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Matthew , ulse pi 7 2021. 27, April Published doi:10.1073/pnas.2021520118/-/DCSupplemental at online information supporting contains article This 2 1 the under Published interest.y competing no declare authors and The Technology; of Institute Massachusetts University. D.E., Columbia performed E.S., Chicago; research, of University designed O.C., Reviewers: S.T. and paper.y the M.O.J., wrote P.G.-P., and research, A.G.C., contributions: Author rs odr,laigt ekg htlmt hi effectiveness. their limits uncoordinated that leakage to multiple, leading often however, borders, quarantined, by be cross to need that conducted regions The these regimes. policies of cer- any a quarantine of has failure network effectiveness. quarantine The the regional limits to structure. 3) substantially network conditions balance” and the “growth of infection, control tain of and leakage knowledge infections, observing sufficient prevent in is delay limited there is there 2) 1) if only and if ease disease. the spread can or leakages (13–18). tiny tracing) necessity way, or contact Either choice inefficient people—by whom some identifying or by in imperfect noncompliance a difficulties (e.g., quarantine of fully quarantine because to to network, infeasible be the may of infec- it part identify Second, quickly 12). to (11, resources tions lacks indi- government a some COVID-19) or either HIV, (8–10) (e.g., because contagious detect, asymptomatically are to viduals difficult are First, diseases challenges. quaran- more two many are face hence infections however, quarantines, observed and only Regional of tined. run), which distance in work, some short within quarantines” of the people “regional days (in are lost alternatives costly (e.g., common, “global Less costly a very etc.). under are school, once those at A but everyone policies. quarantine,” single-regime quarantine consider can first government We policies. these study to when deciding short. for when well “proactive” rates as or infection quarantine, their jurisdiction forecast their to of within as outside infections tracking looking, owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence work. y whom this To to equally contributed S.T. and M.O.J., P.G.-P., A.G.C., o vroe u ehiussol eueu nstudying in useful be domains. other should across interactions techniques networked Our repeat- outcome the everyone. that worsen jurisdictions for substantially and create However, disease a governments magnitude. incubate lax of edly order few then a an jurisdictions, just by informa- other reduced share from are governments infections infections if anticipate Instead, and disease. tion repeated coordinate a to to leading of fail, failure waves to and quarantines cause populace neighbors pri- We own with governments’ their pandemic. how understand of a to an oritization curbing that by in use effective exemplified and are outbreak well policies has quarantine are which contagions, identify that financial costs to systemic change large on climate governments from between problems, coordination of lack The Significance ete xmn uidcinlplce,wihaeregional are which policies, jurisdictional examine then We dis- a of spread the curb quarantines regional that show We to network a through contagion of model general a use We NSlicense.y PNAS y a,e,1,2 https://doi.org/10.1073/pnas.2021520118 n aulThau Samuel and , . y https://www.pnas.org/lookup/suppl/ e at eInstitute, Fe Santa f,1 c National | f7 of 1

ECONOMIC SCIENCES As we show, jurisdictional policies that are reactive do much ner (α = 0.1). First, consider a part of the network in which each worse than proactive ones, as they do not forecast the impact infected person infects 3.5 others on average. If we monitor all of neighboring jurisdictions’ infection rates on their own popu- nodes within distance k = 3 of an infected node, a “typical” path lation. Moreover, a few lax jurisdictions, which wait for higher of infection would lead to roughly 3.5 + 3.52 + 3.53 = 58.625 infection rates before quarantining, worsen outcomes for all expected cases before it reaches the edge of the region. The jurisdictions. chance that this goes undetected is tiny: 0.958.625 = 0.002. In con- trast, suppose the infection starts in a part of the network where A Model each infected person infects just one other, on average, so that Consider a large network of n individuals or nodes. Our the- the local reproduction number here is R0 = 1 rather than 3.5. ory is asymptotic, stating properties that apply with a probability Now a path of length 3 leads to 1 + 1 + 1 = 3 (expected) infec- approaching one as the population grows (n → ∞). Our theoret- tions. The chance that such a spread remains undetected is much ical results consider sequences of networks as n grows, while the higher 0.93 = 0.72. simulations are on given networks with thousands of nodes. Speaking loosely, many different networks can lead to the An infectious disease begins with an infection of a node i0, same average reproduction number, but have very different the location of which is known, and expands via (directed) paths structures. If the distribution of reproduction numbers around from i0. In each discrete time period, the infection spreads from the network has no pockets in which they are too low—i.e., if each currently infected node to each of its susceptible contacts the growth structure of the disease around the network is well independently with probability p. A node is infectious for θ “balanced” and not too low—then it is highly likely that any periods, after which it recovers and is no longer susceptible or early infection will be detected before it gets too far from the contagious, although our results extend to the case in which a first infected node. If, instead, the distribution of reproduction node can become susceptible again. numbers gives a nontrivial chance that the disease starts out on The disease may exhibit a delay of τ ≤ θ periods during which a path with all low reproduction numbers, like the 1, 1, 1, path, an infected and contagious person does not test positive. This can then there is a high chance that it can travel far from the seed be a period of asymptomatic infectiousness, a delay in testing, or before being detected. This highlights the fact that a reproduc- limits to healthcare access (8, 10–12, 19, 20). After that delay, tion number R0 alone is a crude concept and that the specifics of each infected node’s infection is detected with probability α < 1 the network structure matter considerably for whether a disease (for simplicity, in the first period after the delay). α incorporates spreads or is containable. In particular, areas with low R0 (but testing accuracy, availability, and decisions to test. above one) can lead to more containment failures and lead to This framework nests the susceptible–infected–recovered broader infections. Given the short distances in many networks (SIR) model and its variations including exposure, multiple (28–30), a lack of growth balance allows a disease to spread far infectious stages, and death (18, 21–24); agent-based models before detection. SI Appendix, Fig. S1 pictures a network that has (25–27); and others. a high average reproduction number, but is not growth balanced and allows the infection to travel far from the initially infected Results node without detection. Baseline: A Single Jurisdiction with Complete Control. We begin In SI Appendix, Theorem 1 we prove that, with no delays in by analyzing a single jurisdiction with complete control or, detection and no leakage, a (k, x) -regional policy halts infection equivalently, the entire network being one jurisdiction with one among all nodes beyond distance k + 1 from i0 with probabil- policymaker. ity approaching 1 (as the population grows) if and only if the A (k, x) -regional policy is triggered once x or more infec- sequence of networks satisfies growth balance with respect to k. tions are observed within distance k from the seed node i0, at In fact, we prove a stronger version in which the quarantine dis- which point it quarantines all nodes within distance k + 1 of the tance k(n), average degree d(n), transmission probability p(n), seed for θ periods. This captures a commonly used policy where delay θ(n), and detection probability α(n) are all allowed to vary regions that are exposed to the disease are shut down in response with n. to detection. We begin by giving the policymaker the advantage Growth balance is satisfied by some, but not all, sequences of knowing which nodes are within distance k + 1 of the seed, of prominent random graph models, provided that the aver- which could reflect rapid and efficient contact tracing supple- age degrees d(n) satisfy d(n)k(n) → ∞ (SI Appendix). However, mented with rich network data. We later explore how errors the additional heterogeneity in human contact networks makes in this knowledge change the results. We also give the policy- the property unlikely to hold in real networks even if average maker knowledge of which node is the seed and study subsequent connectivity is high. Indeed, if contact networks have some low containment efforts. In practice, policymakers must estimate the degree nodes (as they tend to empirically), then, unless quar- origin of infection, which presents an additional challenge. antine regions are large (k(n) grows sufficiently as n grows), Whether a regional policy halts infection is fully character- growth balance fails and a regional quarantine will be ineffective ized by whether the sequence of networks satisfies what we at halting a spread. call growth balance. This requires that as the networks become Next, we show that the effectiveness of a regional policy breaks large, all paths along which the disease might escape beyond the down, even if a network is growth balanced, once there is suffi- regional quarantine of distance k + 1 are such that at least some cient delay in detection or leakage (due to imperfect information, nodes have many neighbors. In particular, for the sequence of enforcement, or jurisdictional boundaries). networks to be growth balanced with respect to k, there must exist m(n) → ∞ such that in a network with n nodes, every path Delays in Detection and Wider Quarantines. To understand how leading from i0 to a node at distance k + 2 has at least one node delays in detection affect a regional policy, consider two with degree at least m(n). This condition ensures that if an infec- extremes. If the delay is short relative to the infectious period, tion does spread along some path that can take it outside of the the policymaker can still anticipate the disease and adjust by sim- region, then, at some point along the way, it is very likely to infect ply enlarging the area of the quarantine to include a buffer. An many nodes and thus be detected before it reaches the edge of easy extension of the above theorem is that a regional policy with the region. a buffer works if and only if the sequence of networks is growth To better understand growth balance, consider an example of balanced and the delay in detection plus k + 1 is shorter than a disease that is beginning to spread with a reproduction number the diameter of the network (SI Appendix, Theorem 2). Given R0 of 3.5 and such that 1 in 10 cases is detected in a timely man- that real-world networks have short average distances between

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the contain could (k IAppendix SI τ , h oe r rsne nagorpi es,wti their within sense, geographic a in presented are nodes the A, x we sd,and used), (when eid sdtce ihprobability with detected is periods 3 1) (3, = ) eso h ewr safnto fdrce distance directed of function a as network the show we B, R Distance 0 3 = peet h outeso the of robustness the presents 2E section , .5 Jurisdiction rgoa urnie h policymaker The quarantine. -regional t ii OI-9 3) n eset we and (35), COVID-19) mimic (to 01234 θ α eid eoe;fial,quaran- finally, recover; periods https://doi.org/10.1073/pnas.2021520118 0 = .1 ab eto 2B) section Appendix, (SI k 1 + et policy- next, α; fte.In them. of PNAS i 0 along , | (k f7 of 3 , x SI )

ECONOMIC SCIENCES Infected (per mill.) A Recovered (per mill.) 600 6,000 infected recovered 400 4,000

200 2,000

0 0 0 50 100 150 200 Time

BC Infected (per mill.) Infected (per mill.) Recovered (per mill.) Recovered (per mill.) 600 6,000 600 6,000 infected recovered 400 4,000 400 infected 4,000 recovered

200 2,000 200 2,000

0 0 0 0 0 50 100 150 200 0 50 100 150 200 Time Time

Fig. 2. We picture daily infections and cumulative recoveries under three scenarios. The entire network is governed by a single policymaker using a (k, x) = (3, 1) -regional quarantine. In A, there is no detection delay and no leakage. In B, we introduce a detection delay of τ = 3. This represents the 3-d presymptomatic window during which an infected node can transmit, as well as an expected delay in seeking healthcare and testing upon symptom onset (SI Appendix, section 2B) (19). C adds leakage to the setup of B, by having a randomly selected fraction  = 0.05 never quarantine. For each plot, we simulate 10,000 times on the same network with random initial infections and present the average number of infections and recovered people over time, scaled per million.

with variants of the transmission and detection parameters— of x cases, and observes at least x cases within the jurisdiction, varying θ, τ, α, and R0 (SI Appendix, sections 2D and 2E). In our the jurisdiction goes into quarantine. In proactive policies, juris- simulations, we primarily focus on x = 1 and k = 3 (other param- dictions track infections in other jurisdictions and predict their eters appear in SI Appendix, section 2E). We use x = 1 because, own—possibly undetected—infections and base their quaran- as we discuss in SI Appendix, a policymaker attempting to min- tines off of predicted infections. Intuitively, jurisdictions are con- imize the number of infected person periods and quarantined stantly estimating infection rates (including undetected cases) person periods should attempt to minimize the number of infec- in each jurisdiction based on the history of observed infections, tions. We choose k = 3 due to the limited size of our simulated using knowledge of the interaction rates within and across bor- network—a larger boundary would cover too much of the overall ders and the infection and latency properties of the disease. network. More specifically, at each time step t, the number of estimated Fig. 2A shows the outcomes for no delay in detection nor infections at time t is the sum of the estimated infections at any leakage. Consistent with SI Appendix, Theorem 1 the pol- t − 1, plus the expected number of new estimated infections icy is effective: On average 277 people per million are infected minus the number of expected recoveries. The expected num- (0.028% of the population), with 803,956 person days of quaran- ber of new infections is calculated using the connection rates tine per million people. Fig. 2B introduces a delay in detection. to nonquarantined jurisdictions (including own jurisdiction) and With a delay of τ = 3, infections increase, with 2,256 people the estimated infections in those jurisdictions at t − 1. If at any per million eventually infected (0.23% of the population) and point it is clear that the actual number of detected infections 2,301,414 person days of quarantine per million people. Adding in some jurisdiction is above the estimated rate, then the esti- a buffer to correspond to the detection delay effectively makes mation is updated. All jurisdictions begin by estimating that the regional policy global, as the buffered region contains 99.98% there are no infections, until at least one infection is observed. of the population on average. Fig. 2C adds leakage to the setup Details of this calculation are in SI Appendix, section 2D. of Fig. 2B, by having 5% of people never quarantine. The num- Proactive jurisdictions quarantine if they infer (or observe) at ber of cumulative infections per million people increases to 5,138 least x cases. (0.50% of the population). The leakage increases the number of We set x = 1 for both the reactive and proactive simulations quarantined person days to 6,478,055 per million nodes. unless otherwise specified. For both reactive and proactive juris- dictions, when a jurisdiction enters quarantine, all connections to Jurisdictional Policies. We now introduce jurisdictions to the same and within the jurisdiction are severed for θ periods. As before, network as before, and each of the 40 locations in the network we set R0 = 3.5, θ = 5, τ = 3, and α = 0.1. The policymaker does becomes its own jurisdiction. not detect i0 for both the reactive and proactive policies, but does We compare two types of jurisdictional policies. In reactive know its location when setting the quarantine. policies, each jurisdiction decides on when it quarantines based Fig. 3 illustrates the improvement that proactive jurisdic- entirely on internal infections. If a jurisdiction has a threshold tional policies offer relative to reactive jurisdictional policies.

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(29.9%) and lax 298,911 to the infected compared out million population), per the people of 340,587 (34.1% are There policies. reactive 3 Fig. Comparing poli- policies. reactive 3D Fig. use in jurisdictions while 36 cies, remaining 3 the Fig. when jurisdictions. all outcomes for outcomes the worsen using tions a quarantining, of have before threshold and infections reactive a internal are of that threshold jurisdictions high are These setting. the Jurisdictions. Lax per quarantine of days of people. person (1.71% million 51,328,755 infected are with million population), per the people 17,105 outcomes Only improves 3B): pop- (Fig. dramatically million the per quarantining quarantine of Proactive of (28.89% nodes. there days person people case, 3B 131,303,638 million Fig. reactive with per ulation), in the infections while In 298,911 policies, policies. are reactive proactive use use jurisdictions jurisdictions 3A, infections Fig. of number In average the present and infections initial random with network million. same per the on scaled times time, 10,000 over simulate people we recovered plot, and each For policies. lax with in In jurisdiction. entire the down 3. Fig. CD AB Infected (permill.) 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ECONOMIC SCIENCES AB Fraction of regions under quarantine Person−Days of Infection 2,000,000 Single jurisdiction, global quarantine 0.4 Single jurisdiction, regional quarantine All Reactive Multiple jurisdictions, reactive All Proactive Multiple jurisdictions, proactive Reactive w/ lax neighbors 1,500,000 Proactive w/ lax neighbors Multiple jurisdictions, reactive 0.3 w/ lax neighbors Multiple jurisdictions, proactive 1,000,000 w/ lax neighbors 0.2

0.1 500,000

0.0 0 0 250 500 750 1000 10,000,000 50,000,000 150,000,000 Time Person−Days of Quarantine

Fig. 4. A displays the dynamics of quarantines for each of the policy configurations. B plots the number of person-day infections (per million) against the number of person-day quarantines (per million) for six key policy scenarios. The global policy does the best on both dimensions, and the second best is the single-jurisdiction reactive strategy (which does worse than the global one because of leakage). With 40 jurisdictions, both proactive policies outperform the internal, reactive policies. By far the worst, on both dimensions, is the internal, reactive policy with some lax jurisdictions. These results come from the same simulations that produce Figs. 2 and 3.

their borders. We have also shown that if governments are more divide the network into small components of diameter less than attentive to their neighbors, there can be substantial improve- k, so that growth balance is satisfied by default. ments to their infection rates. However, the existence of some We note that the effect of lax jurisdictions can be mitigated if lax jurisdictions imposes a significant cost on everyone else. other jurisdictions eliminate contact with that jurisdiction, e.g., We view the (k, x) quarantine model as a stylized approxima- with travel restrictions. Then, for the nonlax jurisdictions, the tion of some policymakers’ policies. These policymakers estimate situation returns to one without any lax jurisdictions, which is the disease’s location and try to shut down the surrounding area. better as we have shown. This will work only if all nonlax juris- This can range from city blocks to subdistricts to larger regions, dictions participate in a travel ban, as otherwise infection will depending on the country. For example, in multiple states across continue to resurge in jurisdictions that continue contact with India, such as Haryana, Karnataka, Andhra Pradesh, Gujarat, the lax jurisdictions, which then pass infection along to others. and West Bengal (with a combined population of around 285 mil- Our results also suggest caution in using statistical models lion as of the 2011 Indian Census), district officials are entrusted to identify regions to quarantine. Although contagion models to define containment zones. This is decentralized and many are helpful for informing policy about the magnitude of an epi- of their efforts are “microtargeted.” More globally, each dis- demic and broad dynamics, the models can give false comfort in trict plays the role of a jurisdiction in the multijurisdiction case. our ability to engage in highly targeted policies, whose results Another example is a sports league where teams or collections can be influenced by small deviations from idealized assump- of players are contact traced and quarantined—parsimoniously tions (e.g., leakage). Our growth balance condition also points modeled as (k, x) from i0. out that not all parts of a network are equal in their poten- Jurisdictional policies tend to be aimed at the welfare of tial for undetected transmission. Growth balance offers insight their internal populations, yet the external effects are large. into when containment will frequently fail. While full contain- Our results underscore the importance of timely information ment is sometimes, but not always, the full policy goal, it helps sharing and coordination in both the design and execution of us understand what features of the network aid or hinder con- policies across jurisdictional boundaries (38). The results also tainment efforts even under ideal conditions. Policymakers must underscore the global importance of aiding poor jurisdictions. be conscious of limited paths of interactions that can introduce Indeed, there is mounting evidence that a lack of coordi- the disease into a larger population, precisely because detec- nation across boundaries has been damaging in the case of tion is very difficult along such paths. Therefore, such paths are COVID-19 (6). important to monitor. In places where the reproduction num- The use of masks (decreasing p), social distancing (decreas- ber is lower, the probability of observing outbreaks is also lower, ing d), and increasing testing (increasing α) and vaccinations enabling a leakage of undetected infections. (decreasing p) all help attenuate contagion, but unless they main- tain the reproduction number below one, the problems identified Data Availability. There are no data underlying this work. here remain. Even tiny fractions of interactions across borders ACKNOWLEDGMENTS. We thank Marcella Alsan, Abhijit Banerjee, Gabriel are enough to lead to spreading in large populations. With Carroll, Bharat Chandar, Dean Eckles, Ben Golub, Hans Haller, Dave Holtz, modern inter- and intranational trade and travel being a siz- Ali Jadbabaie, Ed Kaplan, Jon Kleinberg, Jacob Leshno, Sudipta Sarangi, able portion of all economies, such interaction is difficult to Devavrat Shah, and Johan Ugander for helpful discussions. We thank Abdul Latif Jameel Poverty Action Lab (J-PAL SA), Tithee Mukhopadhyay, Shreya avoid. Nonetheless, our analysis also offers insights into manag- Chaturvedi, Vasu Chaudhary, Shobitha Cherian, Arnesh Chowdhury, Anoop ing infections at smaller scales, e.g., within schools, sports, and Singh Rawat, and Meghna Yadav for research assistance. The computations businesses. By creating a network of interactions that is highly in this paper were run on the Faculty of Arts and Sciences Research Comput- modular, keeping cross-modular interactions to a minimum and ing (FASRC) Cannon cluster supported by the Faculty of Arts and Sciences (FAS) Division of Science Research Computing Group at Harvard University. making sure that they are highly traceable, together with aggres- We gratefully acknowledge financial support from the NSF under Grants sive testing (especially of cross-module actors), one can eliminate SES-1629446, SES-2018554, and RAPID 2029880. A.G.C. thanks the Alfred P. leakage and effectively bound the set of interactions. This would Sloan foundation for support.

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