Interacting Regional Policies in Containing a Disease
Arun G. Chandrasekhar, Paul Goldsmith-Pinkham, Matthew O. Jackson, Samuel Thau
IInteracting Regional Policies in Containing a Disease
Arun G. Chandrasekhar ∗ , Paul Goldsmith-Pinkham † ,Matthew O. Jackson ‡ , and Samuel Thau § August 2020
Abstract
Regional quarantine policies, in which a portion of a population surrounding infec-tions are locked down, are an important tool to contain disease. However, jurisdictionalgovernments – such as cities, counties, states, and countries – act with minimal coor-dination across borders. We show that a regional quarantine policy’s effectivenessdepends upon whether (i) the network of interactions satisfies a balanced-growth con-dition, (ii) infections have a short delay in detection, and (iii) the government hascontrol over and knowledge of the necessary parts of the network (no leakage of be-haviors). As these conditions generally fail to be satisfied, especially when interactionscross borders, we show that substantial improvements are possible if governments areproactive: triggering quarantines in reaction to neighbors’ infection rates, in some caseseven before infections are detected internally. We also show that even a few lax gov-ernments – those that wait for nontrivial internal infection rates before quarantining –impose substantial costs on the whole system. Our results illustrate the importance ofunderstanding contagion across policy borders and offer a starting point in designingproactive policies for decentralized jurisdictions. ∗ Department of Economics, Stanford University; J-PAL; NBER. † Yale School of Management. ‡ Department of Economics, Stanford University; Santa Fe Institute. § Harvard University.We thank Abhijit Banerjee, Gabriel Carroll, Bharat Chandar, Dean Eckles, Ben Golub, Dave Holtz, AliJadbabaie, Ed Kaplan, and Johan Ugander for helpful discussions. We gratefully acknowledge financialsupport from the NSF under grant SES-1629446 and RAPID a r X i v : . [ phy s i c s . s o c - ph ] S e p ntroduction Global problems, from climate change to disease control, are hard to address without policycoordination across borders. In particular, pandemics, like COVID-19, are challenging tocontain because governments fail to coordinate efforts. Without vaccines or herd immunity,governments have responded to infections by limiting constituents’ interactions in areaswhere an outbreak exceeds a threshold of infections. Such regional quarantine policies areused by towns, cities, counties, states, and countries, and trace to the days of the blackplague. Over the past 150 years, regional quarantines have been used to combat cholera,diphtheria, typhoid, flus, polio, ebola, and COVID-19 [1, 2, 3, 4], but rarely with coordinationacross borders.Decentralized policies across jurisdictions have two major shortcomings. First, govern-ments care primarily about their own citizens and do not account for how their infectionsimpact other jurisdictions: the resulting lack of coordination can lead to worse overall out-comes than a global policy [5, 6, 7]. Second, some governments only pay attention to whatgoes on within their borders, which leads them to under-forecast their own infection rates.We examine three types of quarantine policies to understand the impact of non-coordination:(i) those controlled by one actor with control of the whole society – “single regime policies,”(ii) those controlled by separate jurisdictions that only react to internal infection rates –“myopic jurisdictional policies,” and (iii) those controlled by separate jurisdictions that areproactive and track infections outside of their jurisdiction as well as within when decidingon when to quarantine – “proactive jurisdictional policies.”We use a general model of contagion through a network to study these policies. Wefirst consider single regime policies. A government can quarantine everyone at once under a“global quarantine,” but those are very costly (e.g., lost days of work). Less costly (in theshort run), and hence more common, alternatives are “regional quarantines” in which onlypeople within some distance of observed infections are quarantined. Regional quarantines,however, face two challenges. First, many diseases are difficult to detect, because individualsare either asymptomatically contagious (e.g., HIV, COVID-19) [8, 9, 10], or a governmentlacks resources to quickly identify infections [11, 12]. Second, it may be infeasible to fullyquarantine a part of the network, because of difficulties in identifying whom to quarantineor non-compliance by some people – by choice or necessity [13, 14, 15, 16, 17, 18, 19]. Eitherway, tiny leakages can spread the disease.We show that regional quarantines curb the spread of a disease if and only if: (i) thereis limited delay in observing infections, (ii) there is sufficient knowledge and control of thenetwork to prevent leakage of infection, and (iii) the network has a certain “balanced-growth”structure. The failure of any of these conditions substantially limits quarantine effectiveness.We then examine jurisdictional policies, which are regional quarantine policies conductedby multiple, uncoordinated regimes. The regions that need to be quarantined cross borders,leading to leakage that limits their effectiveness. As we show, myopic policies do much worsethan proactive ones, as they do not forecast the impact of neighboring infection rates on1heir own population. Moreover, a few lax jurisdictions, which wait for higher infection ratesbefore quarantining, substantially worsen outcomes for all jurisdictions.
A Model
Consider a large network of nodes (individuals). Our theory is asymptotic, applying as thepopulation grows (details in the SI). An infectious disease begins with an infection of a node i , the location of which is known, and expands via (directed) paths from i .In each discrete time period, the infection spreads from each currently infected node toeach of its susceptible contacts independently with probability p . A node is infectious for θ periods, after which it recovers and is no longer susceptible, though our results extend tothe case in which a node can become susceptible again.The disease may exhibit a delay of τ ≤ θ periods during which an infected and contagiousperson does not test positive. This can be a period of asymptomatic infectiousness, a delayin testing, or healthcare access [8, 10, 20, 11, 12, 21]. After that delay, the each infectednode’s infection is detected with probability α < α incorporates testing accuracy, availability, and decisions to test.This framework nests the susceptible-infected-recovered (SIR) model and its variationsincluding exposure, multiple infectious stages, and death [22, 23, 24, 18, 25], agent-basedmodels [26, 27, 28], and others. Results
Baseline: an ideal setting
We begin by analyzing a single jurisdiction with complete control.A ( k, x )-regional policy is triggered once x or more infections are observed within distance k from the seed node i ; at which point it quarantines all nodes within distance k + 1 of theseed for θ periods. This captures a commonly used policy where regions that are exposed tothe disease are shut down in response to detection. We give the policymaker the advantageof knowing which node is the seed and study subsequent containment efforts. In practice,the estimation of the infection origin is an additional challenge.Whether a regional policy halts infection in this setting is fully characterized by what wecall growth-balance (formally defined in the SI). This requires that the network have largeenough expansion properties and that the expansion rate not drop too low in any part ofthe network.To better understand growth-balance, consider an example of a disease that is beginningto spread with a reproduction number of 3.5 and such that one in ten cases are detected in atimely manner ( α = 0 . k = 3 of an infected node,2 “typical” chain of infection would lead to roughly 3 . . + 3 . = 58 .
625 expected cases.The chance that this goes undetected is tiny: 0 . . = 0 . . = 0 . k, x )-regional policy halts infection among all nodes beyond distance k + 1 from i with probabilityapproaching 1 (as the population grows) if and only if the network satisfies growth-balance.Growth-balance is satisfied by many, but not all, sequences of random graph models,provided that the average degree d satisfies d k → ∞ (Corollary 1, SI). Without growthbalance, a regional policy fails non-trivially even under idealized conditions.The effectiveness of a regional policy breaks down, even if a network is growth-balanced,once there is leakage (due to imperfect information, enforcement, or jurisdictional bound-aries) or sufficient delay in detection. Delays in Detection and Wider Quarantines
To understand how delays in detection affect a regional policy, consider two extremes. If thedelay is short relative to the infectious period, the policymaker can still anticipate the diseaseand adjust by enlarging the area of the quarantine to include a buffer. An easy extension ofthe above theorem is that a regional policy with a buffer works if and only if the networkis growth-balanced and the delay in detection is shorter than the diameter of the network(Theorem 2, SI). Given that real-world networks have short average distances between nodes[32], non-trivial delays in detection allow the disease to escape a regional quarantine.
Leakage
Next, we consider how leakage – inability to limit interactions [13] or mistakes in identifyingportions of a network to quarantine [17, 18] – diminishes the effectiveness of regional poli-cies. Although minimizing leakage increases the chance that a regional quarantine will be3uccessful, we show (Theorem 3, SI) that even a small amount of leakage leads to a nontrivialprobability that a regional policy will fail.
Jurisdictions and Leakage
We can use the results from regional quarantines as a starting point to understand jurisdic-tional policies. For instance, leakage generally applies when interactions cross jurisdictions.Figure 1 pictures two jurisdictions that fail to nicely tessellate the network.
Figure 1: Inconsistency of Jurisdictions and Distances l Distance l l l a b (a) Jurisdictions with interactions that do not align l Distance l l l a b (b) Figure 1a, but based on distance from infection
Figure 1:
Nodes in two jurisdictions do not align with the distances from the initial infection.In Panel (a), the nodes are presented in a geographic sense, within their jurisdictions, and theinteraction network does not comply with the jurisdictional boundaries. In Panel B, we show thenetwork as a function of directed distance from the initial infection. A coordinated quarantine ofdistance 2 over the network in Panel (b) could contain the infection; however, if it is only executedby the infected nodes jurisdiction in panel (a) then it would fail for cross-jurisdictional connections.
Given leakage across jurisdictional borders, unless policies are fully coordinated acrossjurisdictions, our theoretical results indicate that they will fail to contain infections.
Simulations
The theory provides insights into the various hurdles that quarantine policies face, but doesnot provide insight into how well different types policies will fare in slowing infection and atwhat costs.To explore this, we simulate a contagion on a network of 140000 nodes that mimics real-world data [33, 34, 35, 21]. These simulations illustrate our theoretical results and also showthe improvements that proactive policies provide relative to myopic ones. The results arerobust to choices of parameters (SI).The network is divided into 40 locations , each with a population of 3500. We generatethe network using a geographic stochastic block model (SI). The probability of interacting4eclines with distance. The average degree is 20.49 and nodes have 79.08% of their inter-actions within their own locations and 20.92% outside of their location (calibrated to datafrom India and the United States, including data collected during COVID-19 [33, 34, 35, 21],SI). We set the basic reproduction rate R = 3 . θ = 5 and α = 0 . k, x ) = (3 , τ = 3, infections increase, with 2256 nodes permillion eventually infected (0.23% of the population) and 2301414 node-days of quarantineper million nodes. Adding a buffer to correspond to the detection delay effectively makes theregional policy global, as the buffered region contains 99.98% of the population on average.Figure 2c adds leakage to the setup of Figure 2b, making only 95% of the intended nodesquarantined. The number of cumulative infections per million nodes increases to 5138 (0.50%of the population). The leakage increases the number of quarantined node-days to 6478055per million nodes. Jurisdictional Policies
We now introduce jurisdictions to the same network as before, and each location becomesits own jurisdiction.We compare two types of jurisdictional policies. In myopic policies each jurisdictionquarantines based entirely on internal infections. In proactive policies, jurisdictions trackinfections in other jurisdictions and predict their own – possibly undetected – infections andbase their quarantines off of predicted infections (calculation details in SI). In both cases, ifa jurisdiction enters quarantine, all links within and to the jurisdiction are removed.Figure 3 illustrates the improvement a proactive policy offers relative to myopic internaljurisdictional policies. In Figure 3a, jurisdictions use myopic policies, while in Figure 3bjurisdictions use proactive policies. In the myopic case, there are 118447 infections per millionnodes (11.85% of the population), with 65634600 person-day quarantines per million nodes.Multiple waves are common: 67.4% of jurisdictions have multiple quarantines. Proactivequarantining dramatically improves outcomes (Figure 3b): only 6300 nodes per million areinfected (0.630% of the population), with 37816130 person-day quarantines per million nodes.Multiple shutdowns are less frequent: 56.9% of jurisdictions quarantine more than once.5 igure 2: The Impact of Detection Delay and Leakage (a) ( k, x ) = (3 , (b) ( k, x ) = (3 , (c) ( k, x ) = (3 , Figure 2:
We picture daily infections and cumulative recoveries under three scenarios. The entirenetwork is governed by a single policymaker using a ( k, x ) = (3 , τ = 3.This represents the 3 day pre-symptomatic window during which an infected node can transmit, aswell as an expected delay in seeking healthcare and testing upon symptom onset ([20], SI). Panel2c adds leakage of (cid:15) = 0 .
05 to the setup of Panel 2b. For each figure, we simulate 10000 times onthe same network with random initial infections, and present the average number of infections andrecovered people over time, scaled per million. igure 3: The Effectiveness of Myopic vs Proactive Policies (a) Each jurisdiction quarantines once it observesany infections internally, ignores other jurisdictions (b)
Each jurisdiction quarantines proactively by es-timating internal infections based on observation ofother jurisdictions (c)
36 jurisdictions quarantine once observing anyinternal infections, 4 lax jurisdictions only quarantineonce they reach 5 internal infections (d)
36 jurisdictions quarantine proactively by esti-mating internal infections based on observation ofother jurisdictions, 4 lax jurisdictions only quaran-tine once they reach 5 internal infections
Figure 3:
We picture daily infections and cumulative recoveries under four quarantine policieswith 40 jurisdictions. When a jurisdiction quarantines, it locks down the entire jurisdiction. InPanel 3a, all jurisdictions use a myopic internal policy. In Panel 3b, all jurisdictions use a proactivepolicy. In Panel 3c, we implement the same policies as Panel 3a, but have four lax jurisdictions thatuse x = 5 (0.14% of the jurisdiction population) instead of x = 1 (SI). Panel 3d has 36 jurisdictionswith proactive policies and four with lax policies. For each figure, we simulate 10000 times onthe same network with random initial infections, and present the average number of infections andrecovered people over time, scaled per million. Lax Jurisdictions
Finally, we also add a few “lax” jurisdictions to the setting. These are jurisdictions that aremyopic and have a high threshold of internal infections before quarantining. We examinehow these few lax jurisdictions worsen the outcomes for all jurisdictions.In Figures 3c and 3d, four lax jurisdictions react only to infections within their ownborders and wait until they have detected five infections before quarantining (SimulationDetails, SI). Figure 3c shows the outcomes when the remaining 36 jurisdictions using my-7pic internal strategies, while in Figure 3d the remaining 36 jurisdictions using proactivestrategies. Comparing Figures 3a to 3c, infections are much worse under the myopic inter-nal policies. 209389 nodes per million are infected (20.9% of the population), and 72.8%of regions shut down multiple times. Of the infections in Figure 3c, 84.2% happen in lowthreshold-jurisdictions. Comparing Figures 3b to 3d shows that things deteriorate less for theproactive jurisdictional policies. The 27312 total infections per million nodes (2.73% of thepopulation) is well below either set of myopic policies: 67.4% of jurisdictions have multiplequarantines; and 73.4% of the infections in Figure 3d happen in the proactive regions.Figure 4a displays the dynamics of quarantines for each of the policy configurations fromFigure 3, and Figure 4b displays the number of person-day infections versus the number ofperson-day quarantines.Global quarantines (closing the entire network at once) and single-jurisdiction regionalquarantines (with leakage) do the best on both dimensions. Once jurisdictions are intro-duced, proactive jurisdictions quarantine earlier and have fewer recurrences than myopicjurisdictions. Lax jurisdictions cause an overall higher number of quarantines, over a longertime. The proactive jurisdictional policy trades off more quarantine days for substantiallyfewer infection days compared to the myopic internal policy, but proactive policies do signif-icantly better than myopic policies on both dimensions when mixed with jurisdictions usinglax policies.
Figure 4: The Impact and Costs of Quarantine Policies with and without LaxJurisdictions
All MyopicAll ProactiveMyopic w/ lax neighborsProactive w/ lax neighbors
Fraction of regions under quarantine (a)
Dynamics of quarantines in each of the policyconfigurations l l Single jurisdiction, global quarantineSingle jurisdiction, regional quarantineMultiple jurisdictions, myopicMultiple jurisdictions, proactiveMultiple jurisdictions, myopic w/ lax neighborsMultiple jurisdictions, proactive w/ lax neighbors
Infected Person Periods (b)
Person-day infections vs. person-day quarantines(per million)
Figure 4:
Figure 4a displays the dynamics of quarantines for each of the policy configurations.Figure 4b 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 onboth dimensions, and the second best is the single-jurisdiction myopic strategy (which does worsethan the global because of leakage). With 40 jurisdictions, both proactive policies outperform theinternal, myopic policies. By far the worst, on both dimensions, is the internal, myopic policy withsome lax jurisdictions. These results come from the same solutions that produce figures 2 and 3. ISCUSSION
We have shown that regional quarantine policies are likely to fail unless leakage and delays indetection are limited. Multiple jurisdictions using independent policies are even less effective,as leakage occurs across jurisdictional borders. We have also shown that there are substantialimprovements from proactive policies, and that a few lax jurisdictions greatly worsen theoutcomes for all jurisdictions.Jurisdictional policies tend to be aimed at the welfare of their internal populations, yetthe external effects are large. Our results underscore the importance of timely informationsharing and coordination in both the design and execution of policies across jurisdictionalboundaries [39]. The results also underscore the global importance of aiding poor jurisdic-tions. Indeed, there is mounting evidence that a lack of coordination across boundaries hasbeen damaging in the case of COVID-19 [6].The use of masks (decreasing p ), social distancing (decreasing d ), and increasing testing(increasing α ), all help attenuate contagion, but unless they maintain the reproductionnumber below one, the problems identified here remain. Even tiny fractions of interactionsacross boundaries are enough to lead to spreading in large populations. With modern inter-and intranational trade being a sizable portion of all economies, such interaction is difficultto avoid. Nonetheless, our analysis offers insights into managing infections at smaller scales;e.g., within schools, sports, and businesses. By creating a network of interactions that ishighly modular, keeping cross-modular interactions to a minimum and making sure that theyare highly traceable, together with aggressive testing (especially of cross-module actors), onecan come close to satisfying the conditions of our first theorem.Our results also suggest caution in using statistical models to identify regions to quaran-tine. Although contagion models are helpful for informing policy about the magnitude of anepidemic and broad dynamics, the models can give false comfort in our ability to engage inhighly targeted policies, whose results can be influenced by small deviations from idealizedassumptions. Our growth-balance condition also points out that not all parts of a networkare equal in their potential for undetected transmission. In places where the reproductionnumber is lower, so is the probability of observing outbreaks, enabling undetected leakageof infections. References [1] Hardy, A. Cholera, quarantine and the English preventive system, 1850–1895.
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Interacting Regional Policies in Containing a Diseaseby Chandrasekhar, Goldsmith-Pinkham, Jackson, Thau
A Modeling An Epidemic and Quarantine Policy
The Model
People and Interactions
There are n > t ∈ N . An initial infected node, indexed by i ∈ V , is the only node infected attime 0. We call this node the seed .We track the network via neighborhoods that expand outwards via (directed) paths from i . Let N k be all the nodes who are at (directed) distance k from node i . We use n k todenote the cardinality of N k .For any node in j ∈ N k (cid:48) , for k (cid:48) < k , let n j be the number of its direct descendants and n jk be the number of its (possibly indirect) descendants in N k that are reached by never passingbeyond distance k from i .All unweighted network models are admitted here. Additionally, all results extend di-rectly to any weighted model in which weights are bounded above and below (e.g., proba-bilities of interaction). Note also, that the network can be directed or undirected.The infection process proceeds as follows. In every time period t ∈ { , , . . . } , an infectednode i transmits the disease to each of i ’s neighbors independently with probability p . Anewly infected node is infectious for θ ≥ delay in the ability to detect the disease. The number of periods ofdelay is given by τ with 0 ≤ τ ≤ θ . Delay is a general term that can capture many things.For example, it can correspond to (a) asymptomatic infectiousness, (b) a delay in accessinghealth care given the onset of an infectious period, (c) any delay in the administration oftesting, and so on.In the first period of an infected node’s infectious period – after delay – there is a prob-ability α that the policymaker detects it as being infected. So, potential detection happensexactly once during the first period in which the node can be detected. Detection is indepen-dently and identically distributed. Our results are easily extended to have a random periodfor detection after the delay.Finally, the policymaker may face some error in their knowledge of the network. Thiscan come from their inability to enforce exactly the interactions they wish to allow or limit,this can come from random variation in data collected to estimate interaction networks, or13his can come from misspecification. If there is error, we will track a share (cid:15) of nodes thatare within a k -neighborhood of the seed but are estimated by the policymaker to be outsidethe k -neighborhood. Regional Quarantine Policy
Let regional policy of distance k and threshold x be such that once there are at least x infections (other than the seed) detected within distance k from the initial seed, then allnodes within distance k + 1 of i are quarantined for at least θ periods. A quarantine impliesall connections between nodes are severed to avoid any further transmission and the infectionwaits out its duration θ and dies out.Implicit in this definition is that a quarantine is not instantaneous, but that infectedpeople could have infected their friends before being shut down, which is why the nodesat distance k + 1 are quarantined. All the results below extend if we assume that it isinstantaneous, but with quarantines moved back one step and path lengths in definitionscorrespondingly adjusted.We have assumed the policymaker knows the “seed,” for simplicity - and which may takesome time in reality. This provides an advantage to the policymaker, but we see substantial.containment failures despite this advantage. Growth Balance
In order to conduct asymptotic analysis, a useful device to study the probabilities of eventsin question in large networks, we study a sequence of networks G ( n ) with n → ∞ andan associated sequence of parameters ( α, p, τ, θ, k ) = ( α ( n ) , p ( n ) , τ ( n ) , θ ( n ) , k ( n )). In whatfollows when we drop the index n , and it is implied unless otherwise stated.Consider a network and a distance k from the initially infected node i . A path ofpotential infection to k + 2 is a sequence of nodes i , i , . . . i (cid:96) with i (cid:96) ∈ N k +1 , i j +1 being adirect descendant of i j for each j ∈ { , . . . , (cid:96) − } , and for which i (cid:96) has a descendant in N k +2 .Consider a sequence of networks and k ( n )s. We say that there are bounded paths ofpotential infection to k ( n ) + 2 if there exists some finite M and for each n there is a pathof potential infection to k ( n ) + 2, i , i , . . . i (cid:96) of length less than M , with n j < M for every j ∈ { , . . . , (cid:96) − } .We say that a sequence of networks is growth-balanced relative to some k ( n ) if there areno bounded paths of potential infection to k ( n ) + 2.Growth balance is essentially a condition that requires a minimum bound of expansionalong all paths from some initial infection. The intuition behind the condition is clear: inorder to be sure to detect an infection, within distance k of the seed, it has to be that many ofthe nodes within distance k have been exposed to the disease by the time it reaches distance k . What is ruled out is a relatively short path that gets directly to that distance without14aving many nodes be exposed along that path. Supplementary Figure C.1 presents an illustration of a network that is not growth-balanced.
Supplementary Figure C.1: Growth Balance (a)
Regional Policy Fails (b)
Regional Policy Succeeds
Figure C.1:
Panel (a) demonstrates the possible failure of growth-balance. The infection escapesup the line undetected beyond the quarantine radius. If the infection happens to spread downwards,as in Panel (b), it is much more likely to be detected. However, that only happens with somemoderate probability in this network, and so growth balance fails.
Results
A Benchmark: No Delay in Detection; Perfect Information and Enforcement
We begin with a benchmark case in which there is no delay in detection ( τ = 0) and thepolicymaker can completely enforce a quarantine at some distance k + 1. We allow the size of the quarantine region k to depend on n in any way, as the theoremstill applies. We work with an arbitrary but fixed threshold x , in order to allow infectionsto be detected. What is important is that x not grow too rapidly, as otherwise there is nochance of observing that many infections within some distance of the seed. This is very different from conditions that concern long paths within short distances, such as [40], asours is ruling out short paths with low expansion. Note that this requires knowledge of the neighborhood structure around the seed node, but no otherknowledge of the network by a policy maker. The theorem extends to allow x to grow with n , provided the growth is sufficiently slow, and then that heorem Consider any sequence of networks and associated k ( n ) < K ( n ) − where K ( n ) is the maximum k for which n k > ; , such that each node in N k ( n )+1 has at least onedescendent at distance k ( n ) + 2 , and let x be any fixed positive integer. Let the sequence ofassociated diseases have α ( n ) and p ( n ) bounded away from 0 and 1, no delay in detection,and any θ ( n ) ≥ . A regional quarantining policy of distance k ( n ) and threshold x halts allinfections past distance k ( n ) + 1 with a probability tending to 1 if and only if the sequence isgrowth-balanced with respect to k ( n ) . Note that the growth-balance condition implies that the number of nodes within distance k ( n ) from i must be growing without bound. Theorem 1 thus implies that in order for aregional policy to work, the region must be growing without bound, and also must satisfy aparticular balance condition. Proof of Theorem 1.
To prove the first part, note that if the infection never reaches distance k then the result holds directly since it can then not go beyond k + 1. We show that if thesequence of networks is growth-balanced relative to k , then conditional upon an infectionreaching level k with the possibility of reaching k + 2 within two periods, the probabilitythat it infects more than x nodes within distance k before any nodes beyond k tends to1. Suppose that infection reaches some node at distance k that can reach a node in N k +1 .Consider the corresponding sequence of paths of infected nodes i , i , . . . i (cid:96) with i (cid:96) ∈ N k +1 , i j +1 being a direct descendant of i j for each j ∈ { , . . . , (cid:96) − } , and note that by assumption i (cid:96) has a descendant in N k +2 . By the growth-balance condition, for any M , there is a largeenough n for which either the length of the path is longer than M or else there is at leastone i j with j ≤ (cid:96) − M descendants. In the latter case,the probability that i j has more than x descendants who become infected and are detectedis at least 1 − F M,m ( x ) where F M,m is the binomial distribution with M draws each withprobability m , where pα > m for some fixed m . Given that x and m are fixed, this tends toprobability 1 as M grows. In the former case, the sequence exceeds length M , all of whichare infected and so given that α is bounded below, the probability that at least x of themare detected goes to 1 as M grows. In both cases, as n grows, the minimal M across suchpaths of potential infection to k + 1 grows without bound, and so the probability that thereare at least x infections that are detected by the time that i (cid:96) − is reached tends to 1 as n grows.To prove the converse, suppose that the network is not growth-balanced. Consider asequence of bounded paths of potential infection to k + 2, with associated sequences of nodes i , i , . . . i (cid:96) of length less than M with i (cid:96) ∈ N k +1 , i j +1 being a direct descendant of i j foreach j ∈ { , . . . , (cid:96) − } , with n j < M for every j ∈ { , . . . , (cid:96) − } , and for which i (cid:96) has adescendant in N k +2 . The probability that each of the nodes i , . . . i (cid:96) − becomes infected and growth-balance condition becomes more complicated, as the M in that definition adjusts with the rate ofgrowth of x . Otherwise, it is actually a global policy. The cases of p or α equal to 1 are degenerate.
16o other nodes are infected within distance k −
1, and that all infected nodes are undetectedis at least ( p (1 − α )(1 − p ) M ) M . This is fixed and so bounded away from 0. This implies thatprobability that the infection gets to nodes at distance k , and i (cid:96) − in particular, withoutany detections is bounded below. Thus, there is a probability bounded below of reaching i (cid:96) before any detections, and then by the time the quarantine is enacted, there is at leasta p times this probability that it escapes past N k +1 , which is thus also bounded away from0. We note that Theorem 1 admits essentially all sequences of (unweighted) networks. Thus,for every type of network, one can determine whether a regional policy of some k, x willsucceed or fail. The only thing that one needs to check is growth-balance. If it is satisfied,a regional policy works, and otherwise it will fail with nontrivial probability.The following corollary details the implications of the theorem for some prominent ran-dom network models. Corollary
1. For a sequence of block models (with Erdos-Renyi as a special case), a regional policywith a bounded k has a probability going to 1 of halting the disease on the randomlyrealized network if and only if the seed node’s expected out degree d is such that d k → ∞ .2. For a regular expander graph with outdegree d , a regional policy works if and only ifthe expansion rate d k → ∞ .3. For a regular lattice of degree d , a regional policy works if and only if d k → ∞ .4. For a rewired lattice with a fraction links that are randomly rewired, a regional policywith a bounded k has a probability going to 1 of halting the disease on the randomlyrealized network if and only if d k → ∞ .5. For a sequence of random networks with a scale-free degree distribution, a regionalpolicy works (with probability 1) if and only if k → ∞ . Thus, whether a regional policy works in almost any network model requires that eitherthe degree of almost all nodes grows without bound, or else the size of the quarantine growswithout bound. For a scale free distribution, there is always a nontrivial probability on smalldegrees, and hence in order for a regional policy to work, the size of the neighborhood mustgrow without bound.In practice, even very sparse networks will have a large d k (e.g., if people have hundredsof contacts, 100 is already a million and even with a very low α many infections will Consider a sequence of block models such that the ratio of expected out degree of a node in one neigh-borhood compared to another in some other block cannot grow without bound.
17e detected within a few steps of the initial node). What the growth-balance conditionrules out is that some nontrivial part of the network have neighborhoods with many fewercontacts - so there cannot be people who have just a few contacts, since that will allow fora nontrivial probability of undetected escape (e.g., 2 = 8 and so with only 8 infections, itis possible that none are detected and the disease escapes beyond 3 steps). As many real-world network structures have substantial heterogeneity, with some people having very lownumbers of interactions, such an escape becomes possible even under idealized assumptionsof no delay in detection and no leakage [41, 42, 43, 44, 45]. Delay in Detection
The detection delay, τ , is distributed over the support { , . . . , τ max } . This includes degener-ate distributions with τ max being the maximal value of the support with positive mass. Thepolicymaker may or may not know τ max and we study both cases. The latter is important asin practice we estimate delay periods so there is bound to be uncertainty. When τ is known,we can simply say τ = τ max .Let a regional policy with trigger k and threshold x and buffer h be such that once thereare at least x infections detected within distance k + h from the initial seed, then all nodeswithin distance k + h + 1 of i are quarantined/locked down for at least θ periods.There are two differences between this definition of regional policy from the one consideredbefore. First, it is triggered by infections within distance k + h (not within distance k ), andit also has a buffer in how far the quarantine extends beyond the k -th neighborhood.We extend the definition of growth balance to account for buffers.Consider a network and a distance k from the initially infected node i and an h ≥
1. A path of potential infection to k + h + 2 is a sequence of nodes i , i , . . . i (cid:96) with i (cid:96) ∈ N k + h +1 , i j +1 being a direct descendant of i j for each j ∈ { , . . . , (cid:96) − } .Consider a sequence of networks, n , and associated k ( n ) , h ( n ). We say that there are bounded paths of potential infection to k ( n ) + h ( n ) + 2 if there exists some finite M andfor each n there is a path of potential infection to k + h + 2, i , i , . . . i (cid:96) of length less than M , with n j < M for every j ∈ { , . . . , (cid:96) − h − } . We say that a sequence of networks is growth-balanced relative to some k ( n ) and buffers h ( n ) if there are no bounded paths ofpotential infection to k ( n ) + h ( n ) + 2. Theorem Consider any sequence of networks and k ( n ) < K ( n ) − h − where K ( n ) isthe maximum k for which n k > , such that each node in N k (cid:48) for k (cid:48) > k has at least onedescendent at distance k (cid:48) + 1 , and let x be any fixed positive integer. Let the sequence ofassociated diseases have α ( n ) and p ( n ) bounded away from 0 and 1, θ ( n ) ≥ , and have adetection delay distributed over some set { , . . . , τ max } with τ max > (with probability on τ max bounded away from 0). A regional policy with trigger k ( n ) , threshold x , and buffer This is still extremely sparse, as having 100 contacts out of millions or billions of potential other nodesis a small fraction. A special case is in which τ max is known. max halts all infections past distance k ( n ) + τ max +1 with a probability tending to 1 if andonly if the sequence is growth-balanced with respect to k ( n ) . The Proof of Theorem 2 is a straightforward extension of the previous proof and so it isomitted.This result shows several things. First, if the detection delay is small relative to thediameter of the graph, one can use a regional quarantine policy – adjusted for the detectiondelay – along the lines of that from Theorem 1 and ensure no further spread. This is trueeven if the period is stochastic as long as the upper bound is known to be small.Second, and in contrast, if the detection delay is large compared to the diameter of thegraph, then a regional policy is insufficient. By the time infections are observed, it is toolate to quarantine a subset of the graph. This condition will tend to bind in the case of realworld networks, as they exhibit small world properties and have small diameters [30, 31].As a result, even short detection delays may correspond to rapidly moving wavefronts thatspread undetected.
Leakage in the Quarantine
Next we turn to the case of in which there is some leakage in the quarantine, which maycome for a variety of reasons. The policymaker may have measurement error in knowing thenetwork structure of the network and who should be quarantined. Second, and distinctly,the policymaker may be unable to control some nodes or interactions. Third, the networkmay leak across jurisdictions and some nodes within distance k of i may be outside of thepolicymaker’s jurisdiction.To keep the analysis uncluttered, we assume no detection delay, but the arguments extenddirectly to the delay case with the appropriate buffer. Theorem Consider any sequence of networks. Let the sequence of associated diseaseshave α and p bounded away from 0 and 1, and be such that θ ≥ , with no detection delay.Consider any k ( n ) < K − where K is the maximum k for which n k > , suppose thateach node in N k ( n ) has at least one descendent at distance k ( n ) + 1 , and let x be any positiveinteger.Suppose that a random share of ε n of nodes within distance k of i are not included ina regional quarantine policy and connected to nodes of distance greater than k + 1 – becauseof a lack of jurisdiction, misclassification by a policymaker, or lack of complete control overpeople’s behaviors. Then:1. If ε n = o (( (cid:80) k (cid:48) ≤ k n k (cid:48) ) − ) and the network is growth-balanced, then a regional policy ofdistance k and threshold x halts all infections past distance k + 1 with a probabilitytending to 1.2. If ε n ≥ min[1 /x, η ] for all n for some η > or the network is not growth-balanced,then a regional policy of distance k ( n ) and threshold x fails to halt all infections pastdistance k ( n ) + 1 with a probability bounded away from 0. roof of Theorem 3. Part 1 follows from the fact that if ε n = o (( (cid:80) k (cid:48) ≤ k n k (cid:48) ) − ) then theprobability of having all nodes in N k correctly identified as being in N k tends to 1, and thenTheorem 1 can be applied.For Part 2, suppose that some x infections are detected. The probability that at leastone of them is misclassified is at least 1 − (1 − ε n ) x . Given that ε n ≥ min[1 /x, η ] for any η >
0, it follows that (1 − ε n ) x is bounded away from 1. There is a probability boundedaway from 0 that at least one of the infected nodes is misclassified, and not subject to thequarantine, and connected to a node outside of distance k + 1.The theorem implies that the effectiveness of a regional policy is sensitive to any smallfixed ε amount of leakage. 20 Simulation Details
To illustrate the processes described in the main text, we run several simulations. First,we construct a large network with many jurisdictions. We directly study the content ofthe theorems with several versions of ( k, x ) quarantines with an SIR infection process on anetwork. We use the same process and network to show the issues with regional containment,studying regional and adaptive policies.
Network Model
We model real world network structure as follows.1. There are L locations distributed uniformly at random on the unit sphere.Each location has a population of m nodes with a total of n = mL nodes in thenetwork.2. The linking rates across locations are given as in a spatial model [41, 48]. The proba-bility of nodes i ∈ (cid:96) and j ∈ (cid:96) (cid:48) for locations (cid:96) (cid:54) = (cid:96) (cid:48) linking depends only on the locationsof the two nodes and declines in distance: q (cid:96),(cid:96) (cid:48) = exp( a + b · dist( (cid:96), (cid:96) (cid:48) ))where dist( (cid:96), (cid:96) (cid:48) ) is the distance between the two locations on the sphere and a, b < d RGG as the desired degree from the the RGG. Nodes are uniformly distributedon the unit square [0 , , and links are formed between nodes within radius r (cid:96) [42]. Toobtain the desired degree we set: r (cid:96) = (cid:114) d RGG m (cid:96) π . The remaining links within location are drawn identically and independently withprobability π = d (cid:96) − d RGG m (cid:96) d (cid:96) is the desired average degree for all nodes within location (cid:96) .4. Next, we uniformly add links to create a small world effect, with identical and inde-pendently distributed probability s = cn , where c is an arbitrary constant and n is thetotal number of nodes in the network [29].5. Finally, we designate a single location as a “hub,” to emulate the idea that certainmetro areas may have more connections to all other regions. To do so, we select ahub uniformly at random and add links independently and identically distributed withprobability h from the hub location to every other location.We first take L = 40 and m = 3500 for all locations. We set a = − b = −
15. Next,we set d (cid:96) = 15 .
5, and d RGG = 13 . c = 2. Finally, we set h = 2 . × − . This process results in a graph that emulates real world networks in theUnited States and India [33, 34, 35, 21]. This includes data from India during the COVID-19lockdowns about interactions within six feet, meaning that it is conservative [21].We fix a graph to use in all versions of the simulations. The network we generate issparse, clustered, and has small average distances, as in real world data. Supplementary Table 1: Graph Statistics
Property ValueAverage Degree 20.49Average Local Clustering Coefficient 0.208Diameter 9Average Path Length 5.33Finally, we recalculate the connection probability matrix to accurately reflect rates ofconnection across regions, which we call q . Disease Process
We set parameters as follows: the duration of infection is θ = 5, detection delay (whenincorporated) is τ = 3, and set thresholds x for quarantine based on the simulation type.We set transmission probability p as p = 1 − (cid:18) − R ¯ d (cid:19) θ where ¯ d is the mean degree. We take R = 3 .
5, based on estimates of COVID-19 [36].Cases are detected i.i.d. with rate α , which we define as α = P (symptomatic) · P (Tested) · P (Test Positive | Truly Positive)22e take the symptomatic rate as 43.2% [56], and the power of the test as 79% [57].Following estimates from the literature (5-15%), we set α = 0 . τ = 0, anydetection occurs as soon as they are infected and when τ > τ + 1thperiod of infection.As outlined in the main text, we begin by using θ = 5 and τ = 3 [20, 37, 38, 61]. Simulation Progression
Each time period in the simulation progresses in four parts, which happen sequentially. Thesimulations run as follows:1. The policy maker sees the detected infections from the previous period, and calculatesif a quarantine is necessary in the next period.2. The disease progresses for a period. This includes new infections and recoveries.3. Infected nodes that have just finished their detection delay of τ periods are indepen-dently detected with probability α .4. New quarantines are enacted based on decisions made in part one of the process.Quarantines that have taken place for θ periods end.A node that becomes infected in period t with a detection delay of τ and total diseaselength θ , is tested in period t + τ , results are processed in t + τ + 1, and they will bequarantined (if necessary) starting at the end of t + τ + 1 (under the fourth item above).This means that they have τ + 1 time periods during which they can infect other nodes. Forinstance, if τ = 0 this allows a node that becomes infected but (that was not already underquarantine for other reasons) one opportunity to infect others. This process reflects thatneither detection nor quarantining of individuals (or jurisdictions) happens instantaneously.In addition, we stipulate that the seed node, i is not counted in the quarantining testing andcalculations. This is meant to reflect that it may be unclear whether the disease is spreadingor not. Nodes that are detected are marked as such until recovery. Containment Policies
A random node i is selected and the epidemic begins there. We study the epidemic curve,the number total node-days of infection, and the number of node-days of quarantine for avariety of containment strategies. 23 k, x ) Policies
We examine a number of scenarios using the ( k, x ) policy model outlined in Theorems 1-3.In the case that the quarantine fails, but there are infections outside of the quarantineradius, the policy maker deals with them individually. The policy maker treats each detectedcase outside of the initial quarantine as a new seed, and immediately quarantines all nodeswith the same radius as the initial quarantine.Begin by using a simple objective function to find the optimal threshold for triggeringthe initial quarantine. We minimize a linear combination of the number of infected personperiods and quarantined person periods. For all linear combinations where some weight isgiven to both terms, the optimal threshold is x = 1. The logic is as follows: if the initialquarantine is successful, the number of quarantined person periods will be fixed and alsothe minimum number of quarantined person periods. Therefore, the problem reduces tominimizing the number of infections, which is done by setting x = 1.We study three versions of a ( k, x ) policy. First, we simulate ( k, x ) = (3 ,
1) with nodetection delay. Then, we incorporate a detection delay of τ = 3, still using a policy of( k, x ) = (3 ,
1) with no buffer. Lastly, we study a (3 ,
1) policy with no buffer and enforcementfailures. In this case, a fraction (cid:15) = 0 .
05 of nodes do not ever quarantine.
Global Quarantine Policy
A global quarantine policy imagines the state as an actor which quarantines every node for θ periods when more than x = 1 infections are detected globally. We study this in the casewith a detection delay, to compare to the ( k, x ), regional, jurisdiction based, and proactivepolicies. Myopic-Internal and Proactive Quarantine Policies
For both the myopic-internal and proactive policies, we take each location as a single juris-diction.
Myopic Internal Quarantine Policies.
Jurisdictions respond only to detections withintheir own borders, setting x independently of one another. In addition, states act indepen-dently: jurisdictions do not take detected infections outside of their borders into account.We set x = 1 for all jurisdictions, the most conservative possible threshold unless otherwisespecified. Proactive Quarantine Policies.
We examine a more sophisticated approach to decidingwhen to quarantine. With this policy, each jurisdiction decides to quarantine based on notonly defections within their borders, but within neighboring jurisdictions as well. In eachperiod, each jurisdiction (cid:96) calculates their expected detected infections w (cid:96) as follows:24 (cid:96),t = max { w (cid:96),t − + y (cid:96),t − r (cid:96),t , z (cid:96),t } We use y (cid:96),t to denote the number of expected new infections in region (cid:96) at time t , anduse r (cid:96),t to denote the number of expected recoveries in (cid:96) at t . Each state calculates y (cid:96),t as: y (cid:96),t = p (cid:88) (cid:96) (cid:48) s.t. (cid:96) (cid:48) not quarantined at t-1 m (cid:96) (cid:48) q (cid:96),(cid:96) (cid:48) w (cid:96) (cid:48) ,t − The summation includes the term for spread from (cid:96) to still within (cid:96) . If (cid:96) is quarantinedat time t , then y (cid:96),t = 0. Expected recovery at each period r (cid:96),t is calculated as: r (cid:96),t = w (cid:96),t − θ − w (cid:96),t − θ − + r (cid:96),t − θ . Finally, we set w (cid:96),t < .
01 to be zero, to avoid implementation issues with floating pointcalculations. Setting a lower value to truncate at would improve the performance of theproactive jurisdiction policies, as they would be more sensitive to detected cases in otherjurisdictions.
Uniform and Lax Policies
We run two simulation variants for both the proactive andinternally-based policy: one in which all states are as conservative as possible, setting x = 1and a second in which four regions set a higher threshold of x = 5; in the proactive case,these lax regions also act myopically, following the internal jurisdiction-based policy.We choose x = 5 to simulate lax thresholds. In the United States, New York stateissued a stay at home order when 0.07% of the state population was infected, which scaledto our populations of 3500 that is equivalent to a threshold of 2.73 [62, 63]. When scaledto match our population of 3500, Florida began re-opening with a threshold of 6.15, andsome countries never locked down [64, 63, 65]. In our stylized model, quarantines are moreaggressive as they cut contact completely. Results and Sensitivity Analysis
We include the results of the simulations detailed in the main text in the tables below. Inaddition, we run simulations with two sets of varied parameters: first, we take α = 0 . θ = 8 and τ = 5. Within the United States, estimates for the detectionrate range from 5% to 15%, and in countries with less developed testing infrastructure, thedetection rate is undoubtedly lower [37]. Because disease parameters are estimated, we usea different estimated of the disease lifespan of COVID-19 [61]. For all simulations, we fix R = 3 .
5. 25 upplementary Table 2: Regional Policy Simulation Results θ τ α (cid:15)
PercentInfected InfectionPerson Days QuarantinedPerson Days EscapeRate5 0 0.1 0 0.0276 1384.05 803955.61 0.09535 3 0.1 0 0.226 11282.19 2301413.60 0.4585 3 0.1 0.05 0.514 25688.08 6478054.64 0.5515 0 0.05 0 0.0684 3421.10 11231131.73 0.2255 3 0.05 0 2.81 140667.17 20297075.03 0.6235 3 0.05 0.05 7.80 390155.83 66067046.93 0.7068 0 0.1 0 0.0277 2213.92 1243574.65 0.09048 5 0.1 0 0.285 22834.58 4187189.53 0.5068 5 0.1 0.05 0.559 44709.41 10653981.92 0.582
Results for the parameters used in the main text are the average over 10000 simulations. Results forthe parameters only used in this section are the average over 2500 simulations. For all simulations,we set k = 3 and x = 1. Infection person days and quarantined person days are scaled to be permillion individuals. The escape rate is defined as the frequency with which the disease escapes theinitial quarantine. Supplementary Table 3: Global Policy Simulation Results θ τ α PercentInfected InfectionPerson Days QuarantinedPerson Days5 3 0.1 0.0456 2278.84 4725000.005 3 0.05 0.0804 4019.67 4698000.008 5 0.1 0.0489 3914.26 7507200.00
Results for the parameters used in the main text are the average over 10000 simulations. Results forthe parameters only used in this section are the average over 2500 simulations. Infection person daysand quarantined person days are scaled to be per million individuals. There are fewer quarantinedperson days on average with α = 0 .
05, rather than α = 0 . upplementary Table 4: Internal and Proactive Policy Simulation Results Policy θ τ α
PercentInfected InfectionPerson Days QuarantinedPerson Days FractionRequarantinedInternal 5 3 0.1 11.85 592234.08 65634600.00 0.674Proactive 5 3 0.1 0.630 31501.43 37816130.00 0.569Internal 5 3 0.05 30.61 1530914.46 133060800.00 0.804Proactive 5 3 0.05 1.93 96626.34 56930840.00 0.719Internal 8 5 0.1 10.13 810885.33 87270880.00 0.684Proactive 8 5 0.1 0.792 63328.75 19992240.00 0.568
Results for the parameters used in the main text are the average over 10000 simulations. Results forthe parameters only used in this section are the average over 2500 simulations. For all simulations,every jurisdiction sets x = 1. Infection person days and quarantined person days are scaled to beper million individuals.Supplementary Table 5: Internal and Proactive Policies with Lax Jurisdictions Simulation Results Policy θ τ α
PercentInfected InfectionPerson Days QuarantinedPerson Days FractionRequarantined Low ThresholdCase FractionInternal 5 3 0.1 20.94 1046945.43 102106475.00 0.728 0.842Proactive 5 3 0.1 2.73 136557.93 41470367.50 0.674 0.734Internal 5 3 0.05 34.02 1700815.97 117682100.00 0.822 0.852Proactive 5 3 0.05 11.64 582078.59 101899270.00 0.806 0.720Internal 8 5 0.1 19.65 1572688.84 155807360.00 0.744 0.840Proactive 8 5 0.1 3.23 258570.58 49832880.00 0.677 0.763
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