Optimising the mitigation of epidemic spreading through targeted adoption of contact tracing apps
Aleix Bassolas, Andrea Santoro, Sandro Sousa, Silvia Rognone, Vincenzo Nicosia
OOptimising the mitigation of epidemic spreading through targeted adoption of contacttracing apps
Aleix Bassolas, ∗ Andrea Santoro, ∗ Sandro Sousa, Silvia Rognone, and Vincenzo Nicosia School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom (Dated: February 26, 2021)The ongoing COVID-19 pandemic is the first epidemic in human history in which digital contact-tracing has been deployed at a global scale. Tracking and quarantining all the contacts of individualswho test positive to a virus can help slowing-down an epidemic, but the impact of contact-tracingis severely limited by the generally low adoption of contact-tracing apps in the population. Wederive here an analytical expression for the effectiveness of contact-tracing app installation strategiesin a SIR model on a given contact graph. We propose a decentralised heuristic to improve theeffectiveness of contact tracing under fixed adoption rates, which targets a set of individuals toinstall contact-tracing apps, and can be easily implemented. Simulations on a large number of real-world contact networks confirm that this heuristic represents a feasible alternative to the currentstate of the art.
Keywords: SIR, contact tracing, optimal mitigation, dynamics on networks, distributed systems
Since the first human infection towards the end of2019, the spread of the SARS-COV-2 virus has causedan unprecedented shock around the world, with seriousrepercussions in all aspects of our social and economicactivities [1–3], and a number of casualties that has al-ready passed the two millions figure and is unfortunatelydue to rise further in the near future [4]. The initialefforts to curb the spread of the disease focused on non-pharmaceutical interventions including travel bans, lock-downs, and curfews[5]. These measures are able to dras-tically reduce the opportunities of contacts between in-fected and susceptible people and thus the spread of avirus [6–10], but also have non-negligible effects on theeconomy and social life [11–13]. After the first wave ofinfections in February-May 2020 and thanks to a betterunderstanding of the specific transmission dynamics ofSARS-COV-2 [14–19], many countries have implementedsome sort of “test-trace-treat” system based on digitalcontact tracing [20, 21]. Some of these systems consiston deploying contact tracing (CT) apps on mobile phoneswhich allow to identify and isolate individuals who havebeen in contact with infected ones, thus disrupting sec-ondary infections paths as early as possible. With con-tact tracing in place, many countries have been able topartially re-open several sectors of their economy and todiminish the damage of prolonged disruptions [22–29]An effective digital contact tracing strategy should aimat maximising the probability of detecting contacts be-tween infected and susceptible individuals, and it wouldcompletely eradicate contagion in the ideal case whereCT apps are installed by the totality of a population [30–39]. However, throughout the SARS-COV-2 pandemicthe percentage of the population with CT apps installedhas remained quite low, between 5% and 20% in mostcountries [40], resulting in a dramatically decreased effi-ciency of contact-tracing. ∗ These authors contributed equally to this paper
Here we focus on the problem of determining the set ofnodes which should install CT apps in order to optimisethe effect of contact tracing, i.e., to maximally slow-downspreading and reduce the incidence of a disease, under theassumption that the rate of CT app adoption is fixed. Weprovide an analytic derivation to quantify the decrease ofthe basic reproduction number caused by a generic CTinstallation strategy, and we show that uniform randominstallation – which is the strategy implicitly adopted bygovernments when people are simply asked to install aCT app – has the worst performance of all. We findthat relatively simple targeting strategies based on thestructure of the contact network are significantly moreefficient in reducing the number of secondary infectionsat low adoption rates, in both synthetic and real-worldsystems.
RESULTS
In Fig. 1 we report a sketch of a fictious contact net-work, where some individuals are infected (pink), someother are susceptible (black), and some have a CT appinstalled (indicated by the mobile icon). A perfect lock-down would remove almost all the links in that graph(with the only exception of those among people belong-ing to the same household), so that infected individualswill eventually be unable to find any susceptible personto pass the disease on. When only contact tracing is inplace, instead, some infections are still unavoidable, ei-ther due to a limited app adoption rate or to a delay inthe notification of test results and in isolation of subjectsexposed to infected ones. As made evident by Fig. 1,maximising the impact of contact tracing corresponds tomaximising the probability that the potential transmis-sion of the disease between two individuals is detected,since only contacts among people with CT apps installedcan be detected and traced back. This intuitively cor-responds to maximising the number of edges among the a r X i v : . [ phy s i c s . s o c - ph ] F e b SusceptibleInfected
Figure 1.
Effect of contact-tracing apps on secondaryinfections on a contact network.
Contact tracing appscan only detect potential contagious contacts if both the in-fected and the susceptible individual have a CT app installed.In this case, A can in principle infect any of his direct neigh-bours, including B who has a CT app installed as well, andthe CT app cannot do anything to avoid this. However, if theCT system detects a contact between A and B and then Atests positive, then B can be contacted and put into quaran-tine, thus disrupting all the potential infection paths to thedirect contacts of B (solid green lines). If C has a contactwith A, instead, C has no way of knowing whether the con-tact resulted in an infection or not, he will not be notifiedwhen A tests positive, and he will not go into quarantine.In this case, all the neighbours of C are at risk of catchingthe disease (solid red lines). Similarly, the fact that B has aCT app installed cannot safeguard her from being infected byD (who does not have a CT app installed) and passing theinfection to her neighbours while she has no symptoms. individuals with CT app installed, i.e., the density of thesubgraph induced by the nodes with CT apps, under theconstraint that only a fraction r of the population willhave the CT app installed. Reduction of R in a SIR+CT dynamics. Weconsider here a SIR+CT model, which is a classicalSusceptible-Infected-Recovered (SIR) model on a staticcontact graph [10], with the addition of ideal contacttracing. This means that any susceptible node with in-stalled CT app is quarantined (recovered) as soon as oneof their contacts with CT app installed gets infected. Theparameters of the model are the probability β that an in-fected individual passes the disease to each of its suscep-tible neighbours, and the probability µ that an infectedindividual is removed (due to either recovery or death).We call the contact graph G ( V, E ), with N = | V | nodesand K = | E | edges, and we denote by G (cid:48) ( V (cid:48) , E (cid:48) ) the sub-graph of G induced by CT app installations, i.e., suchthat V (cid:48) is the set of nodes in G with CT apps and E (cid:48) is the set of edges among nodes in V (cid:48) . We quantify theeffect of the installation of CT apps in a certain subset V (cid:48) of nodes by computing the reduction of the basic re-production number R , that is the expected number ofsecondary infections caused by a single contagion event.Let us assume that the generic node (cid:96) is infected andhas passed the disease to its neighbour i . The expectednumber R i of secondary infections caused by i while it re-mains infected depends on whether i is in V (cid:48) , and on howmany of its k i neighbours are in V (cid:48) as well. In particular,if i / ∈ V (cid:48) , R i = βµ ( k i −
1) as in the classical SIR (we have toremove (cid:96) from the count, hence the k i −
1) [10]. If i ∈ V (cid:48) ,instead, there are two possible cases: a) if (cid:96) / ∈ V (cid:48) , thecontact between (cid:96) and i remains undetected, and i canpotentially infect R i = βµ ( k i −
1) more nodes, as in theclassical SIR. b) If instead (cid:96) ∈ V (cid:48) as well, then the contactwith i gets detected by the CT system and i goes intoself-isolation immediately, thus avoiding any secondaryinfection. If we denote by k (cid:48) i the degree of node i in G (cid:48) ,the expected number of infections potentially caused bythe infection of i is equal to R i = βµ ( k i − k i − k (cid:48) i k i and the average number of secondary infections poten-tially caused by each node infected by (cid:96) is given by: R (cid:96) = βµ k (cid:96) (cid:88) i a (cid:96)i ( k i − k i − k (cid:48) i k i where a (cid:96)i are the entries of the adjacency matrix of thecontact graph G . By averaging R (cid:96) over all the nodes of G we obtain the value of the basic reproduction numberin presence of contact tracing (see Methods for details): R (cid:48) = R − N βµ (cid:80) (cid:96) k (cid:96) (cid:80) i a (cid:96)i k (cid:48) i k i ( k i −
1) (1)where R is the basic reproduction number of the classi-cal SIR dynamics on G [10]. As made clear by Eq. (1),we can minimise the value of R (cid:48) by using a generic opti-misation algorithm to compute:max G (cid:48) F ( G (cid:48) ) = (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i ( k i −
1) (2)over the ensemble of possible choices of G (cid:48) . Notice thatif the entire population installs CT apps (i.e., if k (cid:48) i = k i ∀ i ∈ V ) we trivially get R (cid:48) = 0 (see Methods andSupplementary Note 1 for details). CT targeting strategies.
If we assume that we can in-stall the contact tracing app only to a fraction r ∈ [0 , G (cid:48) nodes havinga high degree in G (so that the ratio ( k i − k i is as largeas possible), and, at the same time, a high number ofconnections to other nodes in G (cid:48) (i.e., so that k (cid:48) i is aslarge as possible). The most basic strategy to select afraction r of the N individuals to install CT apps con-sists in asking the population to install a CT app in their r R / R a r k b RND RND theo
DEG DEG theo
DTI SA r ( k ) c Figure 2.
Effect of different CT strategies on the induced subgraph as a function of app adoption rate r . (a) Theratio R (cid:48) /R as a function of the CT adoption rate r for the Random (RND), degree-based (DEG), distributed targeting (DTI),and Simulated Annealing (SA) strategies, calculated using Eq. (1). (b) Average degree of G (cid:48) as a function of the adoptionrate r for each of the strategies. The theoretical predictions for the RND and DEG strategies (solid lines) were obtained usingEq. (7) and Eq. (17), respectively. (c) Second moment of the degree distribution of G (cid:48) as a function of the adoption rate r foreach of the strategies. The plots correspond to an ensemble of configuration model graphs with degree distribution P ( k ) ∼ k − and N = 10 nodes. Results averaged over 100 realisations. mobile phones, under the assumption that each individ-ual will comply with probability equal to r , irrespectiveof any of their specific social or behavioural characteris-tics. In this case, the total number of installations willbe distributed according to a Binomial, with mean equalto rN . In the following, we call this strategy “uniformrandom installation” (RND).A second strategy consists in explicitly targeting allthe potential super-spreaders [41, 42]. In practice, weask the rN individuals with the largest number of con-tacts (links) in G to install the app, assuming that theywill all comply with probability 1. This strategy is indeedutopistic since it requires full knowledge of the contactnetwork and full compliance by the selected nodes. Inthe following, we call this strategy “degree-based instal-lation” (DEG).Here we propose and study a constructive strategy tomaximise Eq. (2) that does not require detailed globalinformation on G , and is thus amenable to a distributedimplementation. We start from a CT set that containsonly the node with the largest degree in G . Then, ateach subsequent step t we add to the CT set one of theneighbours i of any of the nodes in V (cid:48) , with probabilityproportional to the total number of neighbours of thatnode that are already in V (cid:48) (see Methods for details).This creates a “social pressure” on individuals with noCT app installed which is proportional to the numberof their contacts already in V (cid:48) . We call this strategy“distributed targeting installation” (DTI).In Fig. 2(a) we plot the ratio R (cid:48) R as a function ofthe CT adoption rate r for the RND, DEG, and DTIstrategies, on an ensemble of configuration model graphswith power-law degree distributions. As a reference, wealso report the results obtained by optimising Eq. (2)by means of Simulated Annealing (SA). It is worth not- ing that RND is the worst-performing strategy overall,characterised by a much slower decrease of R (cid:48) with r .Conversely, degree-based installation is close to the theo-retical limit established by SA, and produces a noticeabledecrease of R already for quite small values of r . Re-markably, the performance of DTI is quite close to that ofDEG, although DTI is not using any global informationabout the structure of G . In Fig. 2(b-c) we show howthe first and second moment of the degree distribution (cid:101) P ( k (cid:48) ) of the subgraph G (cid:48) vary with r for each of the fourstrategies. More details on the derivation of the full de-gree distribution of G (cid:48) in RND and DEG are reported inMethods, while Supplementary Fig. 1 shows the perfectagreement between the empirical and analytical degreedistributions for these two strategies.It is worth noting that under the DEG strategy (cid:104) k (cid:48) (cid:105) increases very sharply with r and is already quite similarto the value of (cid:104) k (cid:105) in G for very small values of r . Onthe other hand, in RND (cid:104) k (cid:48) (cid:105) increases only linearly with r (see Methods for details), while the performance of DTI isin between those two. However, these plots make it clearthat the sheer density of G (cid:48) is not the only importantingredient for CT app installation. Indeed, SA can attainconsistently lower values of R (cid:48) R than DEG, although thevalues of (cid:104) k (cid:48) (cid:105) produced by SA are almost identical to thatprovided by DEG (see Fig. 2b). SIR+CT in real-world graphs.
In Fig. 3 we show theratio of infected nodes I ( t ) for the four strategies withdifferent values of r on two real-world contact networks,respectively the network of friendship in a high school(top panels) and at a workplace (bottom panels)[43]. Inthe inset of each panel we report a typical compositionof V (cid:48) for each strategy. It is evident that, even at lowadoption rates, the DEG and DTI strategy can heavilymitigate the incidence of the disease better than RND. No AppApp r = 0.3
Figure 3.
Impact of CT strategy and adoption rate on the epidemic peak of SIR+CT in real-world networks .The evolution of the disease in a SIR+CT model (here for β = 0 . µ = 0 .
05) depends heavily on the adoption rate r and onthe strategy used to select which individuals will have a CT app installed. We show here the results on two real-world socialnetworks, namely, the high-resolution face-to-face contact data respectively recorded in a high school (a-d) and a workplace(e-h). We have applied a threshold to both contact networks: keeping only contacts larger than 240 seconds for the High Schooland 10% of the links with the largest weight for the Workplace (See Methods for details). At adoption rates r (cid:38) .
4, the RNDstrategy (a, e) displays a higher percentage of infected compared to DTI (b, f), DEG (c, g) and SA (d, h). These differences arelikely linked to the structure of the subgraph induced by each CT strategy. Typical examples of those graphs for each strategyand r = 0 . This is most probably because DEG and DTI are tar-geting different sets of nodes than RND, and in generalend up selecting nodes with high degree which result ina higher edge density in G (cid:48) .The dynamics of SIR+CT in the two systems ex-hibits some noticeable qualitative differences when dis-tinct strategies are adopted, with respect to the heightof the infection peak (the maximum incidence of the dis-ease), the actual position of the peak (the time at whichit occurs), and the overall duration of the epidemic. In-terestingly, the position of the peak shifts to the right(delays) at small values of r for the DEG, DTI and SAstrategies. Conversely, the peak starts to recede (it is an-ticipated) with respect to the baseline when r becomeslarger than a certain threshold, which depends on theparticular structure of the contact network. While lowvalues of r lead to a delay in the dynamics – the peakshifts to the right – large enough values of r effectivelybreak the network into a number of disconnected com-ponents, resulting in a considerable disruption of thespreading – shift to the left.To better understand these qualitative differences, welook at three key properties of the epidemic curve, namelythe total number of individuals recovered R ∞ , the maxi-mum number of individuals infected across the durationof the epidemic I ( t peak ) and the time to reach the in-fection peak t peak . In particular, we compute the rela-tive performance of each strategy s (being it either DEG, DTI, and SA) with respect to RND using the quantities:∆ R s ∞ = 1 − R s ∞ R RND ∞ ∆ I s ( t peak ) = 1 − I st peak I RND ( t peak) ∆ t s peak = 1 − t s peak t RNDpeak (3)The results are shown in Fig. 4. We found that in thehigh school network an adoption rate of r = 0 . r = 0 . R ∞ , I ( t peak ) and t peak .We simulated SIR+CT in 84 unique real-world con-tact network data sets, filtered by applying two differentthresholds, for a total of 168 undirected graphs [43–50](See Methods for details). In Fig. 5(a-c), we report forthe RND, DEG and DTI strategies, and several adoptionrates r , the spearman correlation between the analytical R (cid:48) and the epidemiological indicators R ∞ , I ( t peak ) and R a R RNDDEGDTISA I ( t p e a k ) b I ( t p e a k ) t p e a k c t p e a k r R d R r I ( t p e a k ) e I ( t p e a k ) r t p e a k f H i g h S c h oo l W o r k p l a c e - I n V S t p e a k Figure 4.
Comparison of epidemic indicators under different CT strategies . Relative decrease with respect to randominstallations of the total number of recovered ∆ R s ∞ , height of the infection peak ∆ I s ( t peak ) and position of the peak ∆ t s peak (see Eq. (3)) for DTI, DEG, and SA targeted installation, in the same contact networks shown in Fig. 3. The inset of eachpanel reports the plot of the raw variable, respectively R ∞ (panel a and d), I ( t peak ) (panel b and e) and t peak (panel c and f). t peak . The correlation with both R ∞ and I ( t peak ) is highfor the three strategies confirming the analytical predic-tions of Eq. (1) despite the small size of the graphs andthe presence of degree-degree correlations. Still, as r in-creases we observe a decrease in the correlations, likelydue to finite size effects. The correlation between R (cid:48) and t peak displays a much richer behaviour: we start witha significant but negative correlation for small r , whichchanges sign until it reaches a maximum. We conjec-ture that the change of sign is related to the movementof the peak: while in the small- r regime lower values of R (cid:48) contribute to a delay of the peak, for larger valuesof r we observe a stronger anticipation of the peak. Thevalue r peak at which the correlation peaks depends on thestrategy in use, the more efficient it is, the lower is thevalue of r peak . The concrete value of r peak seems thusrelated to a certain structural cutoff of the graphs.In a realistic scenario, in which the adoption rate isnot fixed but needs to be promoted, we might be moreinterested in the minimum adoption rate r ∗ needed oneach network to obtain a given reduction of the infectionpeak with respect to the absence of contact tracing. InFig. 5(d-f) we report the histograms of the value of r ∗ in DTI and RND across the 168 networks when we seta reduction in the peak I ( t peak ) of 10%, 30% and 50%,respectively. We found that DTI can achieve a reduc-tion of 30% of the peak in 85% of the networks with anadoption rate smaller than 0 . . Conclusion.
We have shown here that the random CTapp installation – considered in recent studies on thetopic [36–38] and adopted by many governments – is theless effective strategy to mitigate the effects of a pan-demic through contact tracing. The theoretical argumentpresented here, which links the reduction of R to thestructure of the subgraph G (cid:48) induced by CT, holds forany graph under any CT strategy. In particular, Eq. 1provides a concrete recipe to maximise the effectivenessof a CT app deployment as we have shown in real-worldsystems.The reduction of a disease incidence attainable by theDTI strategy is comparable with degree-based target-ing, which performs similarly to the optimal targetingobtained through simulated annealing. A notable advan-tage of using DTI over DEG is that it does not requireany global information about the graph G and it can beimplemented on a distributed manner. For instance, onecould ask every new individual who installs the CT appto broadcast a message to all his contacts, asking themto install it as well. By doing so, each contact with noapp installed will be subject to a level of “social pressure”linearly proportional to the number of contacts who al-ready have the CT app installed (in agreement with theheuristic algorithm of which DTI is based), consequentlyincreasing the likelihood that he will also install it.Although several effective vaccines have been made r Sp e a r m a n a RND RI ( t peak ) t peak r b DTI r c DEG r P ( r ) d
10% peak reduction
RNDDTI 0.0 0.2 0.4 0.6 0.8 r e
30% peak reduction r f
50% peak reduction
Figure 5.
Correlations with network structural measures and performance of CT strategies to mitigate anepidemic . For the 168 real-world contact networks analysed, panels (a-c) report as a function of r the Spearman rankcorrelation between the analytical value of R (cid:48) (Eq. (1)) and the epidemiological indicators R s ∞ (purple), I s ( t peak ) (orange) and t s peak (blue) for the RND (a), DEG (b) and DTI (c) strategies. Panels (d-f) show the distribution of minimum adoption ratios r ∗ needed to produce a 10% (d), 30% (e) and 50% (f) for the DTI (red) and RND (green) strategies. Overall, DTI largelyoutperforms RND, which is the strategy currently adopted by many governments. available recently [51], mass vaccination campaigns areat their initial stage in many countries and might lastfor several months before a sufficient percentage of thepopulation is vaccinated. Moreover, SARS-COV-2 vari-ants may develop vaccine resistance and prolong the du-ration of the epidemic, creating an unsustainable loop ofvaccine updates and vaccination campaigns. Hence, re-ducing the spreading of the virus by detecting potentialinfected individuals and limiting their contacts – throughdigital contact tracing – is still essential [52]. The resultsshown here suggest that governments could significantlyimprove the effectiveness of contact tracing programs byimplementing targeting CT app installations, not only forthe ongoing COVID-19 but any major epidemic event inthe future. METHODSDistributed Targeting Installation strategy
The proposed heuristic constructive algorithm to opti-mise Eq. (1), denominated DTI, starts with a set V (cid:48) ( t =0) containing the node of largest degree in G . At eachstep t , we consider the set S ( t ) of nodes which have atleast one neighbour in V (cid:48) ( t ), then, a node i is selected at t + 1 from S ( t ) and added to V (cid:48) ( t ) with probability: P ( i ; t ) = (cid:80) j ∈ V (cid:48) ( t ) a ij (cid:80) i ∈ S ( t ) (cid:80) j ∈ V (cid:48) ( t ) a ij (4)i.e., node i is selected linearly proportional to the numberof neighbours it has in V (cid:48) ( t ). Properties of the subgraph induced by appinstallation
We provide here a sketch of the derivation of the firsttwo moments of the degree distribution of the subgraph G (cid:48) obtained from a graph G by considering only thenodes which have the contact-tracing app installed andthe edges among them. The full derivations are providedin Supplementary Note 1. In the case of random instal-lation strategy, the probability that a node installs thecontact tracing app is uniform across all nodes. As a con-sequence, the probability that a node with degree k in G has degree k (cid:48) in G (cid:48) is given by the Binomial distribution: P rob ( k (cid:48) i = k (cid:48) | k i = k ) = (cid:18) kk (cid:48) (cid:19) r k (cid:48) (1 − r ) k − k (cid:48) . (5)This means that the expected degree in G (cid:48) of a node thathas degree k in G is just: E [ k (cid:48) i ] = rk i (6)The degree distribution of the subgraph G (cid:48) can be ob-tained by summing the probability in Eq. (5) over allpossible values of k , from which we obtain: (cid:101) P RND ( k (cid:48) ) = N − (cid:88) k =0 (cid:18) kk (cid:48) (cid:19) r k (cid:48) (1 − r ) k − k (cid:48) Finally, for the first two moments of (cid:101) P RND ( k (cid:48) ) we get: (cid:104) k (cid:105) RND = N − (cid:88) j =0 j (cid:101) P RND ( j ) = r (cid:104) k (cid:105) G (7)and: (cid:104) k (cid:105) RND = N − (cid:88) j =0 j (cid:101) P RND ( j ) = r (cid:104) k (cid:105) G + r (1 − r ) (cid:104) k (cid:105) G . (8)To compute the degree distribution of G (cid:48) for degree-based installations, we start from the observation thata node is in G (cid:48) only if its degree is k i ≥ (cid:101) k , where (cid:101) k isobtained by solving the inequality: N (cid:88) (cid:101) k P ( k ) ≥ r (9)where P ( k ) is the degree distribution of G . Now, theprobability that one of the k i neighbours of i is in G (cid:48) isequal to: Q (cid:101) k ( i ) = N − (cid:88) k = (cid:101) k P ( k | k i ) (10)where P ( k | k i ) is the conditional probability of finding in G a node with degree k by following one of the edges of anode with degree k i , chosen uniformly at random. In thespecial case of graphs with no degree-degree correlations, P ( k | k (cid:48) ) = kP ( k ) (cid:104) k (cid:105) = q k , so we have: Q (cid:101) k ( i ) = N − (cid:88) k = (cid:101) k q k = (cid:101) r ∀ i (11)In the absence of degree-degree correlations, the proba-bility of any two nodes to be connected does not dependon their degree, by definition. Hence, the probabilitythat a node of G (cid:48) has a degree equal to k (cid:48) is given againby the Binomial distribution: P rob ( k (cid:48) i = k (cid:48) | k i = k ) = (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) , k ≥ (cid:101) k (12)while P rob ( k (cid:48) i = k (cid:48) | k i = k ) = 0 if k < (cid:101) k . In particular,this means that the expected value E [ k (cid:48) i ] is equal to: E [ k (cid:48) i ] = (cid:101) rk i (13) Notice that (cid:101) r has the same role that r has in the equationsfor uniform random installation. With an argument in allsimilar to that used for random installation, we obtain: (cid:101) P ( k (cid:48) ) = N − (cid:88) k = (cid:101) k P ( k ) (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) (14)Here (cid:101) P ( k (cid:48) ) represents the probability to find a node of G which has degree k (cid:48) in the subgraph induced by appinstallations. To obtain the actual degree distribution inthe induced subgraph, i.e., the probability that one of thenodes of G (cid:48) has degree k (cid:48) , we must rescale (cid:101) P ( k (cid:48) ) to thenodes in G (cid:48) , i.e., we consider the probability distribution: (cid:101) P DEG ( k (cid:48) ) = 1 r (cid:101) P ( k (cid:48) ) (15)It is easy to show that (cid:101) P DEG ( k (cid:48) ) is correctly normalised: rN − (cid:88) k (cid:48) =0 (cid:101) P DEG ( k (cid:48) ) = 1 r rN − (cid:88) k (cid:48) =0 N − (cid:88) k = (cid:101) k P ( k ) (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) = 1 r N − (cid:88) k = (cid:101) k P ( k ) rN − (cid:88) k (cid:48) =0 (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) = 1 r N − (cid:88) k = (cid:101) k P ( k ) = 1 (16)The average degree in the induced graph is obtainedas follows: (cid:104) k (cid:105) DEG = rN − (cid:88) k (cid:48) =0 k (cid:48) (cid:101) P DEG ( k (cid:48) )= 1 r N − (cid:88) k = (cid:101) k P ( k ) rN − (cid:88) k (cid:48) =0 k (cid:48) (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) = 1 r N − (cid:88) k = (cid:101) k P ( k ) k (cid:101) r = (cid:101) rr N − (cid:88) k = (cid:101) k kP ( k ) = (cid:101) r r (cid:104) k (cid:105) (17)where we have used the fact that (cid:80) N − k = (cid:101) k kP ( k ) = (cid:101) r (cid:104) k (cid:105) as per the definition of (cid:101) r in Eq. (11). Similarly, for thesecond moment we obtain: (cid:104) k (cid:105) DEG = N − (cid:88) k (cid:48) =0 k (cid:48) (cid:101) P DEG ( k (cid:48) )= 1 r N − (cid:88) k = (cid:101) k P ( k ) rN − (cid:88) k (cid:48) =0 k (cid:48) (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) = 1 r N − (cid:88) k = (cid:101) k P ( k ) (cid:2) k (cid:101) r + k ( k − (cid:101) r (cid:3) = (cid:101) r r (1 − (cid:101) r ) (cid:104) k (cid:105) + N − (cid:88) k = (cid:101) k k P ( k ) (18) Reduction of R under ideal CT. We derive here a general expression for the effectivevalue of the expected number of secondary infectionscaused by a single infection in a graph with perfect con-tact tracing, under the assumption that a fraction r of thenodes has installed a CT app. Assuming that a genericnode (cid:96) is infected, we want to estimate what is the num-ber of secondary infections caused by a node i infectedby (cid:96) . The number of neighbours R i that can be infectedby i depends on whether i has the CT app installed, andon how many of its neighbours have their app installed.In particular, if i does not have the app, R i = βµ ( k i − i gotthe disease. If i has a CT app installed, instead, thereare two possible cases:1. If the node (cid:96) who infected i has the CT app, thenthe infection has been “detected” by the app and i goes into self-isolation immediately. If this hap-pens, i will not produce any secondary infection inthe graph2. If i got infected by a neighbour without CT app,then the infection remains undetected, and i canpotentially infect R i = βµ ( k i −
1) more nodes.So in the end, the expected number of infections poten-tially caused by the infection of i is: R i = βµ ( k i − k i − k (cid:48) i k i (19)where the term k i − k (cid:48) i k i is the probability that the infectionof node i does not get detected by the CT system. Hence,the expected number of secondary infections potentiallycaused by (cid:96) is given by: R (cid:96) = βµ k (cid:96) (cid:88) i a (cid:96)i ( k i − k i − k (cid:48) i k i (20)where a (cid:96)i are the entries of the adjacency matrix of G .By averaging R (cid:96) over all the nodes of the graph we getthe value of the basic reproduction number with contacttracing: R (cid:48) = βµN (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96),i R i = βµN (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i ( k i − k i − k (cid:48) i k i = βµN (cid:34)(cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i ( k i − − (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i ( k i − (cid:35) (21)This equation holds in general for any graph with anyCT strategy. Notice that the quantity:1 N (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i ( k i −
1) (22) is the expected excess degree of the neighbours of a ran-domly sampled node of G . In other words, it is equal to (cid:104) k nn ( i ) (cid:105) −
1, where k nn ( i ) is the average degree of theneighbours of node i . The basic reproduction number ofthe original graph G is equal to R = βµ N (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i ( k i −
1) (23)that is, the average degree of the neighbours of a ran-domly selected nodes of G , multiplied by βµ . Hence wecan conveniently rewrite Eq. (21) as: R (cid:48) = R − N βµ (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i ( k i −
1) (24)In the special case when G has no degree-degree correla-tions, we have: (cid:104) k nn ( i ) (cid:105) = (cid:104) k (cid:105)(cid:104) k (cid:105) ∀ i (25)hence we can write: R = βµ (cid:20) (cid:104) k (cid:105)(cid:104) k (cid:105) − (cid:21) (26)and we can rewrite Eq. (21) as: R (cid:48) = βµ (cid:34) (cid:104) k (cid:105)(cid:104) k (cid:105) − − N (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i ( k i − (cid:35) (27)As expected, the effect of contact tracing is to reducethe basic reproduction number of the original graph. Ingeneral, Eq. (24) (or Eq. (27) in uncorrelated graphs)provides a recipe to maximise the impact of CT app in-stallation. Indeed, given a certain adoption rate r , we canuse any optimisation algorithm to maximise the fitnessfunction max G (cid:48) F ( G (cid:48) ) = (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i ( k i −
1) (28)over the ensemble of of the possible choices of G (cid:48) . Noticethat F ( G (cid:48) ) can be decomposed in two terms. The firstone is (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i (29)that is, the sum of the expected degrees in the inducedsubgraph G (cid:48) of the neighbours of nodes in G , while thesecond one is: (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i (30)i.e., the sum of the average fraction of degree in G (cid:48) anddegree in G of all the neighbours of nodes in G . In the fol-lowing we denote the fraction of degree in G (cid:48) and degreein G for node i by c i = k (cid:48) i k i Data description and set of networks studied
In this work we considered 84 unique contact networkdata sets constructed from two different types of data:(i) temporal network data, which provide informationregarding the different contacts between individuals andthe duration of each interaction; (ii) static network data,where the contacts have been already aggregated for thewhole duration and a corresponding weight is associatedto each link. We reconstructed each network consideringtwo distinct filtering thresholds by either time – in sec-onds for type (i) – or by fraction of links retained – weightvalues for type (ii) –, resulting in 168 unique graphs.For the networks of type (i), we considered a hospi-tal [45] and a high school [53] temporal data sets, fromwhich we filtered the contacts by applying thresholds of240 and 360 seconds, i.e., each temporal snapshots re-sulted in a distinct network. For an art gallery [54] weused the thresholds of 0 and 20 seconds. Notice thatwe selected these threshold values since they provide thelargest connected component for each network. The type(ii) networks obtained from the “Sociopatterns” projectinclude the contacts between individuals with a weightthat corresponds to either the number of contacts or theirduration [43, 47, 48]. Given that most type (ii) networksare densely connected and a significant proportion of theweights have small values, we filter the networks by keep-ing the top 25% and 10% links with the largest weights. A list of all networks considered here is reported in Sup-plementary Material Table 1.
ACKNOWLEDGEMENTS
A.B. and V.N. acknowledge support from EPSRC NewInvestigator Award grant no. EP/S027920/1. A.S.and V.N. acknowledge support from the EPSRC ImpactAcceleration Award – Large Award Competition pro-gramme. This work made use of the MidPLUS clus-ter, EPSRC grant no. EP/K000128/1. This researchused Queen Mary’s Apocrita HPC facility, supported byQMUL Research-IT. doi.org/10.5281/zenodo.438045.
AUTHOR CONTRIBUTIONS
All the authors conceived the study. A.B. and A.S. per-formed the numerical simulations, analysed the results,and prepared the figures. S.S. and S.R. contributed tothe methods for the analysis of the results and preparedthe figures. V.N. provided methods for the analysis ofthe results and performed the analytical derivations. Allthe authors wrote the manuscript and approved it in itsfinal form.
COMPETING INTERESTS
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Supplementary Note 1.
PROPERTIES OF THE SUBGRAPHS INDUCED BY APPINSTALLATION.
We provide here the full derivation of the degree distribution of the subgraph G (cid:48) induced by a specific set of appinstallations, i.e., the subgraph of G obtained by considering only the nodes which have installed the app and theedges among them. We consider both the case of uniform random and degree-based installation. Uniform random installation
We start by considering the case of uniform random installation on a graph G with assigned degree distribution P ( k ), i.e., when a fraction r of the N nodes, chosen uniformly at random, installs the app. The probability that agiven node i that has installed the app has degree k (cid:48) in G (cid:48) given the fact that it has degree k in the original graph G can be expressed as: P ( k (cid:48) i = k (cid:48) | k i = k ) = (cid:18) kk (cid:48) (cid:19) r k (cid:48) (1 − r ) k − k (cid:48) . (1)Indeed, since app installation is performed uniformly at random with probability r , the probability that a specificneighbour j of a node i has installed the app is just r , and does not depend on whether i has installed the app ornot. Consequently, the probability that exactly k (cid:48) of the neighbours of i have installed the app is given by a Binomaldistribution with success probability r . This also implies that the expected degree of node i in G (cid:48) across the ensembleof realisations of random installation is: E [ k (cid:48) i ] = rk i (2)By using Eq. (1), we can express the probability distribution (cid:101) P RND ( k (cid:48) ) that a generic node with app installed has k (cid:48) neighbours which have installed the app as: (cid:101) P RND ( k (cid:48) ) = N − (cid:88) k =0 P ( k ) (cid:18) kk (cid:48) (cid:19) r k (cid:48) (1 − r ) k − k (cid:48) (3)Notice that this equation holds independently of the specific degree distribution of G and of other local propertiesof the graph, such as the presence of degree-degree correlations or clustering. We can derive the value of (cid:104) k (cid:105) RND byusing the definition: (cid:104) k (cid:105) RND = N − (cid:88) k =0 k (cid:101) P RND ( k ) = N − (cid:88) k =0 k N − (cid:88) j = k P ( j ) (cid:18) jk (cid:19) r k (1 − r ) j − k = N − (cid:88) j =0 P ( j ) N − (cid:88) k =0 k (cid:18) jk (cid:19) r k (1 − r ) j − k = N − (cid:88) j =0 P ( j ) j (cid:88) k =0 k (cid:18) jk (cid:19) r k (1 − r ) j − k = N − (cid:88) j =0 P ( j ) rj = r (cid:104) k (cid:105) G (4)Similarly, for (cid:104) k (cid:105) RND we obtain: (cid:104) k (cid:105) RND = N − (cid:88) k =0 k (cid:101) P RND ( k ) = N − (cid:88) k =0 k N − (cid:88) j = k P ( j ) (cid:18) jk (cid:19) r k (1 − r ) j − k = N − (cid:88) j =0 P ( j ) N − (cid:88) k =0 k (cid:18) jk (cid:19) r k (1 − r ) j − k = N − (cid:88) j =0 P ( j ) j (cid:88) k =0 k (cid:18) jk (cid:19) r k (1 − r ) j − k = N − (cid:88) j =0 P ( j ) (cid:2) j ( j − r + rj (cid:3) = r (cid:104) k (cid:105) G + r (1 − r ) (cid:104) k (cid:105) G (5)2 Degree-based strategy and rich-club coefficient
For degree-targeted app installations, i.e., when the top rN nodes in the ranking by degree have their app installed,Eq. (1) does not hold, since the probability of a node being in G (cid:48) is not uniform and depends instead on its degree in G . If we call (cid:101) k the smallest of the degrees of the nodes in G (cid:48) , the subgraph induced by app installation correspondsto the subgraph among nodes whose degree is ≥ (cid:101) k . The fraction of existing edges among nodes with degree ≥ (cid:101) k is,by definition, the (unnormalised) rich-club coefficient [1] of G : φ ( k ) = 2 e ≥ k, ≥ k N ≥ k ( N ≥ k −
1) (6)computed for k = (cid:101) k . Here we denote by N ≥ k the number of nodes whose degree is larger than or equal to k , and by e ≥ k, ≥ k the number of edges among those nodes. Consequently, the average degree of G (cid:48) can be written as: (cid:104) k (cid:105) DEG = ( N ≥ (cid:101) k − φ ( (cid:101) k ) (7)The general expression for the rich club in networks depends only on the joint degree-degree distribution P ( k , k ),i.e., the probability of finding an edge between two nodes having degree k and k , and reads [2, 3]: φ ( k ) = N (cid:104) k (cid:105) (cid:80) N − k ,k = k P ( k , k ) (cid:104) N (cid:80) N − k = k P ( k ) (cid:105) (cid:104)(cid:16) N (cid:80) N − k = k P ( k ) (cid:17) − (cid:105) (8)So in the case of degree-based targeting, even the first moment of the degree distribution of G (cid:48) depends heavily onthe presence of degree-degree correlation in G , at stark difference with the case of uniformly random installation seenabove.In the special case of uncorrelated random graphs, such as in the configuration model ensemble, the joint degree-degree distribution factorises as: P ( k , k ) nc = q k q k = k k P ( k ) P ( k ) (cid:104) k (cid:105) (9)and it is easy to show that the rich club coefficient can be written as: φ nc ( k ) = (cid:80) N − k = k k P ( k ) (cid:80) N − k = k k P ( k ) (cid:104) k (cid:105) (cid:104)(cid:80) N − k = k P ( k ) (cid:105) (cid:104) N (cid:80) N − k = k P ( k ) − (cid:105) (10)By using Eq. (7) we obtain: (cid:104) k (cid:105) ncDEG = N N − (cid:88) k = (cid:101) k P ( k ) − φ nc ( (cid:101) k ) = (cid:104)(cid:80) N − k = (cid:101) k k P ( k ) (cid:105) (cid:104) k (cid:105) (cid:104)(cid:80) N − k = (cid:101) k P ( k ) (cid:105) (11)It is quite interesting to find that the first moment of the degree distribution of G (cid:48) is indeed connected with therich-club coefficient of the graph at the critical degree.It is actually possible to compute the full degree distribution of G (cid:48) in the case of degree-based installation. If anode i is in G (cid:48) , then we have k i ≥ (cid:101) k . Now, the probability that one of the k i neighbours of i is in G (cid:48) is equal to: Q (cid:101) k ( i ) = N − (cid:88) k = (cid:101) k P ( k | k i ) (12)where P ( k | k i ) is the conditional probability of finding a node of degree k by following one of the edges of a nodeof degree k i . In the special case of graphs with no degree-degree correlations, P ( k | k i ) does not depend on k i , andfactorises as: P ( k | k i ) nc = kP ( k ) (cid:104) k (cid:105) = q k (13)3so we have: Q (cid:101) k ( i ) nc = N − (cid:88) k = (cid:101) k q k = (cid:101) r ∀ i (14)In the absence of degree-degree correlations, the probability of any two specific nodes to be connected does not dependon their degree, by definition. Hence, the probability that a node of G (cid:48) has a degree equal to k (cid:48) is given again by theBinomial distribution: (cid:101) P ( k (cid:48) i = k (cid:48) | k i = k ) = (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) , k ≥ (cid:101) k (15)while (cid:101) P ( k (cid:48) i = k (cid:48) | k i = k ) = 0 if k < (cid:101) k . Notice that (cid:101) r has the same role that r has in the equations for randomassignment. In particular, this means that the expected value E [ k (cid:48) i ] across all the configuration model graphs with apre-assigned degree sequence is equal to: E [ k (cid:48) i ] = (cid:101) rk i (16)With an argument in all similar to that used for random installation, we obtain: (cid:101) P ( k (cid:48) ) = N − (cid:88) k = (cid:101) k P ( k ) (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) (17)Notice that (cid:101) P ( k (cid:48) ) represents the probability to find a node of G which has degree k (cid:48) in the subgraph induced byapp installations. To obtain the actual degree distribution in the induced subgraph, i.e., the probability that one ofthe nodes of G (cid:48) has degree k (cid:48) , we must rescale (cid:101) P ( k (cid:48) ) to the nodes in G (cid:48) , i.e., we consider the probability distribution: (cid:101) P DEG ( k (cid:48) ) = 1 r (cid:101) P ( k (cid:48) ) = 1 r N − (cid:88) k = (cid:101) k P ( k ) (cid:18) kk (cid:48) (cid:19)(cid:101) r k (cid:48) (1 − (cid:101) r ) k − k (cid:48) (18)It is important to stress here that the expression for (cid:101) P DEG ( k (cid:48) ) provided above is valid only in uncorrelated graphs,due to the assumption we made in Eq. (13) and Eq. (14).In Supplementary Fig. 1 we report the the empirical degree distributions of the subgraph G (cid:48) induced by the random(RND) and degree-based (DEG) CT strategies when considering three different adoption rates r = 0 . , . , . P ( k ) ∼ k − and N = 10 nodes. Notice that the numerical simulations are in perfect agreement with the analytical predictions inEq. (3) and Eq. (18) k P ( k ) r = 0.1 a RNDRND theo
DEGDEG theo k r = 0.3 b k r = 0.5 c Supplementary Figure Empirical and theoretical degree distributions of the induced subgraph for differentvalues of adoption rate r.
We report the comparison between the empirical degree distributions of the induced subgraph G (cid:48) for the random (RND) and degree-based (DEG) strategies when considering three different adoption rates r . Solid linesrepresent the theoretical predictions for the RND and DEG strategies calculated using Eq. (3) and Eq. (18), respectively. Theplots correspond to an ensemble of configuration model graphs with degree distribution P ( k ) ∼ k − and N = 10 nodes.Results averaged over 10 realisations. Simulated annealing
The simulated annealing (SA) procedure is commonly used to find the global optimum of a certain cost/energyfunction. In a SA algorithm, an energy function usually attributes certain values to each configuration of the system.The best configuration, which is the one that optimises the energy function, is usually identified by searching thephase space of the system considering Markov Chain Monte Carlo moves that allow to switch from one configurationto another. In our context, we are interested on finding the maximum of the following cost function: F ( G (cid:48) ) = (cid:88) (cid:96) k (cid:96) (cid:88) i a (cid:96)i k (cid:48) i k i ( k i −
1) (19)Hence, for a given app adoption rate r , the SA algorithm tries to find the nodes which provide the maximum value of F in order to reduce the expected number of secondary infections caused by a single contagion in presence of contacttracing. Each configuration is represented by the couple { A, f } , where A is set of nodes ID which adopts the CT appin the network, while f is the energy associated with the configuration, i.e., the value computed using Eq. 19.For a given app adoption rate r , we start at time t = 0 from a random configuration A and evaluate the cor-responding energy f . Then, at each step t of the algorithm, we randomly replace a node i of the set A t − with arandomly selected node j that does not belong to the same set, i.e. A t = ( A t − ∪ { j } ) \ { i } . After calculating theenergy f t of the new configuration, we accept it with probability: p = (cid:40) f t > f e − f − ftT otherwisewhere T has the role of temperature in the simulated annealing procedure. In particular, the initial temperature forthe simulations was set to T max = 1 and in every step it was reduced by δT until T min = 0 .
01 was reached. A stepof δT = 10 − was used for all our numerical simulations.5 Supplementary Note 2.
CONTACT TRACING STRATEGIES BASED ON LOCALINFORMATION
We report in Supplementary Table 1 the real-world networks analysed in the main manuscript along with thethresholds values used for filtering nodes or edges in the networks and basic statistics (resulting number of nodes andedges). For each of the 84 unique graphs analysed, we report the thresholds used and the resulting number of linkswith larger weights.
Network Thres. Nodes Edges Network Thres. Nodes Edges Network Thres. Nodes Edgescontacts-Hospital 240 sec. 67 291 contacts-Hospital 360 sec. 63 228 highschool-2011 240 sec. 117 332highschool-2011 360 sec. 111 252 highschool-2012 240 sec. 171 496 highschool-2012 360 sec. 158 372gall-2009-04-28 0 sec. 190 703 gall-2009-04-28 20 sec. 61 127 gall-2009-04-29 0 sec. 198 736gall-2009-04-29 20 sec. 112 231 gall-2009-04-30 0 sec. 144 486 gall-2009-04-30 20 sec. 59 132gall-2009-05-01 0 sec. 201 558 gall-2009-05-01 20 sec. 37 60 gall-2009-05-02 0 sec. 213 966gall-2009-05-02 20 sec. 89 163 gall-2009-05-03 0 sec. 305 1847 gall-2009-05-03 20 sec. 211 513gall-2009-05-05 0 sec. 78 147 gall-2009-05-05 20 sec. 18 26 gall-2009-05-06 0 sec. 176 745gall-2009-05-06 20 sec. 37 65 gall-2009-05-07 0 sec. 194 801 gall-2009-05-07 20 sec. 72 204gall-2009-05-09 0 sec. 216 993 gall-2009-05-09 20 sec. 156 312 gall-2009-05-10 0 sec. 168 625gall-2009-05-10 20 sec. 138 242 gall-2009-05-12 0 sec. 56 114 gall-2009-05-12 20 sec. 11 15gall-2009-05-13 0 sec. 166 590 gall-2009-05-13 20 sec. 58 82 gall-2009-05-14 0 sec. 132 620gall-2009-05-14 20 sec. 54 134 gall-2009-05-15 0 sec. 241 1301 gall-2009-05-15 20 sec. 127 334gall-2009-05-16 0 sec. 241 1504 gall-2009-05-16 20 sec. 216 577 gall-2009-05-17 0 sec. 187 1347gall-2009-05-17 20 sec. 172 470 gall-2009-05-19 0 sec. 49 112 gall-2009-05-19 20 sec. 13 23gall-2009-05-20 0 sec. 89 507 gall-2009-05-20 20 sec. 75 276 gall-2009-05-21 0 sec. 43 193gall-2009-05-21 20 sec. 38 107 gall-2009-05-22 0 sec. 131 864 gall-2009-05-22 20 sec. 65 183gall-2009-05-23 0 sec. 238 1075 gall-2009-05-23 20 sec. 208 396 gall-2009-05-24 0 sec. 31 68gall-2009-05-24 20 sec. 10 13 gall-2009-05-26 0 sec. 131 513 gall-2009-05-26 20 sec. 57 183gall-2009-05-27 0 sec. 116 395 gall-2009-05-27 20 sec. 45 118 gall-2009-05-28 0 sec. 141 1054gall-2009-05-28 20 sec. 100 495 gall-2009-05-29 0 sec. 93 272 gall-2009-05-29 20 sec. 65 132gall-2009-05-30 0 sec. 127 397 gall-2009-05-30 20 sec. 70 121 gall-2009-05-31 0 sec. 89 267gall-2009-05-31 20 sec. 22 56 gall-2009-06-02 0 sec. 16 61 gall-2009-06-02 20 sec. 11 41gall-2009-06-03 0 sec. 62 174 gall-2009-06-03 20 sec. 16 46 gall-2009-06-04 0 sec. 53 382gall-2009-06-04 20 sec. 37 137 gall-2009-06-05 0 sec. 88 267 gall-2009-06-05 20 sec. 32 56gall-2009-06-06 0 sec. 142 696 gall-2009-06-06 20 sec. 127 324 gall-2009-06-07 0 sec. 155 563gall-2009-06-07 20 sec. 113 203 gall-2009-06-09 0 sec. 74 238 gall-2009-06-09 20 sec. 21 70gall-2009-06-10 0 sec. 35 74 gall-2009-06-10 20 sec. 18 49 gall-2009-06-11 0 sec. 77 161gall-2009-06-11 20 sec. 13 44 gall-2009-06-12 0 sec. 58 158 gall-2009-06-12 20 sec. 25 101gall-2009-06-13 0 sec. 102 264 gall-2009-06-13 20 sec. 21 40 gall-2009-06-14 0 sec. 138 433gall-2009-06-14 20 sec. 105 181 gall-2009-06-16 0 sec. 67 391 gall-2009-06-16 20 sec. 48 198gall-2009-06-17 0 sec. 72 212 gall-2009-06-17 20 sec. 48 81 gall-2009-06-18 0 sec. 74 275gall-2009-06-18 20 sec. 23 66 gall-2009-06-19 0 sec. 125 412 gall-2009-06-19 20 sec. 64 112gall-2009-06-20 0 sec. 149 495 gall-2009-06-20 20 sec. 109 203 gall-2009-06-21 0 sec. 166 676gall-2009-06-21 20 sec. 123 244 gall-2009-06-23 0 sec. 57 128 gall-2009-06-23 20 sec. 19 31gall-2009-06-24 0 sec. 79 369 gall-2009-06-24 20 sec. 22 40 gall-2009-06-25 0 sec. 79 321gall-2009-06-25 20 sec. 31 152 gall-2009-06-26 0 sec. 78 152 gall-2009-06-26 20 sec. 9 13gall-2009-06-27 0 sec. 35 99 gall-2009-06-27 20 sec. 10 17 gall-2009-06-28 0 sec. 107 397gall-2009-06-28 20 sec. 35 71 gall-2009-06-30 0 sec. 128 435 gall-2009-06-30 20 sec. 34 55gall-2009-07-01 0 sec. 167 814 gall-2009-07-01 20 sec. 127 336 gall-2009-07-02 0 sec. 60 180gall-2009-07-02 20 sec. 21 39 gall-2009-07-03 0 sec. 121 321 gall-2009-07-03 20 sec. 32 60gall-2009-07-04 0 sec. 127 526 gall-2009-07-04 20 sec. 103 214 gall-2009-07-05 0 sec. 95 314gall-2009-07-05 20 sec. 30 40 gall-2009-07-07 0 sec. 220 1187 gall-2009-07-07 20 sec. 166 477gall-2009-07-08 0 sec. 186 820 gall-2009-07-08 20 sec. 159 356 gall-2009-07-09 0 sec. 114 373gall-2009-07-09 20 sec. 61 121 gall-2009-07-10 0 sec. 157 776 gall-2009-07-10 20 sec. 103 359gall-2009-07-11 0 sec. 161 673 gall-2009-07-11 20 sec. 102 188 gall-2009-07-12 0 sec. 148 580gall-2009-07-12 20 sec. 114 215 gall-2009-07-14 0 sec. 275 1633 gall-2009-07-14 20 sec. 195 566gall-2009-07-15 0 sec. 410 2765 gall-2009-07-15 20 sec. 351 1205 gall-2009-07-16 0 sec. 318 1441gall-2009-07-16 20 sec. 250 567 gall-2009-07-17 0 sec. 221 1073 gall-2009-07-17 20 sec. 180 405InVS13 25% 93 967 InVS13 10% 87 391 InVS15 25% 217 4150InVS15 10% 208 1656 LH10 25% 62 339 LH10 10% 42 116LyonSchool 25% 240 6637 LyonSchool 10% 225 2654 SFHH 25% 392 17709SFHH 10% 384 7048 Thiers13 25% 326 10873 Thiers13 10% 69 1223bt 25% 634 18727 bt 10% 617 7449 call 25% 12 11call 10% 5 5 enterprise 25% 84 183 enterprise 10% 50 66Hospital 25% 67 284 Hospital 10% 44 110 sms 25% 19 18sms 10% 6 5 student 25% 1452 17134 student 10% 956 6670
Supplementary Table
1. Networks analysed and main statistics. i in the network has the possibility to indicate oneof its neighbours as a candidate for app installation. Then, we rank each node i according to the total number ofvotes v i it received.Although the voting system requires each node to have access to just local information, it is easy to show that ifthe graph has no degree-degree correlations, then the number of votes received by a node i is actually proportional toits degree k i . In particular if the probability that a node j casts a vote to a neighbour i is 1 /k j , the number of votesreceived by a node i is given by: v i = k i (cid:88) j A ji k j , (20)where a ji are the entried of the adjacency matrix of the graph. Despite the number of votes received is proportional tothe degree of a node, the inflow also plays a very significant role – similarly to what happens in PageRank – which canlead to some nodes with higher degree actually receiving a smaller number of votes than nodes with a lower degree.In Supplementary Fig. 2 we compare the average number of votes received by each of the nodes in an ensemble ofconfiguration model graphs with degree distribution P ( k ) ∼ k − and N = 10 nodes and the theoretical predictionprovide by Eq. (20). The agreement between the simulations and the prediction is perfect, and confirms that indeedthe number of votes received by a node is somehow proportional to its degree. k i j A ji k j v i Supplementary Figure Comparison between the votes received by a node and the theoretical predictionwhen each node casts a single vote.
We considered an ensemble of configuration model graphs with degree distribution P ( k ) ∼ k − and N = 10 nodes. The votes received, and the theoretical prediction, have been averaged over all the nodes withthe same degree and the black line indicates a perfect match between both. The assumption that individuals will install the CT app if they are directly targeted, and with probability equalto 1, is rather simplistic and difficult to achieve in real-world systems. Thus, we also explored how effective themitigation of an epidemic would be when instead of following a strict adoption based on rankings, a node i has acertain probability σ i to adopt the app, which depends on the number of votes it has received. One of the simplestapproaches to model the adoption of technologies is based on the usage of sigmoid adoption functions [9–11] in the7 No AppApp r = 0.3
Supplementary Figure Impact of CT strategy and adoption rate on the epidemic trajectory in real-worldnetworks . The evolution of the disease in a SIR model with contact tracing depends heavily on the adoption rate and onthe strategy considered to select the nodes with CT apps. Here, we show the results on two real-world systems, namely,the high-resolution face-to-face contact data recorded for a high school (a-c) and a workplace network (d-f). The strategiesanalysed are the recommendation-based (a,d), the sigmoids σ i ( v i , a = 1) (b,e), and σ i ( v i , a = 5) (c,f). For these simulationswe set β = 0 . µ = 0 . form: σ i ( v i , a ) = 11 + exp( − a ( v i − (cid:104) v (cid:105) )) (21)where (cid:104) v (cid:105) is the average number of votes received by nodes in a given realisation, v i is the number of votes receivedby node i , and a tunes the intensity of the sigmoid. In our simulations we considered either a = 1 or a = 5.It is important to note that the function σ i ( v, a ) needs to be properly normalised by (cid:80) i σ i ( v i , a ) to obtain aprobability function. In Supplementary Fig. 3(a-c) and (d-f), we report the temporal evolution of the number ofinfected individuals I ( t ) for the recommendation-based strategy, respectively for the high school and the workplacenetwork analysed in the main manuscript.In the high school network we observe a more drastic reduction of cases and larger difference between the strategiesas the level of app adoption increases. Furthermore, even though the mitigation effect with the sigmoids is lesspronounced than the best case scenarios relying on global information, we still observe a drastic reduction whencompared to the random case. In particular, to obtain a strong decrease in the total number of infected individualswe need to consider an adoption rate r above 0 .
5. To complement the previous results, Fig. 4 displays the relativeimprovement over RND for the recommendation-based strategy and the two sigmoids as a function of r for thesame two networks. In particular, we focus on the total number R ∞ of people who got the disease, the maximumnumber I ( t peak ) of infected across the duration of the epidemic, and the time t peak at which that number is achieved.Remarkably, for the high school network there are no strong differences in performance between σ i ( v i , a = 5) and anassignment based on the ranking of votes received. Finally, the position of the peak does not seem heavily affectedby the sigmoid strategies, so that there is still a visible shift of the epidemic peak towards the left for high values of r in the high school network.In the following, we refer to the local strategy based on the number of votes v received by a node at therecommendation-based strategy (REC). Considering the 168 networks, Supplementary Fig. 5(a-c) reports the re-sults with the correlations between the epidemiological indicators ∆ R REC ∞ , ∆ I REC ( t peak ) and ∆ t REC peak and structuralmeasures as a function of the adoption rate r . Overall, for low/moderate values of r , the performance of the RECstrategy – measured by ∆ R REC ∞ and ∆ I REC ( t peak ) – displays a moderate correlation only with the density and globalefficiency of the contact networks. Conversely, the rich club coefficient and number of maximal cliques show a highercorrelation only when considering ∆ t REC peak . As done for the DTI, we have also compared how the REC strategy isable to mitigate I ( t peak ) when compared to RND, as can be seen in Supplementary Fig. 5(d-f) where we report thedistribution of r ∗ needed for a reduction of of 10%, 30% and 50%, respectively.8 R a R REC ( REC , 5)(
REC , 1) I ( t p e a k ) b I ( t p e a k ) t p e a k c t p e a k r R d R r I ( t p e a k ) e I ( t p e a k ) r t p e a k f H i g h S c h oo l W o r k p l a c e - I n V S t p e a k Supplementary Figure Comparison of epidemic indicators under different CT strategies . Relative decrease withrespect to random installations of the total number of recovered R ∞ , height of the infection peak I ( t peak ) and position of thepeak t peak for the recommendation-based, the sigmoids σ i ( v i , a = 1), and σ i ( v i , a = 5) installation strategies in the high school(a-c) and workplace networks (d-f). The inset of each panel reports the plot of the raw variable, respectively R ∞ (panel a andd), I ( t peak ) (panel b and e) and t peak (panel c and f). Finally, to compare all the different decentralised CT strategies presented in this work, we report in SupplementaryFig. 6 the epidemic trajectory for the two real-world systems analysed in the main manuscript when consideringdifferent CT adoption rates r . Overall, DT I is the best performing decentralised strategy, which provides a significant r R a R REC
DensityGlobal EfficiencyRich Club CoefficientNum. Maximal Cliques r b I REC ( t peak ) r c t RECpeak r P ( r ) d
10% peak reduction
RNDREC r e
30% peak reduction r f
50% peak reduction
Supplementary Figure Correlations with network structural measures and performance of CT strategies tomitigate an epidemic . For the 168 real-world contact networks analysed, panels (a-c) report the correlation ( R ) between theepidemiological indicators ∆ R REC ∞ (a), ∆ I REC ( t peak ) (b) and ∆ t REC peak (c) and the network density (red), the global efficiency(blue), the rich club coefficient (purple) and the number of maximal cliques (orange) for different values of r . All measureswere computed considering the full graphs. In panels (d-f), we report the distribution of minimum adoption ratios r ∗ neededto produce a 10% (d), 30% (e) and 50% (f) for the REC (red) and RND (green) strategies. Overall, the REC strategy performsbetter than RND and requires lower adoption ratios to produce an equivalent reduction of the peak. r . Such effect is notably visible in the high school network. t I ( t ) r a High School 2011
RND(
REC , 1)(
REC , 5)
RECDTI t I ( t ) r b Workplace-InVS15
RND(
REC , 1)(
REC , 5)
RECDTI
Comparison between decentralised strategies
Supplementary Figure Mitigation of the epidemic trajectories using decentralised CT strategies . We reportthe epidemic trajectory when considering different decentralised CT strategies and levels of app adoption r for the two real-world systems considered in the main text. Overall, the CT decentralised strategies consistently lead to a significant reductionof the epidemic trajectory with respect to the random case even for small values installation rate r , with DT I being thebest-performing strategy. Supplementary Note 3.
SIR MODEL WITH MAXIMUM DELAY
The SIR model with the ideal contact tracing presented in the main manuscript is based on the assumption that anysusceptible node with CT app installed is immediately quarantined (recovered) as soon as one of their contacts withCT app installed gets infected. However, such assumption seems too simplistic when applied to real-world systems.Indeed, people may receive the app notification after several hours/days from an infectious contact, or worse, theymay ignore the app notification and continue to have contacts with their acquaintances until they develop symptoms.As a result, in this section we report the results obtained when considering a different variant of the SIR model − SIR d − which accounts for the maximum delay in self-isolation, so that the neighbours of an infected node onlyget recovered if an infection event takes place. In other words, if a node i and its neighbour j have both the contacttracing app installed, j will only become recovered if i succeeds into infecting j . This mimics the fact that people arein general unwilling to move into self-isolation if they have no symptoms, and represents the maximum possible delaybetween a positive test and self-isolation of it contacts.To compare the two SIR variants across the 168 real-world networks analysed in the main manuscript, we reportin Supplementary Fig. 7 the RND and DTI distributions of app installation rates r ∗ required to respectively observea 10, 30, 50% peak reduction with respect to having no contact tracing in place. Interestingly, also when consideringthe SIR d model, the DTI strategy consistently performs better than the RND. However, all things being equal, onaverage we need slightly higher CT adoption rates r to achieve similar performances in the peak reduction whenconsidering the SIR d model.Furthermore, to easily investigate the impact of the contact tracing strategies when considering the SIR d model,we ran our simulations on the same two real-world systems considered in the main manuscript, the high school andworkplace networks. In particular, in Supplementary Fig. 8 we report the comparison of three different epidemicindicators for the SIR d model. Overall, the results are in agreement with the one presented in the main manuscript,so that we observe a great improvement of the targeted CT strategies with respect to the random case in all the threeepidemic indicators even for low values of adoption rate r . However, we need higher adoption rates to reach levels ofreduction of total number of infected and infected in the peak similar to the results presented in the main manuscript, P ( r ) a
10% peak reduction
RNDDTI b
30% peak reduction
SIR with ideal CT c
50% peak reduction r P ( r ) d
10% peak reduction
RNDDTI r e
30% peak reduction
SIR d SIR with maximum delay r f
50% peak reduction
Supplementary Figure Performance of CT strategies in mitigating the epidemics for the two variants of theSIR model . For the 168 real-world networks analysed, we report the distribution of app installation ratios r (cid:63) required toobserve a 10, 30, and 50% peak reduction with respect to no contact tracing for the DTI (red) and RND (green) strategiesin the SIR model with ideal contact tracing (a-c) and in the SIR model with delay (d-f). Overall, the DTI strategy performsfar better than the RND as it requires lower values of r to induce a similar reduction of the peak in both SIR model variants.However, panels (d-f) clearly show that we need slightly higher CT adoption rates r to achieve similar performances in thepeak reduction with the SIR d model. R a R RNDDEGDTISA I ( t p e a k ) b I ( t p e a k ) t p e a k c t p e a k r R d R r I ( t p e a k ) e I ( t p e a k ) r t p e a k f H i g h S c h oo l W o r k p l a c e - I n V S t p e a k Supplementary Figure Comparison of epidemic indicators under different CT strategies for the SIR modelwith delay . Relative decrease with respect to random installations of the total number of recovered R ∞ , height of the infectionpeak I ( t peak ) and position of the peak t peak for DTI, DEG, and SA targeted installation when considering the SIR model withdelay. For all the three synthetic indicators, the targeted strategies provide a level of reduction similar to the one presented inthe main paper, yet achieved with a slightly higher values of adoption rate r (10% increase). The inset of each panel reportsthe plot of the raw variable, respectively R ∞ (panel a and d), I ( t peak ) (panel b and e) and t peak (panel c and f). at least 10% difference of installation rate respectively.Finally, we show that, even in the SIR d model, the decentralised strategies presented in the main paper (DTI) andin the previous sections (REC and variants) allow to obtain a substantial reduction of the epidemic trajectory withrespect to the random case. We report in Supplementary Fig. 9 the comparison between the decentralised strategieswhen considering the SIR d model for the two real-world systems considered in the main manuscript. Remarkably,even for low values of adoption rate r and also when considering the maximum delay model SIR d , the decentralisedCT strategies provide a consistent reduction in the epidemic trajectory in respect to the random case. t I ( t ) r a High School 2011
RND(
REC , 1)(
REC , 5)
RECDTI t I ( t ) r b Workplace-InVS15
RND(
REC , 1)(
REC , 5)
RECDTI
Comparison between decentralised strategies
Supplementary Figure Mitigation of the epidemic using decentralised strategies in the SIR model with delay .We report the epidemic trajectory when considering different CT decentralised strategies and levels of adoption rate r forthe two real-world systems considered in the main manuscript using the SIR d model. Interestingly, the CT decentralisedstrategies consistently lead to a significant reduction of the epidemic trajectory with respect to the RND, with DT I being thebest-performing strategy. [1] Colizza, V., Flammini, A., Serrano, M. A. & Vespignani, A. Detecting rich-club ordering in complex networks. Naturephysics , 110–115 (2006).[2] Latora, V., Nicosia, V. & Russo, G. Complex networks: principles, methods and applications (Cambridge University Press,2017).[3] Newman, M.
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