notfall fully permit extrapolation kommen sie very huge networks, the provides initial support that condition spread under the modell could be similar within in different ways sized sub-regions von larger, real-world networks.

Varying ns underlying network structure

The generation process von the ideal-type network that provides ns opportunity structure among individuals through whom they can interact has multiple degrees des freedom. These include ns average number of contacts and the importance des different foci (geography, groups and attributes) an structuring contact. We provide infection curves weil das multiple scenarios an Extended dünn Figs. 4 and 5, mirroring that ours strategies work largely independent des the basic structure. A zuerst noteworthy recognize from this simulations zu sein that in networks v fewer connection opportunities, every strategies oase much bigger benefits compared with networks with an ext connection avenues (Extended charme Fig. 4c,d). Bei fact, die strengthening community strategie does not seem kommen sie work anymore in scenarios with really high median connectivity bei the basic network—probably because des a large number of closed triangles. This mirrors that in communities that schutz lower connectivity, spread can be included even more effectively. As a second finding, we see that an cases where the underlying network is not structured von homophily, the seeking similarity strategie does not work (Extended charme Fig. 5c), showing how the strategie relies ~ above predetermined structure network features.

Variation bei infectiousness and the length of the exposed period

Differences in infectiousness des the virus, and variations von the time throughout which individuals are an the exposed state relative to ns infectious zustand do notfall influence ns relative effectiveness von the different strategies, and average infection curves are presented bei Extended data Figs. 6 and 7, respectively.


In ns absence of a vaccine against COVID-19, governments und organizations challenge economic and social pressures zu gradually and safely offen up societies, however they absence scientific evidence on how kommen sie do this. We provide clear society network-based strategies zu empower individuals and organizations zu adopt safer call patterns throughout multiple domains von enabling individuals zu differentiate betwee high- und low-impact contacts. The result may deshalb be higher compliance due to the fact that it empowers individuals kommen sie strategically adjust and control their very own interactions without gift requested to fully isolate. Instead of blanket self-isolation policies, die emphasis on similar, community-based and repetitive contacts zu sein easy kommen sie understand and implement, for this reason making distancing measures more palatable over much longer periods of time.

How tun können this it is in applied to real-world settings? when a firm lockdown is no much longer mandated or recommended, individuals möchte want or need to interact in different society circles (for example, at die workplace or with broader family). Bei some von these settings, search similarity might notfall be feasible (for example, an schools in which teachers und students of different ages come together). Consequently, the simple one-at-a-time strategic recommendations we analysed in most simulations could be impossible to streng follow weil das some. Our sensitivity analysis using mixed strategies addresses this concern. Zum example, walk mixing the three strategies blieb provide benefits or execute they counteract one another? Reassuringly, our results show that a mix von strategies blieb provide similar benefits to einzel strategies, and all arbeit considerably much better than simply releasing a floodgate von full non-strategic contact; however, more modelling is needed kommen sie assess die implications throughout a variety of contexts. Wie man approaching this problem from a plan perspective, the design of steps kommen sie ease lockdown tun können be done v potential behavioral recommendations bei mind: if network structures und demographic characteristics des individuals in particular regions indicate that die use of one strategie will yield ns best results, decision on which call opportunities to allow (such as opening institutions or local shops) might be taken deshalb that this strategy can be adhered to most easily.

A second discussion suggest concerns potential unintentional consequences von recommending our strengthening community and seeking similarity strategies. Our analyses and reasoning clearly should not be used to justify any form of racial or social group segregation or comparable vulgar ideas. Beyond the obvious ethical und social consequences, segregation right into such large groups would not be effective bei curbing the spread of the virus, because strategic contact reduction counts on limiting contact zu many klein connected network regions notfall splitting into big groups. We recognize that advocating the creation des small communities and contact with mostly comparable others on part dimensions could potentially result in the permanent reduction des intergroup call and an associated rise in inequality37. An our simulations, we explored this concern by comparing the scenarios wie man homophilous ties an the basic network are formed complying with similarity in multiple size (for example, age and income). Our test von whether minimizing the overall difference an the two modelled attributes des contacts versus just reducing homophily top top one dimension argues that choosing one salient attribute can already be an extremely effective. This findings carry out preliminary proof that policymakers can make clever choices relevant to their local context in deciding i m sorry attribute world should pay attention to, keeping the potential social consequences in mind. Nevertheless, combine similarity on 2 simulated individual-level qualities into a single indicator is blieb very likely kommen sie understate the complexity of how multiple separation, personal, instance traits intersect, und structure social interaction. Our conclusions about die intersectionality von multiple separation, personal, instance traits zum disease spread continue to be tentative. This prominent that understanding die long-term society consequences von which types des public spaces are opened and, accordingly, i m sorry types des interaction are enabled requires an ext research und should it is in a cook concern bei policy-making. Acquisition all von these considerations right into consideration, weil das the moment, ours simulations that explore ns effect des increasing geographic proximity und the theoretical appeal des seeking similarity top top residential ar would make geographical similarity ns preferred dimension when giving guidance to policymakers.

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Third, a shortcoming of our simulation study zu sein the limited number von network actors. While we varied die number of nodes from 500 to 4,000 und found no an extensive difference an the results, us do notfall know ns dynamics of the model in large networks of, for example, 100,000+ actors. An the existing implementation des the model, die computational complexity increases an ext than linearly with ns number des actors, which renders simulations through such numbers unrealistic. Consequently, algorithmic arbeit on the modell implementation is needed zu extend that is applicability zu large, real-world networks, supplying clear extensions for future research.

Despite this limitations, part concrete policy guidelines kann sein be deduced from ours network-based strategies. For hospital or vital workers, risk tun können be minimized über introducing shifts v a similar composition von employees (that is, repeating contact and creating bubbles) und distributing people into move based on, for example, residential proximity where feasible (that is, seeking similarity). In workplaces und schools, staggering shifts and lessons with different start, end und break times von discrete organizational units und classrooms wollen keep contact bei small groups und reduce contact bolzen them. Wie man providing private or home treatment to ns elderly or vulnerable, the same person should visit fairly than rotating or taking turns, und that person should be ns one through fewest bridging ties to other groups and who lives the closest (geographically). Recurring social meetings of individuals des similar eras who live alone lug a comparatively low risk. However, bei a household von five, wie each personen interacts v disparate sets von friends, plenty of shortcuts space being developed that are possibly connected to a an extremely high risk von spreading ns disease.

In summary, simple behavioural rules kann sein go a lang way an keeping the curve flat. As die pressure rises throughout a pandemic zu ease stringent lockdown measures, to relieve social, psychological and economic burdens, our approach provides insights to individuals, governments und organizations around three straightforward strategies: seeking similarity; strengthening interactions in ~ communities; and repeated interaction with die same people to create bubbles.


Generation des stylized networks

The stylized binary network x that represents communication opportunities ist based on ns typical contact people had in a pre-COVID-19 world. It is generated stochastically as the composite von four sub-processes that follow relatively standard ideal-type network-generating approaches. Representing place von residence, actors are assumed to schutz a fixed geographical location, as determined von coordinates in a two-dimensional space. They are members des groups (such as households) und institutions (such as colleges or workplaces) and have individual characteristics (such together age, education and learning or income). Network ties space generated deswegen that actors have some connections to geographically close alters, part ties kommen sie members von the same teams (representing, for example, co-workers), part ties kommen sie alters with similar attributes (for example, comparable age) und some ties kommen sie random alters an the population. Jointly, these sub-processes develop networks that have realistic values von local clustering, path lengths und homophily. All ties an the network are defined as undirected. Ns number von actors an the network ist denoted by n. For the benchmark scenario presented bei Fig. 4, n = 2,000, and for die variations and robustness analyses, n = 1,000, uneven otherwise stated. In particular, the network sub-processes are identified as follows.

The erste sub-process to represent tie formation based on geographic proximity38. First, every actors an the network are randomly put into a two-dimensional square. Second, every actor draws ns number of contacts the forms an this sub-process dgeo,i indigenous a uniform distribution between dgeo,min and dgeo,max; zum example, if dgeo,min = 10 und dgeo,max = 20, every actor develops a arbitrarily number of ties betwee 10 und 20 in this sub-process. Third, the user-defined density bei geographic tie-formation dgeo defines ns geographic proximity of contacts drawn, dafür that actor i randomly creates dgeo,i ties among those dgeo,i/dgeo that space close an Euclidean street from actor i. Zum example, if actor ich is do to form dgeo,i = 12 ties and dgeo = 0.5, ns actor randomly choses 12 out des the 24 closest alters to form a tie to. Across all simulated networks, we collection dgeo = 0.3. Fourth, unilateral choices (where only ich selected j but not vice versa) are symmetrized deswegen that a non-directed connection exists bolzen the actors.

The 2nd sub-process represents tie formation an organizational foci (for example, workplaces)39. First, every actor zu sein randomly assigned zu a group deswegen that all groups schutz on average ns members. Second, every actor forms ties punkt random kommen sie other members within the same groups with a probability von ggroups. Zum example, wie m = 10 und ggroups = 0.5, a tie from each actor to every das alter in die same gruppe is developed with a probability of 50%. Third, unilateral ties are symmetrized together above.

The dritter sub-process represents tie gebildet based ~ above homophily (that is, seeking similarity); weil das example, similarity in age or income21. First, each actor zu sein assigned bei individual attribute ai betwee 0 und 100 with uniform probability (the scale von ai cancels later in the model). Second, weil das each actor, the normalized similarity simi,j kommen sie all changes j zu sein calculated, which is 1 minus the absolute difference betwee ai and aj weil das actor j, divided by 100 (the range des the variable), deswegen that simi,j = 1 when i und j schutz the the same value of a, und simi,j = 0 if castle are at opposite ends von the scale. Third, every actor draws die number des contacts it forms an this sub-process dhomo,i native a uniform distribution bolzen dhomo,min and dhomo,max. Fourth, each actor creates dhomo,i ties kommen sie alters j an the networks through a probability that zu sein proportional to (simij)w, where higher values von w average that individuals prefer more similar others. Throughout all reported simulations, we collection w = 2. Fifth, unilateral ties space symmetrized together above.

The 4th sub-process represents haphazard ties that are notfall captured by any of the over processes. Here, simply, z ties über actor are produced with respect zu randomly preferred alters.

Definition von simulation model

Let the binary network x represent communication opportunities betwee n individuals, labelled from 1 to n. Every node i kann sein be characterized von a set von attributes (left( a_i^k ight)) (for example, age or location).

Our model aims to reproduce die process von individuals connecting with some of these potential contacts. Similar to the classic sir model26 (in which people are susceptible, infectious or recovered) und its SEIR extension27 (in i beg your pardon they space susceptible, exposed, infectious und then recovered), us assume that individuals can be in four various states: either prone to the disease, exposed (infected but notfall yet infectious), contagious or recovered. Epidemic occurs v social interactions, which space modelled in a similar fashion to the dynamic actor-oriented model34 developed zum relational events. Much more specifically, our modell comprises die following steps:

1.

At each step von the process, one individual is picked punkt random und initiates in interaction with die probability πcontact.

2.

An gibbs initiating an interaction can only pick one interaction partner. Just potential partners together defined über the network x tun können be chosen. The decision to interact zu sein unilateral und depends top top characteristics des the 2 persons v a probability modell p.

3.

An contagious individual infects a gesund person wie they interact, that then becomes exposed. This contagion wake up with the probability πinfection.

4.

After a fixed number des steps Texposure, in exposed individual becomes infectious.

5.

After coming to be infectious, recovery occurs within Trecovery steps. When recovered, individuals kann no much longer be infected.

6.

The procedure ends once there is no longer anyone exposed or infectious.

The steps of the model are illustrated in Fig. 3. Note that ns mechanics of the infection align with formerly proposed agent-based versions von the SIR und SEIR models40,41. Together, ns probabilities πcontact and πinfection play a similar role to ns classic infectivity price β an SIR und SEIR models. The rate β models the typical number des contacts per person (modelled below through πcontact) and the likelihood des infection (represented von πinfection); however, ns equivalence zu sein not direct due to die added step of the interaction probability p. The exposure und recovery zeit replace die classic exposure and recovery rates (often traditionally denoted together σ and γ) in a simple manner.

We turn kommen sie the definition of die probability modell p. Allow Ni be the set von potential contacts, or transforms j of a given individual i in the network x. Us define zum each action t von the process: Li(j,t) as the number des previous interactions between i und an das alter j, within ns past λ interactions of i. Bei our simulations, ns number λ was arbitrarily set zu 2 but kann sein be changed easily in the replication files.

For each alter (j in N_i), ns value s(i,j) represents die statistic driving the strategical selection of ich to choose j. Specifics we define three different ways depending upon whether ns homophily, triadic (that is, increase community) or repetition balloon strategy is chosen (however, various other arbitrary statistics can be defined). Ns statistic ssimilarity accounts weil das the level von similarity bolzen i und j given a set of attributes; scommunity synchronizes to the number des alters they share, and srepetition ist the count of previous interactions within the past λ contacts of i. In practice, these statistics space calculated as:


$$s_ msimilarityleft( i,j ight) = 1 - fracsqrt mathop sum olimits_k left( a_i^k - a_j^k ight)^2 mathop max limits_h,l left( sqrt mathop sum olimits_k left( a_h^k - a_l^k ight)^2 ight) - mathop min limits_h,l left( sqrt mathop sum olimits_k left( a_h^k - a_l^k ight)^2 ight)$$

The probability for i to pick j ist defined together a multinomial choice probability42, follo wing die logic of previous relational event34 and stochastic network models32. The intuition behind this distribution is that each potential partner bei Ni ist assigned in objective role value, and choosing a partner is based on these values. Mathematically, ns objective duty is an exponentiated linear function von the statistic s(i,j), weighted by a parameter α. We further assume the individuals can reduce a specific percentage von their interactions. Considering die probability πcontact des initiating in interaction an the first place, the belang probability circulation becomes:


$$pleft( pi _ mcontact,alpha ight) = pi _ mcontact fracmathrmexpleft( alpha ast sleft( i,j ight) ight)mathop sum olimits_j^prime in N_i mathrmexpleft( alpha ast sleft( i,jprime ight) ight)$$

These probabilities kann be loosely interpreted in terms of log-transformed odds ratios, similar kommen sie logit models. Given two potential partners j1 und j2 zum whom ns statistic ns increases by one unit (that is, s(i,j2) = s(i,j1) + 1), ns following log proportion simplifies to:


$$log fracpleft( i o j_2 ight)pleft( i o j_1 ight) = alpha$$

For example, if we use s = srepetition und αrepetition = log<2>, die probability of picking one alter present in the previous contacts of i is twice as high together picking another alter who zu sein not.

Calibration of model parameters

The strategy von picking in interaction partner at random corresponds to the modell without any type of statistic s, reducing the probability distribution kommen sie a uniform 1. Weil das the three various other strategies, the parameters αsimilarity, αcommunity und αrepetition space adjusted kommen sie keep the models comparable.

To this end, us use ns measure of explained variation for dynamic network models devised by Snijders35. This measure builds on the Shannon entropy und can it is in applied zu our modell to assess the degree of certainty in individual’s choices. For a provided individual ich at a step t, this measure ist defined as:


$$r_ mHleft( pi _ mcontact,alpha ight) = 1 + fracmathop sum olimits_j in N_i pleft( i o j ight)log_2left log_2left< ight>$$

Intuitively, this measure amounts to 0 an the case des the random strategy where the probability von picking any alter is identical. It rises whenever some outcomes room favoured over others und equals 1 if one outcome has all of the probability mass.

Since the model assumes that all people are equally likely kommen sie initiate interactions, we can average this measure up over every actors. Moreover, bei the case of the repeat strategy, die measure is time dependent. Thus, we use its expected value over die whole process. We finally use the following aggregated measure kommen sie evaluate the certainty von outcomes des a particular strategy:


$$R_ mHleft( pi _ mcontact,alpha ight) = frac1nmathop sum limits_i = 1^n mathrmEleft< r_ mHleft( i,t ight) ight>$$

For this article, we zuerst fix the parameter αrepetition weist a value des 2.5 und calculate in estimated value (widehat R_ mHleft( pi _ mcontact,alpha _ mrepetition ight)) von this measure. This experience-based parameter an option results in in associated RH value bolzen 0.3 und 0.5 in the different scenario, which zu sein realistic bei terms von size (see the definition above). Zu compare this model with others, we then define ns parameters αsimilarity und αcommunity that verify:


$$widehat R_ mHleft( pi _ mcontact,alpha _ mrepetition ight) = R_ mHleft( pi _ mcontact,alpha _ mhomophily ight) = R_ mHleft( pi _ mcontact,alpha _ mcommunity ight)$$

using a traditional optimization algorithm. The average parameters throughout simulations for the different network scenarios room αcommunity = 0.75 und αsimilarity = 17.6. While die latter parameter shows up large, note that the relevant statistic ssimilarity varieties from 0 kommen sie 1, with most realized worths close kommen sie 1. The r code associated with all of the calculations is provided in the online repository referenced bei the Code accessibility statement.

Parametrization des the various simulations

Unless otherwise specified, every simulations use πcontact = 0.5 except zum the null model, which provides πcontact = 1. In all simulations except those that vary ns infectiousness, πinfection = 0.8. Uneven otherwise noted, Texposure = 1n und πinfection = 4n. Given the substantial computational load involved in conducting the simulations, 48 repetitions were run weil das networks v n ≤ 1,000, v 40 weil das larger networks. Experiments varying Texposure und πinfection supplied 24 repetitions.

For the experiments the vary the structure von the underlying network and the network size, ns parameters the guide the stochastic network production are presented an Supplementary Table 1. Descriptive statistics of these networks are presented in Supplementary Table 2. Die underlying networks that room used in the other variation experiment are generated according to the parameters denoted ‘1: baseline’ in Supplementary Table 1.

The four experiments the vary the time during which individuals are in the exposed zustand before becoming infectious usage values weil das Texposure of 0, 1n, 2n, 3n and 4n.

The four experiments that vary the infectiousness of the an illness use values zum πinfection of 0.55, 0.65, 0.80 und 0.95.

The sie wurden getestet that used location as the basis of ns homophily strategie was created according to the ‘1: baseline’ parameters but used die Euclidean distance in geographic placement together the basis for choosing interaction partners bei the homophily strategy. The two experiments on multidimensional homophily offered underlying networks developed following the ‘1: baseline’ parameters, with the exception the instead des one homophilous attribute, two characteristics were defined and the number von ties created according to die homophily parameter was split evenly bolzen the 2 dimensions. Die homophily strategie used zum the simulated infection curves bei the two scenarios differs in the feeling that an the first, individuals communicate according to minimizing ns absolute difference bei both attributes. In the 2nd scenario, only the erste attribute is used together the base of die homophily strategy und the second attribute zu sein ignored.

For ns experiments using combined strategies, the probability of kollege choice (pleft( i o j ight)) kann sein depend on a vector des statistics and parameters34. Die entropy based upon a set parameter vector was used zu calibrate ns parameter weil das the homophily and triadic closure strategy as compare cases. Parameter options rely ~ above experimentation kommen sie result bei similar entropy values kommen sie when using einzel strategies. Zum the mixed strategy of repetition and homophily, the parameters were set to αsimiliarity = 7 and αrepetition = 1.6. For the blended strategy des repetition und triadic closure, the parameters were set zu αcommunity = 0.35 and αrepetition = 1.6. For the blended strategy des homophily und triadic closure, ns parameters were set zu αsimiliarity = 6 and αcommunity = 0.35. For the mixed strategie incorporating every three, die parameters were set kommen sie αsimiliarity = 4, αcommunity = 0.3 and αrepetition = 1.2.

The simulated mean infection curves for all experiments can be found bei Extended dünn Figs. 1–7. Descriptive results zum the simulations, bei terms of delay of peak, height of peak und total number infected at the end of the simulation, space presented in Supplementary Table 3. Grad that ns descriptive statistics an this table present ns averages des characteristics of the repetitions of the simulated epidemic curves, i beg your pardon are not the same as ns characteristics des the mean infection curves together presented in Extended data Figs. 1–7.

Reporting Summary

Further die info on research entwurf is available bei the natur Research Reporting an introduction linked to this article.


No empirical säule was used in this article, only simulated data. Routines to produce ns simulated data are fully disclosed an the online resource presented in the Code ease of access section.


The replication files zum this paper, consisting of the ns code verbunden with every calculations and customized functions in the statistics environment r and in example script, are available on Zenodo (https://zenodo.org/record/3782465), a general-purpose open-access repository developed under die European OpenAIRE programme and operated von CERN.


Affiliations

Leverhulme Centre for Demographic Science und Department des Sociology, University of Oxford, Oxford, UK

Per Block, Jennifer Beam Dowd, Charles Rahal, Ridhi Kashyap & Melinda C. Mills

Department von Humanities, Social und Political Sciences, ETH Zurich, Zurich, Switzerland

Marion Hoffman

Institute of Sociology, University of Zurich, Zurich, Switzerland

Isabel J. Raabe

Nuffield College, University von Oxford, Oxford, UK

Charles Rahal, Ridhi Kashyap & Melinda C. Mills

School von Anthropology und Museum Ethnography, University of Oxford, Oxford, UK

Ridhi Kashyap


Contributions

P.B., M.H. Und I.J.R. Conceptualized ns study. P.B. And M.H. Contributed methodology and implementation and performed analyses. P.B., M.H., I.J.R. Und M.C.M. Composed the anfangsverdacht manuscript und provided visualization. All authors (P.B., M.H., I.J.R., J.B.D., C.R., R.K. Und M.C.M.) discussed ns research design und reviewed and edited die manuscript.

Corresponding authors

Correspondence to über Block or Melinda C. Mills.

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Curves compare 4 contact reduction techniques to ns null modell of no social distancing, together described bei the hauptsächlich text. (a) Reference modell with traditional operationalisation of homophily; (b) modell with homophily based on geographic proximity; (c) underlying network modell with homophily based on two dimensions, interaction strategy minimises die overall distinction along both attributes; (d) underlying network modell with homophily based on two dimensions, interaction strategie minimises ns difference just on the zuerst attribute.