Digital Proximity Tracing in the COVID-19 Pandemic on 1 Empirical Contact Networks

Digital contact tracing is increasingly considered as a tool to control infectious disease outbreaks. 15 As part of a broader test, trace, isolate, and quarantine strategy, digital contract tracing apps have been proposed 16 to alleviate lock-downs, and to return societies to a more normal situation in the ongoing COVID-19 crisis. 1,2 Early 17 work evaluating digital contact tracing 1,3 did not consider important features and heterogeneities present in real-world 18 contact patterns which impact epidemic dynamics. 4,5 Here, we ﬁll this gap by considering a modeling framework 19 informed by empirical high-resolution contact data to analyze the impact of digital contact tracing apps in the COVID- 20 19 pandemic. We investigate how well contact tracing apps, coupled with the quarantine of identiﬁed contacts, can 21 mitigate the spread of COVID-19 in realistic scenarios such as a university campus, a workplace, or a high school. 22 We ﬁnd that restrictive policies are more effective in conﬁning the epidemics but come at the cost of quarantining a 23 large part of the population. It is possible to avoid this effect by considering less strict policies, which only consider 24 contacts with longer exposure and at shorter distance to be at risk. Our results also show that isolation and tracing 25 can help keep re-emerging outbreaks under control provided that hygiene and social distancing measures limit the 26 reproductive number to 1 . 5 . Moreover, we conﬁrm that a high level of app adoption is crucial to make digital contact 27 tracing an effective measure. Our results may inform app-based contact tracing efforts currently being implemented 28 across several countries worldwide. 1,6–11 29

correspond to a contagion event. Any contact tracing policy is thus based on thresholds on the 116 duration and proximity of a contact to define the associated infection risk. Among 'risky' contacts, 117 some correspond to infections while others do not. The latter correspond to "false positives", i.e., 118 non-infected individuals who will be quarantined. Similarly, among the contacts considered as non-119 risky by the contact tracing, some might actually be infected ("false negatives"). The use of real-120 world data makes it possible to evaluate the number of false positives and negatives for various 121 policies, together with their effectiveness in containing the spread. These outcomes represent 122 crucial information as they might determine the usefulness of contact tracing apps. On the one 123 hand, a low number of quarantined, or mis-calibrated policies, can unwittingly omit many potential 124 spreaders. On the other hand, highly restrictive policies might require to quarantine large numbers 125 of individuals, including non-infected people, with a consequent high social cost. 126 Overall, our approach allows us to evaluate the effect of different contact tracing policies, not only 127 on the disease spread but also in terms of their impact on the fraction of quarantined individuals. 128 We develop our main analysis adopting the most recently described epidemiological characteristics 129 of COVID-19, and we consider a scenario with a reproduction number R 0 = 1.5, representative of a 130 situation of a re-emerging outbreak that may be faced after the release of the lock-down measures.
Identifying a tracing policy that is able to contain the epidemic is a non-trivial task, and our aim 140 is to quantify and describe the properties that makes a policy effective, in terms of duration and 141 signal strength thresholds, tracing period of time and isolation efficiency. Furthermore, we show 142 that isolation and tracing measures are only as effective as the technology they rely upon: No 143 tracing is possible if adequate proximity and time resolutions are not available. 144 Even if it is clear that the choice of a particular policy should be primarily guided by its effectiveness 145 in containing the virus, we find that not all successful policies are equal. In particular, we demon-146 strate that beyond a certain accuracy, stricter policies do not improve the containment. Hence, 147 comparing policies also at this level allows one to improve their design and to reduce their side 148 effects. 149 2 Results and Discussion 150 We evaluate the effect of measures based on the deployment of a digital contact tracing app on the 151 mitigation of the Covid-19 pandemic. As we do not consider geography nor large-scale mobility, 152 our modeling can be considered as referring to a limited geographical area, similarly to previous 153 modeling efforts. 1, 35 154 In particular, we assume two types of interventions to limit the spread of the virus. First, infected 155 individuals are isolated when they are either symptomatic and self-reporting or if they are identified 156 through randomized testing. Second, individuals who have had a potentially contagious contact 157 with identified infected individuals can be preventively quarantined, following an exposure notifica-158 tion via an app on their smartphone. Schematically, if a detected infected individual has the app, 159 the anonymous keys that her/his device has been broadcasting through Bluetooth in the past few 160 days are exposed. The app of the individuals with whom s/he has been in contact in the past 161 days recognizes these keys as stored on their own device and calculates a risk score. If the risk 162 score obtained by an individual is above a certain threshold (determined by the considered policy), 163 the contact is "at risk" and the individual is assumed to go into quarantine. We refer to Troncoso 164 et al. 6 for more details on the implementation on privacy preserving proximity tracing. Our re-165 search question is whether or not it is possible to contain a COVID-19 outbreak by means of such 166 measures. 167 We introduce the concept of potentially contagious contact into a mathematical framework where 168 the epidemic evolution is governed by a model based on recursive equations, inspired by the work 169 of Fraser et al., 3 and recently adapted to the Covid-19 case. 1 This model quantifies the number 170 of newly infected people at each time interval, given a characterization of the disease in terms 171 of infectiousness and manifestation of symptoms. The model is specifically designed to consider 172 the two interventions described above, whose effectiveness are quantified by two parameters ε I , ε T 173 varying from 0 to 1 (where ε I = 0 means "no isolation" and ε I = 1 represents a perfectly successful 174 isolation of all individuals who are found to be infected, either via self-reporting or testing; ε T 175 quantifies instead the efficacy of contact tracing). 176 Here, particular attention is devoted to the study of the dependence of ε T on ε I , and to assess 177 which policies are achievable given the present technology and resources. To this aim, we couple 178 this model with a realistic quantification of the effect of these two measures based on real-world 179 contact and interaction data. The following results are indeed obtained by simulations using the 180 Copenhagen Networks Study (CNS) dataset 50 that describes real proximity relations of smart-181 phone users measured via Bluetooth (see Section 4.1). Moreover, we present in the  tary Information simulations performed using two other datasets collected by the SocioPatterns 183 collaboration with a different type of wearable sensors. 51, 52 184 We emphasize that in each case, a realistic quantification of the tracing ability is obtained by 185 simulating the epidemics on a dataset, but the controllability of the disease is assessed by the 186 general mathematical framework and is therefore not bounded by specific datasets. 187 True positive (infected) False positive (not infected) False negative (infected) True negative (not infected) Quarantined Contagious contact Tracing Figure 1: The contacts among users of the contact tracing app are registered through via the app.
As soon as an individual is identified as infected s/he is isolated, and the tracing and quarantine policy is implemented. Depending on the policy design, the number of false positives and false negatives may vary significantly.
We consider five different policies ( Table 1) that correspond to different threshold levels on the 188 signal intensity, considered as a proxy of distance, and on the duration of the contact. Recall that 189 Bluetooth does not measure distances per se, therefore all real-world implementations are based 190 on thresholds on Received Signal Strength Indicator (RSSI) values. We additionally assume that 191 each individual app stores the anonymous IDs received from other apps, representing a history of 192 the past contacts over the past n days. Here, we consider n = 7 days, as we found (Supplementary Information C.1) that a longer memory does not produce any significant improvement. Overall,194 this implies that we consider a simplified version of the app, which does not compute risk scores 195 but is simply able to remember the contacts corresponding to a sufficiently close and long-lasting 196 proximity during n days, while contacts below the thresholds are not stored. In addition, each policy 197 is tested with the isolation efficiency values ε I = 0.2, 0.5, 0.8, 1, which encode isolation capacities 198 ranging from rather poor to perfect isolation of any symptomatic or tested positive person. 199 Signal strength Duration Contact  Table 1: Parameters defining the policies, and fraction of the total number of interactions of the CNS dataset that they are able to detect. A larger value of the magnitude of the signal strength tends to correspond to a larger distance, such that in the second column the thresholds go from the least to the most restrictive policy.
A fundamental difference between the policies we consider is related to the fraction of contacts 200 that are stored by the app. Figure  Finally, we consider two additional policies (  Figure 2: Growth or decrease rate of the number of newly infected individuals, assuming either that all the infected people can eventually be identified and isolated ( Figure 2a); or that only symptomatic people can be isolated with 20% of infected individuals asymptomatic ( Figure 2b); or that only symptomatic people can be isolated with 40% of infected individuals asymptomatic ( Figure   2c). In all settings the cases are reported with a delay of 2 days. 250 We have run numerical simulations of the disease spread on the CNS dataset, implementing the 251 different policies to determine their impact in mitigating the epidemic. tiousness, but a non-negligible number of contacts occur while the individual is highly infectious.

257
By considering the five different policies of Table 1 in terms of which contacts are considered at 258 risk and thus kept by the app, and running corresponding simulations for ε I = 0.2, 0.5, 0.8, 1, we 259 obtain in each case the actual value of ε T which quantifies the quality of the tracing policy (see 260 Section 4.1). This value therefore ceases to be an arbitrary parameter: it is a direct consequence 261 of the policy, the value of ε I and the contact data. We then plug the values (ε I , ε T ) into the 262 idealized model, observe in which region of the diagrams of Figure 2 they fall, and deduce whether 263 containment is achieved or not by this policy and this value of ε I .

264
The results, reported in Figure 4 (center right), reveal that not all parameter configurations are 265 feasible. In particular, the largest value of the tracing efficacy ε T can be reached only for Policies 266 4 and 5. Policies 1, 2 and 3 still manage to reach the epidemic containment phase if ε I is large 267 Figure 3: The top left panel shows a scatterplot of signal strength vs duration for all contact events in the CNS dataset, and displays the thresholds defining the various policies: the contacts identified as "at risk" are those included in the areas identified by the colored lines. Top right and bottom left panels separately depict the distributions of signal strength and duration, together with the infectiousness functions ω dist and ω exposure respectively (black curves), see Table 3  ple assuming that symptomatic people can be isolated and that an additional 50% of asymptomatic can be identified via randomized testing. The points correspond to the parameter pairs such that ε I is an input and ε T an output of the simulations on real contact data, for the five policies. The different scenarios are defined by an app adoption level of 60%, 80%, or 100% (from left to right), and by a value of R 0 equal to 2, 1.5, or 1.2 (from top to bottom). Until now, we have assumed that the only limit to reach a perfect contact tracing resides in the 276 technology specifications of each policy. This implies however that the totality of the population 277 adopts the app, something which is clearly unrealistic in practice. We thus repeated our simulations 278 assuming that only a fraction of the population uses the app, while the remaining individuals are 279 outside the reach of the tracing and quarantining policies, but they are still isolated whenever 280 detected because symptomatic or through random testing (see Section 4.1.5). 281 We found that reducing the app adoption implies an important reduction in the tracing policies 282 effectiveness. The first two columns of panels in Figure 4 report the results for an adoption of 60% 283 and 80% respectively. If R 0 = 2, practically none of the policies is able to stop the spreading. 284 However, this pessimistic scenario changes under the current working hypothesis of R 0 = 1.5 285 (second line of panels in Figure 4). An app adoption of 80% or even 60% is then sufficient to 286 obtain good results: all policies except for Policy 1 manage to contain the spread for ε I = 0.8, and 287 all of them for ε I = 1 (Figure 4, center row). The situation is even better with a smaller value of 288 R 0 = 1.2. In this case, even in the case of an app adoption of only 60%, all policies are effective 289 as soon as the isolation efficacy is at least 0.5 (bottom left panel in Figure 4). 290 We observe that the tracing efficiency, which clearly varies considerably with different levels of 291 app adoption, practically does not depend on R 0 . Indeed, ε T only accounts for the fraction of 292 secondary infections that are correctly traced, independently on the spread of the virus and the 293 amount of infected people in the population. 294 We also note that the effect of a limited app adoption on the tracing efficiency ε T appears to be 295 quadratic: a 60% app adoption reduces the efficiency roughly to its 40%, while an 80% adoption 296 reduces it to the 70% (see Supplementary  of a different set of contacts considered as at risk. In some cases this produces the desirable 308 effect of containing the spread, but side effects emerge as well. Indeed, some of the "at risk" 309 contacts do not actually lead to a contagion event, while contacts classified as non risky might, 310 since the spreading process is inherently stochastic. It is thus important to quantify the ability 311 of each policy to discriminate between contacts on which the disease spreads and the others, in

338
In this study, we have analyzed the ability of digital tracing policies to contain the spread of Covid-339 19 outbreaks using real interaction datasets to estimate the key effective parameters and to shed 340 light on the practical consequences of the implementation of various app policies. 341 We found that the set of parameters that allow containment of the spread is strongly influenced 342 by the fraction of asymptomatic cases. By first assuming an ideal setting where any pair of pa-343 rameters ε I , ε T is possible, we showed ( Figure 2) that the area of the phase space representing 344 the setting where it is possible to control the epidemic is reduced when considering 20% or, worst 345 case scenario, 40% of asymptomatic individuals in the population, i.e. infected people that we 346 cannot isolate nor contact trace and who therefore continue spreading the virus to their contacts. 347 We remark that this is in contrast with the scenario considered in Fraser et al. 3 and in Ferretti et 348 al., 1 where the entire infected population is assumed to become symptomatic eventually. 349 We tested five policies to define risky contacts that should be traced (Table 1), with different restric-350 tion levels. When implemented on real contact data measured by Bluetooth, this approach allows 351 us to estimate, for each value of ε I , the actual value of ε T and thus to determine the efficacy of 352 each policy (colored points in Figure 4). Using these implementations on real data restricts the 353 available values of the control parameters. This added layer of realism reveals that only the most 354 restrictive policies can lead to epidemic containment. 355 Moreover, even for these policies, tracing is effective only if the isolation is effective: Policy 2 356 requires a perfectly effective isolation (ε I = 1), while an 80% isolation is sufficient for Policies 3, 357 4, and 5 ( Figure 4). In particular, better tracing policies may work with a less effective isolation 358 strategy. 359 Our results highlight how isolation and tracing come at a price, and allow us to quantify this price 360 using real data: the policies that are able to contain the pandemic have the drawback that healthy The overall output of the model is the predicted number λ(t) of newly infected individuals at time 441 t, and we are interested to study policies that contain the epidemic, i.e., such that λ(t) → 0 as t 442 grows. 444 We describe here the network simulation that leads to the estimation of the parameter ε T .

451
At each time instant, and for each node in the graph which is currently infected, a probability distribution is used to decide whether or not the virus is spread to each of its contacts. This probability is the product of three components, i.e., and it quantifies what contacts are relevant for the disease transmission. The three components • ω dist (s s ), the probability for an infected individual to transmit the disease given the signal 458 strength s s of a contact, appearing in Figure 3 (top right panel). 459 We refer to Table 3 in Supplementary Information for the definition of each of these distributions.  • Each τ i value is incremented by δ.

469
• If an individual i is neither isolated nor quarantined, s/he can infect each of her/his neighbors j of with a probability ω data (τ i , s i,j , e i,j ) (see previous Section). 471 • Newly infected people are assigned τ = 0 and a time to onset symptoms t + t o , where t o is 472 extracted from the distribution onset time(·) defined in Table 3. 473 • If an individual is recognized as infected (either as symptomatic or by testing) but still not 474 isolated, we isolate him/her with probability ε I and we quarantine all her/his contacts accord-475 ing to a policy, that is, all her/his contacts above a spatio-temporal threshold (see the next 476 Section for a precise description of this policy).

477
• If a quarantined individual becomes symptomatic, we quarantine all her/his previous contacts 478 (i.e., before entering quarantine) according to the above-mentioned policy. to the limitation of the tracing efficiency. This is quantified by e T (t), which takes values between 0 502 and 1 and is in general time-dependent. 503 The value of e T is estimated from the numerical simulations on the real temporal networks of 504 contacts as follows. Once an individual is isolated we trace her/his contacts according to the 505 chosen policy, then we count the fraction of people that s/he has actually infected who remained 506 outside of the quarantine. By averaging on individuals and time we obtain e T . The obtained 507 value thus encodes the contribution of the chosen policy, adoption rate, duration of the memory of 508 contacts and potentially the warning of only the direct contacts or also of contacts of contacts. 509 The tracing efficiency can therefore be defined as the product of the two independent factors: such that we obtain the maximum efficiency only if isolation is perfect and the quarantine error e T 511 is zero. When modeling different levels of adoption of the app we implement the following procedure: we 514 extract at the beginning of each simulation a random list of users, that will act as non adopters. 515 During the simulation these agents will contribute to the spread of the virus and will be subject 516 to isolation whenever detected as infected, as any other individual, but in that case their contacts 517 cannot be traced. Moreover they do never appear in any contact list, and thus they are never 518 quarantined. In practice we simulate the fact that a contagious contact is recorded only if both the 519 infectious and the infected have the app. 520 We make the simplifying assumption that the app influences only the quarantining of individuals, 521 but not the isolation policy. Namely, we assume to be able to detect and thus isolate an infected 522 individual independently of the app, while we are able to trace the contacts only between pairs of 523 app adopters.    A Characteristic parameters of the disease 761 The infectiousness β(τ ) of an infected individual in the continuous model is assumed to be given 762 by the product of R 0 and the curve ω(τ ) described in gious. An alternative shape of the curve is discussed in Section A.3.

771
In the simulation we do not make use of the general infectiousness β(τ ) but we consider the con-  Table 3. can thus be set to the desired value), the relation between the infectiousness of a contact ω data 790 and the reproduction number is more implicit. Indeed, ω data depends on contagion probabilities as 791 functions of the distance and duration of a contact (see Table 3), and the reproduction number is 792 a consequence of the form of ω data and the contact data. We thus tune the parameters defining 793 ω exposure so that the empirical reproduction number R data 0 coincides with the actual R 0 used in the 794 continuous model. 795 Although it is known that R 0 has a large variability 42, 49 and some recent works 42, 69 suggest that the 796 relationship between R 0 and the real size of an outbreak is not trivial, the procedure to estimate  Table 4 the results of the simulation. 804 Since the simulations are stochastic in nature, we report the mean and standard deviation of R data   , and the corresponding estimates of the mean and standard deviation of R data 0 . 809 tiousness probability 811 We consider here another infectiousness curve that has been derived in the recent literature by He 812 et al. 56 . We show that, although this curve is different from the curve ω that we use in this paper, 813 the predictions of the model do not change significantly. This means that the model predictions are 814 robust with respect to changes in the assumed infectiousness curve.  Figure   824 6a. 825 From these g, f , we can reconstruct a PDF ω He (τ ) to be used in our model. This can be done simply by sampling two values from g and f and adding them (the total time from infection to secondary infection is simply split into two intervals separated by the time of symptoms onset). A numerical PDF of this distribution ω He , computed over the same 10 5 samples, is in Figure 6b. This function ω He may also be obtained analytically by convolution as using the analytically known f and g. The discretized convolution is also shown in Figure 6b, and 826 it coincides indeed with the numerical values of ω He .

827
Observe that this distribution assigns a small but positive probability (1.42%, see below) also to in-828 fectiousness at negative time (i.e., an individual may infect another one before being itself infected). 829 We ignore this small probability, and we assume that this is due to the fact that the two distributions 830 f and g are estimated from two different populations (according to 56 ), and thus statistical errors 831 may be present. 832 Figure 6b shows also the PDF ω that we used in the paper. Both distributions ω and ω He peak at 833 around 5 days, and they have similar support. The main difference is that the right tail of ω He is 834 larger, meaning that it models a non negligible probability of secondary infection also several days 835 after the infection of the spreader. PDFs f and g ( Figure 6a); estimated PDF ω H e, and PDF ω He (Figure 6b); fir of ω He with a lognormal distribution.
To have an analytical expression of ω He we try to fit shifted lognormal, gamma, and Weibull distri-837 bution to ω He (by least-squares minimization over the numerically computed PDF). The best results 838 are obtained with a lognormal distribution with µ = 2.087, σ = 0.457, and shifted by 2.961, which 839 is plotted in Figure 6c. This allows also to derive an explicit cumulative density function CDF He of 840 ω He , which gives an estimate of CDF He (0) = 0.0142 (the fraction of negative-time infections). 841 We can now use this modified infectiousness ω He in our model and compare the results with the 842 ones of Figure 4. First, we estimate the parameters defining ω exposure as in Section A.2 (see Table   843 5. The chosen values is also in this case β 0 = 0.001, corresponding to a value R data 0 = 1.5.    correspond to the parameter pairs such that ε I is an input and ε T an output of the simulations on real contact data, for the policies of Table 1 conclusions would be introduced by adopting this ω He in place of the current one.

852
The epidemic model form 1, 3 (to which we refer for a precise derivation) provides a quantification of 853 the number Y (t, τ, τ ) of people at time t that have been infected at time t − τ by people who have 854 in turn been infected at time t − τ .

855
The model characterizes Y as a function of s(τ ) and β(τ ) (see Section A in Supplementary Information). Observe that both are quantities in [0, 1], and that s(τ ) is a non decreasing. The model then states that Y (t, 0, t) is a given initial value and that for 0 ≤ τ < t it holds In the two cited papers the values of ε I , ε T ∈ [0, 1] are fixed, while we assume from now on that 856 they depend on τ .

857
Observe that in the absence of containment policies (i.e., ε I = ε T = 0) the model predicts a As mentioned before, the model was analyzed in 1, 3 by considering its asymptotic behavior as t 864 grows to infinity. We instead need a finite-time model that allows a flexible treatment of real data. 865 To this end, it is convenient to use the variable Λ(t, τ ) := Y (t, 0, τ ) (see 3 ) which represents the 866 number of people which are infected at time t by people who have been infected for time τ ≤ t.

867
With straightforward manipulations, equation (2) can be rewritten for 0 ≤ τ < t as follows where we changed the integration variable to ρ := τ − τ , and we used the translational invariance of Y . In the variable Λ, this reads as Observe that this is an evolution equation that requires to define an initial number of infected 868 people, i.e., we assume that the quantity Λ(0, 0) := Λ 0 is a given number. 869 The quantity of interest is then the total number λ(t) := t 0 Λ(t, τ )dτ of newly infected people at 870 time t.

872
We fix a value T > 0 as the maximal simulation time and take n + 1 points in [0, T ] i.e., τ i := i T n , 873 0 ≤ i ≤ n. 874 We will approximate the values of Λ(τ k , τ i ) for k = 1, . . . , n and i = 0, . . . , k − 1, while, according 875 to, 3 we set Λ(τ k , τ i ) = 0 for all i ≥ k. Moreover, we assume that the value Λ(τ 1 , τ 0 ) is given. 876 Observe that this discretization is equivalent to assume that the number of new cases is measured 877 only at equal discrete times (e.g., at the end of each day) rather than measured continuously. 878 We show in the next section that the continuous model (3) can be approximated by defining a suitable value for Λ(τ 1 , τ 0 ), and then iteratively computing the values of Λ(τ k , τ i ) by applying the simple formula where the matrix A ε I ,ε T ∈ R n×n is defined for 0 ≤ i, j ≤ n − 1 as to be solved, and only the asymptotic state can be estimated. 882 Moreover, we can use Λ to compute (4) 883 We fix a value T > 0 as the maximal simulation time and take n + 1 points in [0, T ] i.e., τ i := i T n , 884 0 ≤ i ≤ n.

888
For 1 ≤ k ≤ n we first evaluate (3) at the points, first in the variable t for 1 ≤ k ≤ n, i.e., and then in the variable τ for τ < t, that is for 0 ≤ i < k ≤ n, i.e., Now observe that for 0 ≤ i < k ≤ n we have which ranges between τ k for i = 0 and τ 1 for i = k−1. The last equation becomes for 0 ≤ i < k ≤ n We can then use the quadrature rule (5) to discretize the integral and obtain

Observe that the upper limit in the sum has values
which ranges between τ i and τ k−1 . Inserting this into the last equation we get for 0 ≤ i < k ≤ n We can define the matrix A ε I ,ε T ∈ R n×n whose entries are defined for 0 ≤ i, j ≤ n − 1 as which has a triangular structure (the first row is nonzero, in the second row the last element is zero, 889 ..., in the last row only the first element is nonzero).

890
With this matrix we can rewrite (6) as which is a recursive equation that determines the evolution of Λ(t, τ ) once an initial condition is 891 given. 892 Assuming for now that these initial conditions are given, we can compute Λ(τ k , τ i ) forward in k and 893 backward in i. That is, after we computed Λ(τ , τ i ) for all = 1, . . . , k − 1, and for 0 ≤ i < , we can 894 use (7) to compute Λ(τ k , τ i ) for 1 ≤ i < k, since in this case the right hand side contains values 895 Λ(τ k−i , τ j ) which have already been computed since 1 ≤ k − i ≤ k − 1 for 1 ≤ i < k.

896
The only remaining case is i = 0, and in this case the formula (7) gives instead and thus .

This term is positive if and only if
Since the left hand side is at most β(τ 0 ), it is sufficient to require that n/T > β(τ 0 ), or n > β(τ 0 ) · T .

897
In this way we defined Λ(τ k , τ i ) for all values 1 ≤ k ≤ n and 0 ≤ i < k. It remains to assign the 898 value Λ(τ 1 , τ 0 ), which can be fixed to the initial value Λ 0 .   The graphs report the mean and standard deviation (shading) over 20 independent runs. The table reports mean and standard deviation of the total number of distinct individuals who have been quarantined over the whole simulation timeline and the percentage of those among them who were effectively infected (true positive), corresponding to the attack rate. 925 An additional possibility is to keep track of contacts in a recursive way. Namely, when an individual 926 is isolated, not only its contacts are quarantined, but also its contacts' contacts. For these reasons we remark that the concept of second-order tracing, a topic of recent discus-958 sions, deserves further investigation and may possibly be expanded in a follow-up of this work.

960
To complement the analysis of Section 2.3 on the effect of a limited app adoption, we provide here 961 a numerical quantification of the reduction in the values of ε T . For the five policies of Table 1 and   962 for ε I = 1, we report in Table 6 the ratio between the values of ε T in the case of a limited app 963 adoption (60% or 80%) and in the case of full app adoption (100%), in all cases with R 0 = 1.5.

964
Observe that the computed values are actually quite close to show a quadratic reduction effect.

966
To additionally verify the robustness of our predictions with respect to the epidemiological mod-967 elling, we assume here that the number of asymptomatic individuals is 20%, and additionally that a 968  randomized testing policy that covers 25% of the asymptomatic population is in place. In this case, 969 our results (see Figure 10) show that the policies of Table 1  +3.5% Figure 10: Tracing policy efficiency with 80% asymptomatic and 25% random testing. Growth or decrease rate of the number of newly infected people assuming that symptomatic people account for the 80% of the infected individuals, that they can be isolated and that an additional 25% of asymptomatic can be identified via randomized testing. The points correspond to the parameter pairs such that ε I is an input and ε T an output of the simulations on real contact data, for the policies of Table 1. C.5 Close-range short-exposure vs long-range long-exposure interactions 973 We test here two additional policies obtained by mixing a low space resolution and a high time 974 resolution, and viceversa. The policies are defined in  Table 7: Parameters defining the two additional policies, and fraction of the total number of interactions of the CNS dataset that they are able to detect.  Figure 11b) of the contacts in the CNS dataset. Figure 11a gives a scatterplot of signal strength vs duration, and displays the thresholds defining the two policies of Table 7.
The values of the parameters (ε I , ε T ) characterizing the numerical simulations for the new policies 981 with R 0 = 1.5 are shown in Figure 12 (see Figure 4, center-right panel, for a comparison with the 982 policies of Table 1), and it is clear that Policy 7 is as effective as the most restrictive policies (Policy 983 4 and Policy 5), while Policy 6 fails to contain the virus for an isolation efficiency smaller than 0.8.

984
The efficiency of quarantines is assessed by the number of false positives and false negatives, 985 reported in Figure 13. 986 We deduce that the ability to control the contagion seems to be more sensitive to duration of con-987 tacts than to their spatial distance. Indeed, policies which capture close range but short exposure 988 interactions happen to be less performative in quarantining people than those signaling long range 989 interactions with long exposure. In other words, quarantining individuals who have had a short in-990 teraction with an infected one, even if at close-range, is unnecessary. On the other hand, it appears 991 to be important to track contacts with a high spatial resolution, including the ones that happens at 992 a rather long distance. 993 However, we remark once more that these results are depending on the infectiousness model that 994 we have defined here, and that they could possibly change in a different setting. -8.0% -5.5% -3.0% -0.5% +2.0% +4.2% Figure 12: Tracing policy efficiency (alternative policies) . Growth or decrease rate of the number of newly infected people assuming that symptomatic people can be isolated and that an additional 50% of asymptomatic cases can be identified via randomized testing. The points correspond to the parameter pairs such that ε I is an input and ε T an output of the simulations on real contact data, for the policies of Table 7.
Figures 14 and 15 refer to the more realistic case where the app adoption is reduced to 80%. We 996 also maintaing the assumption that 20% of the infected individuals are asymptomatics or, equiva-997 lently, that they are instead the 40% and an additional 20% (that is 50% of the asymptomatics) is 998 identified through random testing. +4.2% Figure 14: Tracing policy efficiency with 80% symptomatic (alternative policies). Growth or decrease rate of the number of newly infected people assuming an 80% app adoption level, with 80% symptomatics. The points correspond to the parameter pairs such that ε I is an input and ε T an output of the simulations on real contact data, for the policies of Table 7.
terns collaboration 1 , which is based on wearable active Radio Frequency Identification (RFID) 1004 devices that exchange radio packets, detecting close proximity (≤ 1.5m) of individuals wearing the  Table 7, assuming an isolation efficiency of ε I = 0.8, an 80% app adoption level, and with 80% symptomatics. The graphs depict the mean and standard deviation over 20 independent runs. The table reports mean and standard deviation of the total number of distinct individuals who have been quarantined over the whole simulation timeline and the percentage of those among them who were effectively infected (true positive), corresponding to the attack rate.
thus defined only as a function of contact durations. 1008 In order to see the effectiveness of the policies and the spreading of the virus, it is needed that the 1009 length of the collected data is larger than 15 days. As the SocioPatterns data have a high temporal 1010 resolution (20 seconds) but were collected for shorter overall durations, we artificially extend the 1011 length of each dataset by replicating it (coping and pasting the entire dataset at the end of the 1012 dataset itself).  For both these datasets, similarly to the CNS dataset, most contacts happen before the infec-1015 tiousness reaches its peak (Figure 16), even if contacts are present for all possible durations. 1016 Nevertheless, these are sufficient to spread the infection. 1017 We further run the simulations on the network for the five policies of  positive reach a maximum between 10 and 15 days and then start decaying. In the case of InVS15 1027 this decay to zero is more evident since the simulation time is sufficiently long (Table 8)