Contact Tracing: Computational Bounds, Limitations and Implications
CContact Tracing: Computational Bounds, Limitations andImplications
Quyu Kong β Australian National University &Data61, CSIRO & [email protected]
Manuel Garcia-Herranz
Ivan Dotu β Moirai [email protected]
Manuel Cebrian β Max Planck Institute for Human [email protected]
ABSTRACT
Contact tracing has been extensively studied from different per-spectives in recent years. However, there is no clear indicationof why this intervention has proven effective in some epidemics(SARS) and mostly ineffective in some others (COVID-19). Here, weperform an exhaustive evaluation of random testing and contacttracing on novel superspreading random networks to try to identifywhich epidemics are more containable with such measures. We alsoexplore the suitability of positive rates as a proxy of the actualinfection statuses of the population. Moreover, we propose novelideal strategies to explore the potential limits of both testing andtracing strategies. Our study counsels caution, both at assumingepidemic containment and at inferring the actual epidemic progress,with current testing or tracing strategies. However, it also bringsa ray of light for the future, with the promise of the potential ofnovel testing strategies that can achieve great effectiveness.
INTRODUCTION
The COVID-19 pandemic has posed significant threats to publichealth globally since 2020. It spreads rapidly around the world andit is hard to control due to a large proportion of pre-symptomaticand asymptomatic cases [40]. Although preventive measures, suchas social distancing [50] and lockdowns [19, 51], have shown tobe effective in slowing down the disease spreading, they comeat the cost of economic downturn and the risk of a second waveafter lifting restrictions. In order to respond to such time-criticalemergencies while avoiding strict measures, testing of potentialinfections via contact data and quarantining of positive cases havebeen proposed as alternative measures [2, 38]. However, limitedtesting/tracing resources have imposed challenges on the massdeployment of such measures.In this work, we tackle an extensive evaluation of different ran-dom testing and contact tracing strategies to ascertain their effi-ciency. To answer this, we first build a simulation tool involvingtwo major components β an extended stochastic epidemic modelsimulator with tracing and quarantining actions, and a contact net-work simulator that takes both π and the dispersion parameterinto account. These allow us to identify the characteristics of eachstrategy for a large set of parameter combinations. Moreover, byintroducing two ideal (despite unreal) strategies, we can propose a β Work done during an internship at the Max Planck Institute for Human Development β Corresponding authors classification of an epidemic (along with the initial time point andtesting resources) with levels: 1) can be contained with classic con-tact tracing (either forward, backward or both), 2) can be containedwith contact tracing with priorities, 3) can be contained with smarttesting strategies and 4) cannot be contained with testing.Overall, we find that: our superspreading networks better char-acterize the actual dispersion effect; backward contact tracing isslightly better at finding positive cases for epidemics with smalldispersion parameter with a small number of tests; and that there isboth a gap between classic contact tracing and ideal contact tracingand between it and ideal testing, that make exploring smart contacttracing and testing strategies worth it. We also show the limitationsof contact tracing as a proxy for the actual epidemic status and pro-pose random testing as a better estimate for such inference. Finally,our models assume full knowledge of the contact network, akinto using a perfect digital tracking app; other modeling decisionsattempt at conveying optimal computational conditions for contacttracing and testing.
Background
Compartmental epidemic models, such as the Susceptible-Infected-Recovered (SIR) model [26], are classical models used for gaininginsights into disease transmission dynamics. Such models defineseveral infection states for individuals in a population, and thedynamics of individuals moving between states can be modeleddeterministically or stochastically [3].Many works have explored epidemic simulations over randomnetworks [28, 43, 44]. The key issues to consider in this context arethe stochasticity of epidemic diffusions and the network architec-ture.Stochastic simulations of epidemics are conducted with the Gille-spie algorithm [28]. The algorithm samples, at each step, from thetotal rates of all possible events, the next event time and an eventtypeβe.g., an infection event or a recovery event. Every simula-tion continues until no infectious individuals can be found in thepopulation.Keeling and Eames [24] provide a review of properties of epi-demics over a variety of common network structures, includingErdΕsβRΓ©nyi random networks, lattices, small-world networks,scale-free networks, and spatial networks. Others [27, 39] attemptto design disease spreading networks where node connections are a r X i v : . [ c s . S I] F e b lustered. On the other hand, other works [2, 36, 56] seek to con-struct networks from real-world contact data. In this work, weemploy random networks with controllable epidemic parameters toanalyze the limits of contact tracing and random testing strategies.The number of secondary infections of individuals has beenshown to be heterogeneous, and numerous βsuperspreading eventsβhave been reported [53]. Lloyd-Smith et al. [37] proposed to modelthis phenomenon with a negative-binomial distribution parame-terized by the basic reproduction number π and a dispersion pa-rameter π . The dispersion parameter π controls the superspreadingeffect of a given disease, e.g., π = .
16 was estimated for SARS [37]where lower π values indicate a stronger superspreading effect.Specifically, modeling efforts on the superspreading effect of theongoing COVID-19 pandemic have been focused on estimatingthe dispersion parameter [12], understanding the causes [46] andevaluating control strategies [23].As a tool for containing infectious epidemics, contact tracingis the process of identifying individuals who have been in contactwith known infected cases. This process usually involves intensivemanual efforts from specialists in interviewing infected patientsand reconstructing their potential contacts [18]. Prior work hastargeted improving the efficiency and accuracy of this processthrough communication traces [16] or dedicated digital apps [1, 5].Effective contact tracing is considered to be critical in control-ling an ongoing epidemic [48, 49], which successfully containedSARS [35]. However, this tool has experienced failures before, suchas the British foot-and-mouth epidemic [17]. Therefore, a rich set ofliterature has studied the effectiveness of applying contact tracingin controlling epidemics [7, 22, 29] and, especially, in the ongoingCOVID-19 pandemic [2, 9, 18, 25, 32]. RESULTSSuperspreading random networks
The simplest random networks, the ErdΕsβRΓ©nyi (ER) networks [15],have been shown to reduce the epidemic models to their classicalfully mixed variants [24]. Therefore, in the context of epidemics,they can be fully characterized given the number of nodes π andthe π , π½ , and πΎ parameters. In the same spirit, we have developed anew method to generate random networks taking into account thesuperspreading effect. Given the number of nodes π , the epidemicmodel parameters and the dispersion parameter π , we can fullydefine the node degree distributions of such networks, which wecall superspreading random networks.We show that the node degree distribution of a superspreadingrandom network is defined as π ( π | π, π , π½, πΎ ) = π π΅ ( π β | π, π ( πΎ + π½ ) π½ ) π (cid:205) β π = π π΅ ( π β | π, π ( πΎ + π½ ) π½ ) π (1)where π π΅ (Β· | π, π ( πΎ + π½ ) π½ ) is a negative binomial distribution param-eterized by the dispersion parameter π and mean π ( πΎ + π½ ) π½ . We thenapply the configuration model [45] to generate random networks.The detailed derivation is explained in Methods.Given the assumption of infection independence (see Methods),we seek to ascertain that desired dispersion parameters are achieved and maintained throughout the exponential phase of the epidemic.Fig. 1a shows that this is the case up to π = π is known to be low (such as COVID-19 orSARS) would lead to a misrepresentation of the distribution of theinfections.Prior works [37, 46] typically model heterogeneous secondaryinfection numbers with variances in individual infectiousness, i.e.,posing a Gamma distribution on infection rates. Supplementary Fig. 2shows that, for low π values and especially for low recovery rates,the difference between the intended dispersion parameter and theachieved dispersion parameter in the first 100 infected nodes issignificant and much larger in this case than for the superspreadingnetwork.Finally, we show in Fig. 1c the node degree distribution of su-perspreading networks for different values of the dispersion pa-rameters, π , infection rates, and recovery rates. Other networkmeasures, such as clustering coefficients, are shown in Supplemen-tary Fig. 3. A study of contact tracing and Random Testing
See Methods for a complete description of the computational setupused throughout this section.
Correlation between positive rates and actual infections
Positive rates from testing are widely adopted for estimating under-lying infection statuses in populations [55]. Therefore, in this part,we evaluate the correlation between positive rates (see Methods)and actual infected numbers on each day. Fig. 2 depicts the aver-aged daily correlations for SIR model (A similar study is depictedin Supplementary Fig. 4 for the SEIR model).First, we can see that, in general, for small π , small π , large πΎ and a small number of tests, the correlations are relatively poor,so positive rates from all strategies are misrepresenting the actualinfection. These correlations become better as the π , π and thenumber of tests increase.For RT (Fig. 2a), we observe close proportions between dailyconfirmed rates and actual infection rates in general. We can con-clude that random testing, for a sufficiently large π and number oftests, is a good estimator of the actual epidemic progress. Note thatcorrelations always improve as the number of daily tests increases,even though the testing is also affecting the epidemic since posi-tives are being quarantined. For a scenario in which testing doesnot affect the epidemic ( π π =
0) see Supplementary Fig. 5a for SIRor Supplementary Fig. 5b for SEIR.Figs. 2b and 2c show the correlations for forward and backwardcontact tracing, respectively. We can see that in both cases, posi-tive rates often overestimate the actual infections, especially forlow π and πΎ values. Moreover, BCT overestimates more than FCT,especially when the number of tests is low and the epidemic has low dispersion parameter. This also means that BCT is better atfinding positive contacts under these conditions.Also, we see instances in which CT daily positive rates increasewhen more daily tests are deployed, whereas actual infections in thepopulation decrease. For example, when π = , πΎ = . , π = . ,
000 daily tests lead to higher proportions of positivecases than 100 daily tests, which results in a false estimation of theepidemic.In conclusion, positive rates from contact tracing can be mislead-ing and should be considered with caution. However, positive ratesfrom random testing with a sufficiently large number of daily testsshow more promise as a proxy for the actual epidemic progress(A similar trend can be seen in Supplementary Fig. 4 for the SEIRmodel).
Analysis on final epidemic infections
Here, in order to classify different epidemics, we introduce twonovel strategies that represent both the ceiling of contact tracingand the ceiling of testing. Fig. 3 shows a cartoon representation ofthe effect of these strategies on a toy infection network and theirdifferences with both random testing and contact tracing. We callthese ideal strategies oracles, and they are defined in Methods.When comparing average total infections, Fig. 4a shows that, forthe networks simulated with π = .
1, around 20% of the populationmay be infected in the worst scenario without any intervention andless than 10% of population when π < .
5. On the other hand, whenintervention operations are applied, epidemics with higher π½ and π are more difficult to contain. Only GOT contains epidemics withina small number of daily tests, which becomes more difficult as π grows. COT performs slightly better than the other contact tracingstrategies (forward and backward). However, BCT does not showobvious advantages over FCT. In general, contact tracing does notseem to have a considerable impact in the course of the epidemic,and much less so random testing. Also, note that all epidemics resultin much larger infected populations when simulations occur over anER network. For SEIR models in Supplementary Fig. 6, it can also befound that incubation periods (when individuals are not contagiousbut they are detectable) result in more containable epidemics whencompared to the same parameter sets in SIR models.The nature of the superspreading events when π is low renders alower portion of the population infected at the end of the epidemic(Figure 3a). However, since superspreading events will have moreimpact in denser communities (connected components in the net-work), especially in those where initially infected individuals werefound, we explore the average percentage of the infected populationin the top 5 largest communities in the network. Figure 3b showsthat this has no effect for large π values or in ER networks butshows noticeable differences, especially for π = .
1. Note that, forexample, π = . πΎ = .
05 with no interventions reaches only10 to 15 percent of infected individuals at the end of the epidemic,but around 80 percent within the densest communities.From Fig. 4c and Supplementary Fig. 6c, we see that high π andsmall πΎ lead to longer epidemic diffusion times, peaking at π = . π and lower π values. Also, contact tracing seems to have agreater impact on times than on the total infected population. When comparing SEIR models to SIR models, longer days are observedoverall due to the incubation periods.Finally, Fig. 4 helps us classify parameter sets in terms of howcontainable they are. Thus, we can see that epidemics with π = π = . π values and as π increases, epi-demics are not containable with RT or any CT strategies. Only theglobal oracle, equipped with large numbers of daily tests, can havean impact. Lastly, for extreme π values (10 and beyond) and large π values (above 1 and for ER networks), there is no interventionthat can contain the epidemic. A mixed strategy for containment and surveillance
Many countries have established daily risk level thresholds that areaccompanied by different measures. For illustration purposes, wechoose the levels defined by the Spanish government (see Methods).Fig. 5 depicts the maximum threat level achieved during anepidemic for each of the epidemic parameter values, the number ofdaily tests, and different contact tracing/testing strategies.Fig. 5a shows the actual levels achieved. As it can be seen, foralmost all cases, the maximum threat level is reached at some pointin time during the epidemic (except for low π values). Fig. 5bshows the maximum threat level when considering all positivetests as the actual total positives in the population. This inferenceconsistently underestimates the maximum threat level. On the otherhand, Fig. 5c shows the maximum threat levels if those are inferredfrom the positive rates found during testing. As expected from ourcorrelations section results, this scenario consistently overestimatesthreat levels (except RT).Finally, we constructed a mixed strategy in which 100 daily testswere devoted to random testing regardless of the strategy (for 10daily tests we devoted 5 to random testing and for 100 daily tests50 were devoted to random testing). Fig. 5d shows the maximumthreat levels reached with this new strategy when only positiverates from random testing are used. Although far from perfect, wecan see that this strategy shows maximum threat levels very similarto the actual ones shown in Fig. 5a.Note that this mixed strategy could potentially result in differentinfection levels, since not all tests are devoted to the given strategy.However, Supplementary Fig. 7 shows that the differences betweenthis strategy and the non-mixed ones (compare with Fig. 4) arenegligible. Known diseases
As an example and proof of concept, we seek to quantify the ef-fect of intervention strategies on several known diseases withtheir reported epidemic parameters, including Measles [6, 47, 54],H1N1 [11, 20, 31], Ebola [4, 34], SARS [10] and COVID-19 [2, 12].Their detailed parameters are listed in Supplementary Table 1. Inour simulations, we fix the population number to π = ,
000 and πΌ =
10 initial infections. Again, we vary the number of daily testsfrom 0 to 10 , ig. 6 depicts the epidemic curve of the simulation results of thefive epidemics under different interventions. Supplementary Fig. 8shows the same results for the cumulative cases.In Figs. 6a to 6d, overall, the operations with oracle strategiescontain epidemics better, while random testing requires much moredaily tests to control the epidemic spreading. In particular, bothcontact tracing operations (forward and backward) lead to similaraverage daily infection curves. When comparing different diseases,SARS (Fig. 6c) is the most containable epidemic, while Measlescannot be controlled due to its extreme π value and its infectionprocesses in all simulations finished shortly compared to otherepidemics as shown in Fig. 6e. These two are the extreme pointswhere an epidemic can be controlled with CT and where it cannot becontrolled even with GOT, respectively. For the rest, H1N1 (Fig. 6c)can be contained with relatively few resources, and Ebola (Fig. 6b)and Covid-19 might need one order of magnitude more to reachcontainment. Note that SEIR models are easier to contain under theassumption that E individuals are not contagious, but they will betested positive. Finally, note that the global oracle can control allepidemics with few resources, except for Measles.Supplementary Fig. 9 and Supplementary Fig. 10 show the sameresults when we disregard the dispersion parameter and run thesimulations over ER networks, both for the epidemic curve andcumulative cases, respectively. It can be seen that final infectionsare much higher than those found in superspreading networks withlow dispersion parameters.Of relevance, with the parameters for COVID-19 from [2, 12],only around 10% of the population is infected. A reasonable amountof daily tests can reduce this proportion to around 2 β DISCUSSION
In this paper, we study the computational bounds of contact trac-ing and random testing. We introduce random networks that takeinto account the superspreading effect with controllable dispersionparameters for the first time. We first find that backward contacttracing is slightly better than forward contact tracing for low dis-persion parameters and a small limit of daily tests. We then findthe limitation of contact tracing as a means to describing the actualepidemic status. Afterward, we provide a classification of epidemicsin terms of how containable they are, in which we find that there isa gap between classic contact tracing and optimal contact tracing,and between this an optimal testing. The implications are exploring both smart contact tracing and smart testing techniques is worth it.We also see that the length of the epidemic can be misleading andthat contact tracing also has an impact on it.Finally, we want to address the fact that other recent works oncontact tracing models [2, 18, 32] for COVID-19 show more opti-mistic results than what transpires from our study regarding theimpact of contact tracing. We believe three main factors contributeto this difference: (1) The superspreading effects controlled by thedispersion parameter are not explicitly considered in these works,while our study is done over networks where we control the dis-persion parameter for the simulations. (2) Testing and/or tracingresources are unlimited for these references, while we have ana-lyzed various levels of available testing or tracing resources up to10% of the total population of daily tests. (3) Lastly, most of these pa-pers ( [2, 18]) consider between 35% and 70% of infected individualsas triggers of contact tracing, while for this work, only hospital-ized individuals (which is 5% of infected individuals throughoutthe main text) and positives found via random testing or contacttracing are considered.For illustration purposes, we turn our focus to the exhaustivework carried out in [2] to try to ascertain objectively whether thesethree issues can explain the differences observed. First, we seethat, when we run the epidemic diffusion process as stated in [2](see Methods) over the network provided in https://github.com/aaleta/NHB_COVID, the resulting dispersion parameter is π = . . πΌ = ETHODSContact networks simulation
Simulate ErdΕsβRΓ©nyi random networks
Bartlett and Plank [8] reveal the analytical connection betweenthe epidemic parameters and network parameters for ErdΕsβRΓ©nyirandom networks [14], i.e., π = π½ ( π β ) πΎπΎ + π½ (2)where a network is parameterized by the average node degree πΎ .When π β β , the node degrees of these networks are Poissondistributed with a mean value π ( πΎ + π½ ) π½ . Simulate networks with dispersion parameters
In this section, we describe the algorithm for simulating randomcontact networks that lead to negative binomial distributed sec-ondary infections with given parameters.To generate a random network, we first seek to derive its nodedegree distribution. We provide here some definitions includingseveral probability generating functions (PGFs) following [44]: β’ Given a random network with its degree distribution P ( π = π ) = π π , its PGF is πΊ ( π₯ ) = (cid:205) β π = π π π₯ π and the average degreeequals < π > = πΊ β² ( π₯ = ) = (cid:205) β π = ππ π . β’ The excess edge of a vertex is defined as the number of remain-ing edges connected to the vertex when follow a randomedge to that vertex. This corresponds the remaining neigh-bor counts when disease diffuses to a new individual. Theprobability that a vertex at the end of a random edge hasexcess degree π β ππ π < π > . Therefore, thePGF for the excess degree of a vertex is πΊ ( π₯ ) = (cid:205) β π = ππ π π₯ π β (cid:205) β π = ππ π (3) β’ More importantly, given the probability that a infected indi-vidual infects his/her neighbor π , the PGF of the number ofinfected neighbors is πΊ β² ( π₯ ) = πΊ ( + ( π₯ β ) π ) (4)Similarly, the PGF for the excess occupied degree is πΊ β² ( π₯ ) = πΊ ( + ( π₯ β ) π ) We are then able to derive the degree distribution. The assump-tion of a negative binomial (NB) distributed secondary infectionsindicates that πΊ β² ( π₯ ) is also the PGF of an NB distribution. Giventhe average secondary infection π and the dispersion parameter π ,we have πΊ β² ( π₯ ) = β βοΈ π = (cid:18) π + π β π (cid:19) (cid:18) π π + π (cid:19) π (cid:18) ππ + π (cid:19) π π₯ π (5) = (cid:18) + π π ( β π₯ ) (cid:19) β π (6)where the second step is due to the binomial theorem. With achange of variable, we get πΊ ( π¦ ) = (cid:18) + π ππ ( β π¦ ) (cid:19) β π (7) where πΊ ( π¦ ) is simply another NB with mean π π and dispersionparameter π . We denote its probability mass function as π π β which,as mentioned above, is equal to ππ π < π > = π π β . Due to (cid:205) β π = π π =
1, weare able to compute the average node degree < π > = (cid:205) β π = π π β π (8) = π π ( β π )( ππππ + π ) π ( π + π π ) β π (9)which can be easily solved numerically.We then need to define the infection probability π given the in-fection rate and recovery rate. For an infected individual, assuminghis/her recovery follows a rate πΎ and he/she is infecting a neighborwith a rate π½ , we can easily derive the probability of the neighborbeing infected as π½πΎ + π½ . If we further introduce a relaxed assumptionthat all neighbors are infected i.i.d., we then have π = π½πΎ + π½ .Given the derived degree distributions, we can then simulatea random network by applying a configuration model [45]. Wenote that any self-loops or parallel edges are removed from thegenerated networks. Explore empirical dispersion parameters in simulations
Contact networks simulated via the method described in Methodsis based on an assumption that one infectious individual infectshis/her neighbors independently. However, this relaxed assumptionmay lead to errors between chosen dispersion parameters and theirempirical values in simulations. Here we explore such discrepanciesvia simulations.
Clustering coefficients
Nelakonda and Rhomberg [42] show that clustering coefficients ofthe configuration model are defined as:1 π ( < π > β < π > ) < π > (10)where π is the number of nodes. This indicates that the clusteringcoefficients are 0 in the limit of large networks. Experimental setup
Here we detail the main concepts of our experimental setup. As itwill be seen, all modeling decisions are geared towards a more favor-able scenario for the impact of contact tracing rather than towardsa more realistic one. Therefore, the following results are best-casescenarios for all strategies. The main modeling considerations areas follows: β’ Compartmental models.
We simulate outbreaks with twoepidemic models, SIR and SEIR. The SIR model assigns threepossible infection statuses to individual nodes, susceptible( π ), infected ( πΌ ), and recovered ( π ), whereas the SEIR modelfurther introduces the exposed ( πΈ ) for modeling incubationperiods of diseases. While most nodes are in the susceptiblestatus ( π ), πΌ number of nodes are initialized as infectiousnodes ( πΌ ) at π‘ =
0, who spread the disease to their neigh-bor nodes at a rate π½ . The dynamics of infectious periods(i.e., from πΌ to π ) and incubation periods (i.e., from πΈ to πΌ ) ofindividual nodes are defined by rates πΎ and π , respectively. esides, we introduce a hospitalized ( π» ) compartment in Ex-perimental setup section to model the possibility of infectedindividuals disclosing their statuses via hospitalization. Therate of πΌ to π» is π , and it can be directly calculated fromthe probability of hospitalization π π» . Moreover, the basicreproduction number, π , is an important epidemic quan-tity that defines the average secondary infections caused bya single infected individual. Throughout the paper, resultsfor SIR models are shown in the main text figures, whilethose for SEIR are depicted in supplementary figures. Themain decision about the SEIR model is that individuals whoare in incubation periods cannot infect but can be detectedas positive in a test. Both models are extended with a Hos-pitalization rate, where some infected individuals becomehospitalized. Hospitalized individuals are considered as au-tomatic positives who trigger contact tracing. β’ Tracing and testing setups.
In our experiments, we ex-plore the limits of contact tracing and random testing incontaining epidemics. Given a fixed number of tests per day,these strategies are applied daily in epidemic simulations,and different individuals are then tested for their infectionstatuses. We assume the test results are provided immediatelywithout false positives and false negatives. We disregard thenumber of tracers and assume instead the limiting resourcesare the daily tests. Calculating the number of tracers to dailytests is not discussed in this work. β’ Random testing (RT).
This operation devotes resources torandomized testing within a population. Random tests canbe performed in real-life scenarios by contacting randompeople and asking them to take a test or by announcingvoluntary tests at specific locations. Here, we assume all in-fection statuses will receive an equal probability of selectionfor tests and also that recovered individuals can be chosenfor such random tests (only individuals that have tested pos-itive and still have not recovered are not selected). Thesetwo decisions together allow us for a unique implementationthat accounts for both real-life scenarios at the same time. β’ Contact tracing (CT).
During simulations, we maintain aqueue of individuals to be tested and traced. Neighbors ofpositive cases, found by contact tracing, random testing, orhospitalization, are recorded in the queue. The probabilityof discovering a contact of an infected node is denoted as π π and is set to 1 (akin to using a perfect contact tracing app [1]).We note that, when the queue becomes empty, all remainingtesting resources are devoted to random testing, so that alltests are performed every day. Two possible strategies arisewhen individuals in the queue are prioritized differently: For-ward contact tracing (FCT) orders the tests for individualsaccording to the times when they are added to the queue.On the other hand, backward contact tracing (BCT) pri-oritizes the tests to neighbors of positive cases, up to onehop in networks. This latter approach has been proposed forepidemics with low dispersion parameter [13, 21, 41], aimingto find sources of the infections. β’ Quarantine.
We assume a scenario where nodes that havetested positive are quarantinedβi.e., removed from the con-tact networkβwith a probability that we denote as π π . Thisprobability is set to 1. β’ Parameter exploration.
We quantify the effect of differentintervention operations on a wide range of possible epi-demics by varying all epidemic parameters. We also explorethe space of initial infected (with no re-introductions) andthe number of daily tests. A complete table of the param-eters tested is presented in Supplementary Table 1. In themain text, we discuss a smaller set of parameters that arerepresentative of the observed trends. Complete results forall parameters can be found in Supplementary Data 1. β’ Simulation setups
Given an epidemic model with chosenparameters, we first simulate 15 different superspreadingrandom networks based on the different network parametersbut with a fixed number of nodes π = ,
000 (some resultswith π = , ,
000 are shown in Supplementary Data 2).On each superspreading network, we randomly select πΌ individuals as initial infections and simulate an epidemicwith applications of all intervention strategies daily. Thisprocess is repeated 30 times on every contact network, whichresults in 450 simulations in total. In all results that follow,the number of initially infected nodes πΌ is set to 10. Resultsfor larger numbers of initially infected nodes are shown inSupplementary Data 1. Positive rates from tracing/testing
Since we cannot select (for testing) individuals that have alreadytested positive but are not yet recovered (which we call ππ‘π below),the positive rates are calculated as follows:
πππ πππ‘π = (( πππ ππ‘ππ£ππ / π‘ππ π‘π ) β ( π β ππ‘π ) + ππ‘π )/ π Oracles
The contact tracing oracle (CTO) prioritizes individuals in thetesting queue so that the actual infected ones are visited (tested)first. This provides an upper bound on the best contact tracingstrategy.The global oracle testing (GOT) assumes the availability ofinformation about all newly infected individuals who are thentested and quarantined. Thus, the contact tracing queue is filledonly with infected individuals. This assumption leads to an upperbound for future smart testing strategies.We note that, although these two oracle scenarios are unrealisticin practice, they provide ideal upper-bounds for potential newstrategies that might arise in the future.
Threat Levels
We use the Spanish Government threat levels, which are basedon the total infection number, in the last 14 days, per 100 , ,
000 individuals. Inpractice, these levels also take into account positive rates and ICUoccupancy, however, we ignore these here for the sake of simplicity. aseline Simulations from the Prior Work We apply the network provided in https://github.com/aaleta/NHB_COVID by Aleta et al. [2]. The provided network is unweighted andhas 10 ,
000 nodes and an average node degree of 10. In terms of thesimulations, we implement the epidemic parameter sets followingthe statistics detailed in [2] where a 50% probability of the discov-ery of symptomatic infections is applied and 40% of contacts aresuccessfully traced. The only two differences with [2] are: testingand quarantining are performed immediately (whenever tests areavailable) and no household contacts are automatically quarantined(since these are unknown in the network provided).
Software Implementation
The simulation algorithm was implemented in Python based onthe sample code provided by Kiss et al. [28] . We extended thesimulation with multiple containment strategies described in thispaper. We use the R programming language for batch running thesimulations by invoking the Python modules and for analyzing andplotting the simulation results. Code availability
The source code is available at https://github.com/qykong/testing-strategies. The simulation code and all param-eters used for related works can be found in the repository. https://github.com/springer-math/Mathematics-of-Epidemics-on-Networks 7 : Ξ³ : Ξ³ : Ξ² : . Ξ² : . Ξ² : Dispersion parameter E s t i m a t ed d i s pe r s i on pa r a m e t e r R0 11.5 22.5 33.5 10 (a) R0 E s t i m a t ed d i s pe r s i on pa r a m e t e r Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : Ξ² : Ξ³ : (b) Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ³ : , R : Ξ² : . Ξ² : . Ξ² : Node degree C o m p l e m en t a r y c u m u l a t i v e den s i t y p l o t Dispersionparameter ERk: 0.1 k: 0.5k: 1 k: 1.5k: 10 (c)
Figure 1: Comparison of network properties given different parameters. (a) shows the difference between chosen dispersionparameters and estimated dispersion parameters in simulations. (b) depicts the estimated dispersion parameters of ER ran-dom networks. (c) presents a complementary cumulative density plot of node degrees for dispersion networks with varyingparameters. : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R -0.50.00.51.0-0.50.00.51.0-0.50.00.51.0-0.50.00.51.0 Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (a) R : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (b) R : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (c)
Figure 2: Correlation plots between daily infections and positive rates for SIR models with different parameters, π½ = . and π π» = . . (a) shows correlation plots for random testing, (b) shows correlation plots for forward contact tracing, and (c) showscorrelation plots for backward contact tracing. Red lines represent correlation values and green lines indicate expected finaltotal infections. Bars represent expected daily infection ratios and daily average positive rates. Note that bars are scaled bythe maximum values.
19 5 4 62 1715 20 Random Testing(a)(b) Contact Tracing
Oracle Tracing
Global Oracle
Figure 3: Cartoon representation of all contact tracing/testing strategies for a toy network. (a) shows a network where somenodes are shaded indicating that they are infected, and some of those include a cross symbol, indicating they are hospitalized,i.e., they are triggers of contact tracing. This network is a representation of a snapshot of the status of an epidemic in a givenmoment in time. (b) shows the four strategies considered for the toy network in (a): Random testing β nodes are selected atrandom, the resulting list of nodes to test contains 1 infected individual that will test positive and be quarantined. Contacttracing β contacts of all hospitalized nodes are included in the list in no particular order, we see that 3 infected individualsare in this list, although some non-infected nodes will be tested before them. Oracle tracing β this is an ideal contact tracingstrategy, where we assume there is an oracle to tell us beforehand who in the testing queue is infected and we prioritize testingthese nodes. Global oracle β in this ideal strategy we assume an oracle who tells us the actual infection status of the wholenetwork so we only add infected nodes to the testing queue. Note that the differences between forward and backward contacttracing cannot be shown in a single day snapshot of the epidemic. Note that the number of positive nodes found for eachstrategy is dependent on the number of tests available per day. In this example, if the test limit were 6, the positives foundwould be 1,3,3 and 5, respectively; if the limit were 5, the positives found would be 1,2,3 and 5, respectively. : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (a) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (b) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (c)
Figure 4: Evaluation of epidemic simulations over a set of parameters for SIR model with π½ = . and π π» = . . (a) shows averagefinal infections, (b) shows average final infections in top 5 communities and (c) shows average days to the last infections. : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (a) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (b) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R global oracleoracle tracerback tracingforward tracingrandomglobal oracleoracle tracerback tracingforward tracingrandomglobal oracleoracle tracerback tracingforward tracingrandomglobal oracleoracle tracerback tracingforward tracingrandom Number of daily tests (c) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R global oracleoracle tracerback tracingforward tracingrandomglobal oracleoracle tracerback tracingforward tracingrandomglobal oracleoracle tracerback tracingforward tracingrandomglobal oracleoracle tracerback tracingforward tracingrandom Number of daily tests (d)
Figure 5: Summary of threat levels of SIR model simulations given different epidemic parameters and intervention strategieswhere π½ = . and π π» = . . (a) shows actual highest risk levels for SIR models, (b) shows highest threat levels from confirmedcases, (c) shows highest threat levels from positive rates and (d) shows highest threat levels for the mixed strategy (positiverates estimated only from random tests). aily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (a) COVID-19 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (b) Ebola Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (c) SARS Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (d) H1N1 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (e) Measles Figure 6: The average daily infection ratios of five known diseases by varying number of daily tests, under different interven-tion strategies. EFERENCES [1] Nadeem Ahmed, Regio A Michelin, Wanli Xue, Sushmita Ruj, Robert Malaney,Salil S Kanhere, Aruna Seneviratne, Wen Hu, Helge Janicke, and Sanjay K Jha. 2020.A survey of covid-19 contact tracing apps.
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3, 3 (2015), 410β419. https://doi.org/10.1109/TETC.2015.2398353 15 upplementary Figures Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Ξ³ : , Ξ² : Dispersion parameter E s t i m a t ed d i s pe r s i on pa r a m e t e r R0 11.522.533.510 !"" (a) k: 0.1 k: 0.5 k: 1 +
02 5e +
02 1e +
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03 1e +
04 2e +
04 4e +
04 8e +
04 1e +
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02 1e +
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04 8e +
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02 5e +
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04 8e + Number of infected individuals for estimation E s t i m a t ed d i s pe r s i on pa r a m e t e r R0 (b) Figure 1: Chosen dispersion parameters and their empirical values. (a) evaluates the effect of computing empirical dispersionparameters over different numbers of first infected nodes. (b) shows the evolution of estimated dispersion parameters overthe course of the whole epidemics for π½ = and πΎ = . . : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Ξ² : , Ξ³ : Dispersion parameter E s t i m a t ed d i s pe r s i on pa r a m e t e r R0 12.53.510Dispersion networksER networks with gammadistributed infection rates
Figure 2: Comparing empirical dispersion parameters obtained from superspreading networks (solid colored lines) and ERnetworks with gamma distributed infection rates (dotted colored lines). The dotted black lines indicate the true dispersionparameters. : : : : :
10 ER Ξ² : . Ξ² : . Ξ² : R Ξ³ (a) R : : : : Ξ³ : . Ξ³ : . Ξ³ : . Clustering coefficients C o m p l e m en t a r y c u m u l a t i v e den s i t y f un c t i on ERk: 0.1 k: 0.5k: 1 k: 1.5k: 10 (b)
Figure 3: Clustering coefficients given different parameters. (a) Expected clustering coefficients given different parameterswhen π = ; (b) cumulative density plots of clustering coefficients given different parameters, π½ = : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R -0.50.00.51.0-0.50.00.51.0-0.50.00.51.0-0.50.00.51.0 Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (a) R : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (b) R : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (c)
Figure 4: Correlation plots between daily infections and positive rates for SEIR models with different parameters, π½ = . , π = . and π π» = . . (a) shows correlation plots for random testing, (b) shows correlation plots for forward contact tracing,and (c) shows correlation plots for backward contact tracing. Red lines represent correlation values and green lines indicateexpected final total infections. Bars represent expected daily infection ratios and daily average positive rates. Note that barsare scaled by the maximum values. : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R -0.50.00.51.0-0.50.00.51.0-0.50.00.51.0-0.50.00.51.0 Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (a) R : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : : , Ξ³ : k : . k : . k : k : E R -0.50.00.51.0-0.50.00.51.0-0.50.00.51.0-0.50.00.51.0 Number of daily tests
AverageIn infection each day AveragePositives each day Averagetotal infection Correlation betweenIn infection and Positives each day (b)
Figure 5: Correlation plots between daily infections and positives of (a) SIR and (b) SEIR when π π = . Note that π½ = . , π = . and π π» = . , and both plots show for results under random testing. Red lines represent correlation values and green linesindicate expected final total infections. Bars represent expected daily infection ratios and daily average positive rates. Notethat bars are scaled by the maximum values. : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (a) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (b) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests
10 30 100 300Average days tothe last infection (c)
Figure 6: Evaluation of epidemic simulations over a set of parameters for SEIR model with π½ = . , π = . and π π» = . . (a)shows average final infections, (b) shows average final infections in top 5 communities and (c) shows average days to the lastinfections. : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (a) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (b) R : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : : Ξ³ : k : . k : . k : k : E R Number of daily tests (c)
Figure 7: Evaluation of epidemic simulations over a set of parameters for SIR model with π½ = . and π π» = . for the mixedstrategy. (a) shows average final infections, (b) shows average final infections in top 5 communities and (c) shows average daysto the last infections. aily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s (a) COVID-19 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s (b) Ebola Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s (c) SARS Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s (d) H1N1 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (e) Measles Figure 8: The average daily cumulative infected populations of five known diseases by varying number of daily tests, underdifferent intervention operations. aily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (a) COVID-19 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (b) Ebola Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (c) SARS Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s (d) H1N1 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age da il y i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (e) Measles Figure 9: The average daily infection ratios of five known diseases by varying number of daily tests, under different interven-tion strategies, over ER networks. aily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (a) COVID-19 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (b) Ebola Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (c) SARS Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (d) H1N1 Daily tests: 10 Daily tests: 100 Daily tests: 1000 Daily tests: 10000 day A v e r age c u m u l a t i v e i n f e c t i on s global oracleoracle tracer back tracingforward tracing randomnone (e) Measles Figure 10: The average daily cumulative infected populations of five known diseases by varying number of daily tests, underdifferent intervention operations, over ER networks. ctual infections Poistives Estimated from all positives Estimated from RT positives global oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnone day
12 3 4 (a) COVID-19
Actual infections Poistives Estimated from all positives Estimated from RT positives global oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnone day
12 3 4 (b) Ebola
Actual infections Poistives Estimated from all positives Estimated from RT positives global oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnone day
12 3 4 (c) SARS
Actual infections Poistives Estimated from all positives Estimated from RT positives global oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnone day
12 3 4 (d) H1N1
Actual infections Poistives Estimated from all positives Estimated from RT positives global oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnoneglobal oracleoracle tracerback tracingforward tracingrandomnone day
12 3 4 (e) Measles
Figure 11: The average daily threat levels of five known diseases by varying number of daily tests, under different interventionstrategies. day A v e r age da il y nu m be r s Average daily positives detectedAverage daily tests conducted (a)
Number of daily tests A v e r age f i na l i n f e c t ed p r opo r t i on s Aleta et al.Our setup (b)
Figure 12: Simulations of COVID-19 on the network provided by Aleta et al. [2] for πΌ = . (a) shows the average number oftests used and daily positives detected when of symptomatic infections are discovered. (b) shows the average final infectedpopulation proportions of the scenario in (a) and, for a scenario in which only hospitalized ( . of symptomatic infections)are discovered (Our setup). Following [2], only of the contacts are discovered during contact tracing. The forward tracingstrategy is considered in both figures, backward contact tracing shows identical results. upplementary Tables Table 1: Parameter used for experiments
Parameters Values π , 1 , , πΌ , 50, 100 π½ .
2, 0 . , 1 πΎ , , , 1 π π , 1 .
5, 2, , 3, , π , , , 1 . π π» , 0 .
1, 0 . , , , , Model π π½ πΎ π π» π COVID-19 [2, 12] SEIR 2 . . . . . . .
15 0 .
125 0 . .
333 0 . .
33 0 .
19 0 .
143 / 0 .
294 8 . . . .
143 0 . . .
932 0 .
274 / 0 .
079 0 .32