vaccine prioritisation using bluetooth exposure notification …...2020/12/14 · vaccine...

Vaccine Prioritisation Using

Bluetooth Exposure Notification Apps

Mark D Penney1,2, Yigit Yargic1,3, Lee Smolin1, Edward W Thommes4,5,6, Madhur Anand7, andChris T. Bauch2

1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada2Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario Canada

3Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada4Vaccine Epidemiology and Modeling, Sanofi Pasteur, Toronto, Ontario, Canada

5Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada6Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

7School of Environmental Sciences, University of Guelph, Guelph, Ontario, Canada

After vaccinating health care workers and vulner-able groups against COVID-19, authorities willneed to decide how to vaccinate everyone else.Prioritising individuals with more contacts can bedisproportionately effective, in theory, but identi-fying these individuals is difficult. Here we showthat the technology underlying Bluetooth expo-sure notification applications, such as used for dig-ital contact tracing, can be leveraged to prioritisevaccination based on individual contact data. Ourapproach is based on the insight that these appsalso act as local sensing devices measuring eachuser’s total exposure time to other users, therebyenabling the implementation of a previously im-possible strategy that prioritises potential super-spreaders. Furthermore, by generalising percola-tion theory and introducing a novel measure ofvaccination efficiency, we demonstrate that this“hot-spotting” strategy can achieve herd immu-nity with up to half as many vaccines as a non-targeted strategy, and is attractive even for rela-tively low rates of app usage.

Since the first recorded cases in late 2019, COVID-19 has spread like wildfire across the globe. In orderto contain the spread of the disease, an array of non-pharmaceutical interventions have been deployed, includ-ing hygienic measures, isolation of infected individuals,quarantine of their exposed individuals, social distancing,and even large scale closures of businesses and institutions[41]. This stimulated efforts to develop COVID-19 vac-cines in record time [24]. In late 2020, the first phaseof COVID-19 vaccination began with a small numberof health-care workers and elderly individuals receivingthe Pfizer/BioNTech vaccine. These groups were recom-mended for vaccine prioritisation [4, 3] based on the high

rate of mortality and other severe outcomes experiencedby elderly individuals [13] and the principle of reciprocity,which states that those who accepted the greatest risks tomitigate the effects of the pandemic, should be vaccinatedfirst [36].

Once the most vulnerable individuals have been vac-cinated, public health authorities will need to decide howto vaccinate the rest of the population[27, 21, 12]. In2021, data on whether available COVID-19 vaccines pre-vent both transmissibility and disease in vaccinated in-dividuals will become increasingly available [18]. Hence,public health authorities may want to consider pivotingto a strategy that interrupts transmission of SARS-CoV-2 most effectively. And in all likelihood, the supply ofvaccines and/or the capacity to administer them will con-tinue to be limited in early 2021[24], necessitating addi-tional prioritisation decisions.

In this work we study how to prioritise COVID-19vaccines in the second phase, where vaccination of vul-nerable groups has been achieved. We assume a goal ofachieving herd immunity through interrupting transmis-sion as efficiently as possible, and we explore a scenariowhere COVID-19 vaccines can prevent transmissibility aswell as disease, as applies to many other existing vaccinesagainst respiratory pathogens [10].

Prioritisation of vaccines is often based on age or otherdemographic factors, and we know that contact patternsdiffer between demographic groups. However, as demon-strated in empirical studies of contact patterns [30], het-erogeneity within demographic groups is much greaterthan the differences between them. Because individualswith more contacts have the potential to fuel the spreadmore than others, prioritising individuals for vaccinationaccording to the risk they pose to further spread has beenfound to be highly effective in previous theoretical mod-

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els [35, 19, 16, 14]. However, such exposure prioritisa-tion strategies are difficult to implement in practice sincepublic health authorities only have information on broaddemographic or regional factors upon which to act.

It is exactly in this regard that Bluetooth exposurenotification apps have opened up a new avenue for publichealth interventions. The core functionality of these appsis the creation of an encounter log between app users.This encounter log is useful for more than just exposurenotification: It also quantifies the user’s exposure to oth-ers. Said another way, these apps are also sensors whichmeasure the duration of exposure: an epidemiologicallysignificant quantity.

Though other digital contact tracing technologies ex-ist, Bluetooth-based solutions have been chosen when theprotection of user’s personally identifiable information wasa core concern. The data in the encounter log can only beused by public health authorities in a manner which doesnot require the centralized collection of user data. In fact,in our proposal each user’s device makes a decision purelylocally, based on the number of encounters reported, as towhether or not they will be prioritised for vaccination. Bypreferentially vaccinating individuals with greater totalexposure to others our proposed strategy is significantlymore efficient than traditional prioritisation strategies.

The success of digital contact tracing technologies hasbeen hampered by inadequate uptake rates [23, 11]. Ourproposal doesn’t suffer from this issue. We show that theefficiency depends only on the fraction of the app-usingpopulation which receive the vaccine, with the greatestrelative reduction occurring with the vaccination of roughly20-40% of app users.

Our modelling approach is based on percolation the-ory: a collection of analytical techniques, coming fromstatistical physics [39], which has been successfully ap-plied to material sciences [37] and to the spread of forestfires [25, 26] and infectious diseases [17]. Here we extendthe theory in two directions: Firstly, we introduce a novelmeasure to compare vaccine prioritisation strategies; andsecondly, we incorporate heterogeneity in the risk of infec-tion (total exposure time) between contacts. These toolsallow us to show that the improved efficiency gained bypiggy-backing vaccine prioritisation strategies on top ofexposure notification apps is a robust phenomenon, as itderives its power from the very heterogeneities in contactpatterns that shape the spread of COVID-19.

Bluetooth Exposure Notification

In March 2020, COVID Watch released a white paperdetailing an anonymous Bluetooth-based system that ex-ploits the ubiquity of Android and iOS smartphones tosupport contact tracing [20]. The idea has seen widespread

adoption, with nearly every developed country having in-corporated it into their digital contact tracing solutions.Both the Google/Apple and BlueTrace frameworks arealso based on this technology, the former of which is avail-able throughout North America and the European Unionand the latter in Singapore and Australia.

Protecting users’ privacy was a fundamental princi-ple underlying the design process. Every 10-20 minutesanonymous tokens are exchanged between app users whoare in close proximity to one another. Each user’s de-vice stores these tokens in an encounter log. The datastored by the app is able to determine neither the num-ber of contacts nor the duration of any individual con-tact. Instead, the number of tokens logged over a giventimeframe is a measure of the user’s total exposure timeto others: the sum, over each contact, of the duration ofthat contact. To model this feature of Bluetooth exposurenotification systems and how it impacts vaccination pri-oritisation strategies, we present new theory in the nextsection that incorporates both the number and durationof contacts. This theory predicts that the total expo-sure time of contacts has a direct causal influence on thespread of infectious disease.

Country Downloads per 100 peopleCanada [2] 20

Germany [8] 29Italy [6] 17

New Zealand [1] 49Scotland [5] 27Singapore [7] 56

Table 1: Exposure notification app download rates in selectedcountries based on official sources.

Percolation on Weighted Networks

Many infectious diseases spread through close contact,and contact patterns in human populations display a highdegree of heterogeneity [9, 22]. One successful approachto understanding the impact of heterogeneity is to modelthe spread of infectious disease as a percolation processon the network of contacts [29, 28, 38]. In the applicationof percolation theory to infectious diseases, the sum totalof human contacts forms a network across the entire pop-ulation. The vertices of this network are the individualsin the modelled population. An edge joins two individualsif they come into potentially infectious contact with oneanother in a typical time window matching the infectiousperiod of the disease. The transmissibility of the infec-tious disease is described by the transmission probability,denoted T .

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In a series of foundational works [32, 33], Newman de-veloped analytical techniques based on probability gener-ating functions. Through these techniques one can deriveformulas for key epidemiological quantities in terms ofsummary statistics of the degree distribution of the con-tact network, that is, the distribution of the number ofcontacts made by individuals in the modelled population.In particular, Newman proved that the basic reproductionnumber, R0, depends on the transmission probability T ,the average degree, and the variance of the degree distri-bution. As a corollary of these results, in particular thepresence of the variance, we learn a fundamental insight:heterogeneity in contact patterns influences the spread ofinfectious disease.

The extent to which contact patterns differ betweenindividuals goes beyond the number of contacts they have.Another key variable is the duration of each contact, asover the infectious period of the disease a typical indi-vidual will accumulate a large time in contact with a fewregular contacts, and a much smaller amount of time witheach of a potentially larger number of irregular contacts.Studies have shown that longer contact duration increasesthe transmission probability of COVID-19 [34]. Both theepidemiological significance of contact duration and thefunctionality of Bluetooth exposure notification systemsimply the need to incorporate contact duration into thepercolation framework.

We have generalized Newman’s analytic framework tounderstand percolation on weighted networks (see Meth-ods). In our model, each edge, which represents a contactbetween individuals, is further equipped with a weightrepresenting the duration. The transmission probabilityis assumed to depend on the duration of contact. Thisleads to replacing T with a distinct Tw for each weight.

We derive a formula showing that total exposure timeof individuals is a driving factor in the spread of disease.As we discussed above, Bluetooth exposure notificationapps measure each user’s total exposure time to otherusers of the app. Our model therefore predicts that to-tal exposure time is an epidemiologically relevant quan-tity with a direct causal relationship to the reproductionnumber. Therefore, vaccine prioritisation via exposurenotification apps is both plausible and potentially effec-tive. To further examine the role of total exposure time inepidemic dynamics, in the next section we develop a quan-titative framework for comparing prioritisation strategies.

Because of heterogeneities in contact patterns, vacci-nation of different individuals will have a varying impacton the rate of spread. In this regard, heterogeneity is aresource: preferential vaccination of those with greatertotal exposure time should lead to a greater reduction ofdisease spread from the same number of vaccine doses.

In order to compare different strategies we must fur-ther expand the percolation toolkit to incorporate vac-

cination. We model vaccination as a stochastic processwhich removes vertices from the original contact network.We assume that the strategies under consideration priori-tise individuals according to the number and duration oftheir contacts, which is the case for strategies leveragingBluetooth exposure notification apps. In the attachedmethods we derive formulas for the reproduction numberon the residual network of susceptible individuals for ageneral vaccination strategy of this type.

Assessing Vaccination Efficiency

To compare two vaccine prioritisation strategies it is notenough to consider the post-vaccination reproduction num-ber without taking into account the number of individualsvaccinated by each strategy. This is especially true if oneis concerned with how best to use a limited vaccine sup-ply. To that end we introduce a novel measure to comparevaccine prioritisation strategies. We define the efficiencyof a vaccination strategy to be:

Ev=1 −Rv/R0

V,

where Rv is the post-vaccination reproduction numberand V is the fraction of the population that receives thevaccine. In other words, the efficiency of a strategy is thepercentage decrease achieved in the reproduction numberper percentage of the population vaccinated. It is im-portant to note that since vaccination is modelled as astochastic process, both Rv and V , and hence the effi-ciency, will vary across different realizations of the vacci-nation process.

The baseline strategy against which we compare pri-oritisation strategies is the uniform strategy, under whichvaccines are distributed uniformly across the populationto achieve a target vaccine coverage without taking intoaccount any form of contact heterogeneity. The efficiencyof the uniform strategy equals 1 since the expected repro-duction number decreases linearly with the vaccine cov-erage from its original value. Strategies for which the ex-pected efficiency is less than 1 are regarded as inefficient,since the same number of vaccines could have achieved agreater impact if they were allocated uniformly. On theother hand, strategies with efficiency greater than 1 arepromising candidates for a significant impact. In the nextsection we introduce our exposure prioritisation strategybased on Bluetooth exposure notification apps whose ef-ficiency is much greater than 1. We will refer to thisvaccination strategy as “hot-spotting”, in reference to afire-fighting practice that focuses on areas with intensefires [31].

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Figure 1: The efficiency for the hot-spotting strategy forR0 = 1.5 with various app usage rates are compared with theuniform vaccination in relation to vaccine coverage. The dotsrepresent the mean efficiency obtained from 10,000 simula-tions for each dot. The shaded regions show the intervals thatcapture 90% of all simulation results. The dashed lines in thebottom figures show the expected efficiency obtained from theformula in Methods.

The Hot-spotting Strategy

Here we propose a “hot-spotting” vaccination strategythat prioritises app-using individuals for vaccination ac-cording to their total exposure time. The strategy oper-ates without the central collection of any user data, asthe prioritisation of each user is decided locally by theirdevice.

The strategy depends on a parameter β encoding theprobability of success in a weighted coin-flip. Each userperforms a coin flip for each encounter stored in their en-counter log over a fixed time period. Vaccines are priori-tised to those who receive at least one success. Since theprobability of obtaining at least one success on n weighted

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Figure 2: The post-vaccination reproduction numbers forthe hot-spotting strategy for R0 = 1.5 with various app usagerates are compared with the uniform vaccination in relationto vaccine coverage. The dots represent the mean efficiencyobtained from 10,000 simulations for each dot. The shadedregions show the intervals that capture 90% of all simulationresults. The dashed lines in the bottom figures show the ex-pected post-vaccination reproduction numbers obtained fromthe formula in Methods.

coin flips is 1 − (1 − β)n, we see that users with a greatertotal exposure time have a greater probability of beingprioritised for vaccination. Note that the extent to whichhigh total exposure time individuals are prioritised canbe increased by requiring a greater number of successes.

We simulate this strategy on a simulated network de-rived from a diary-based population contact survey (seeMethods) [30]. In our model, individuals using the appform a subnetwork of the weighted network describingthe population’s contact patterns. The number of entriesin a user’s encounter log is their weighted degree in theapp-user subnetwork. The results of the prioritisationstrategy depend on the rate of app usage in the popula-

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tion, denoted U . In particular, the vaccine coverage V(proportion of individuals vaccinated) is bounded by theusage rate U .

In Figure 1 we show the simulation results for the effi-ciency as a function of vaccine coverage assuming a basicreproduction number of R0 = 1.5. The choice of basic re-production number has little impact on the efficiency, aswe show in the Supplementary Information by consider-ing R0 = 2.2 as well. For very low vaccine coverage theexpected efficiency is near 4 for all U values, in agree-ment with our theoretically derived value of 3.8. Thereis also a large variance in efficiencies due to small num-ber statistics. As one expects from a strategy prioritisinghigh exposure individuals, the efficiency decreases as vac-cine coverage increases. This is because high exposureindividuals are likely to be vaccinated first. Nonetheless,across the full range of vaccine coverage rates our pro-posed strategy is one to four times more efficient thanthe uniform vaccination strategy.

We observe that the expected efficiency curves for dif-fering usage rates are simply the same curve rescaled.This implies that the important quantity to consider forassessing the efficiency is not the overall vaccine cover-age V but rather the fraction of the app-using populationwhich are vaccinated.

Figure 2 shows the simulation results for the post-vaccination reproduction number, assuming again an ini-tial R0 = 1.5. The convexity of the curves is a witness tothe superior efficiency of our proposed strategy. We seethat as the fraction of the app-using population whichare vaccinated increases the slope decreases, in alignmentwith the decreasing efficiency. The greatest relative im-pact is achieved by vaccinating 20-40% of the app users.

Finally, when app usage rates are high enough, hot-spotting achieves herd immunity with fewer than half asmany doses. In Figure 3 we show that the significantlylower herd immunity thresholds persist across a range ofR0 values, and applies even when only 40% of the popu-lation uses the app.

Concluding Remarks

Implementing our proposal in practice is surprisingly easyon the technical front, although it requires the participa-tion of key players. The exposure prioritisation scheme it-self is straightforward and requires no modifications of thecore Bluetooth exposure notification framework. Sincethe Google/Apple system is implemented at the operatingsystem level it is Google and Apple who must incorporatethis new functionality. Public health authorities who wishto use hot-spotting must also ensure that their vaccine al-location systems are coordinated with the functionality ofthe app. Our proposal is, moreover, not strictly limited to

10% 20% 30% 40% 50% 60%5% 15% 25% 35% 45% 55%

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Figure 3: The necessary vaccine coverage to bring the re-production number from its initial value R0 = 1.5 (black) orR0 = 2.2 (red) to the herd immunity threshold Rv

= 1 isdemonstrated for the uniform and hot-spotting strategy forvarious app usage rates. The dots represent the mean valuefrom our simulations, whereas the intervals show the rangethat captures 1σ (68.27%) of all simulation results.

If R0 = 2.2 and U = 40%, vaccinating only the app users wouldnot be enough to reach herd immunity.

digital contact tracing solutions based on Bluetooth. Thehot-spotting strategy can be added on to any technologywhich creates an encounter log for each user.

As for any vaccine prioritisation strategy, hot-spottingcomes with ethical concerns, including questions of access,equity, and vaccine program objectives. In addition, ifnot carefully integrated as part of a holistic approach topublic health measures, a naive implementation of thisapproach could lead to perverse incentives for people toincrease their exposure or circumvent the system. Weare proposing hot-spotting as part of a broader approachthat addresses equity and is implemented in a way thatpre-empts harmful behavioural responses [40].

An important limitation of our approach is the un-derlying data used to construct the model network, whichwas collected before the pandemic [30]. Moreover, in bothour theoretical work and our simulations we have assumedthat there is no distinction between a potentially infec-tious contact and the contacts detected by Bluetooth ex-posure notification apps. Finally, our modelling does notpredict the time evolution of the epidemic in response tovaccination.

These simplifying assumptions could be relaxed in fu-ture work with more detailed agent-based simulations thattest the generality of the new theory we have introduced.However, we note that a diverse collection of previous lit-erature already finds that prioritising individuals basedon their number of contacts can be highly effective [35,19, 16, 14]. We have built upon this literature by (1)proposing a measure Ev of the relative efficiency of differ-ent strategies, (2) showing that existing COVID-19 digital

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contact tracing technology allows the measurement of epi-demiologically important quantities without violating pri-vacy, (3) proposing a new vaccination strategy based onthese measures that is technically feasible to implement,and (4) introducing novel analytical techniques and simu-lations that (5) demonstrate how hot-spotting is dramati-cally more efficient than uniform allocation. We concludethat hot-spotting could enable public health authoritiesto greatly reduce the social and economic costs of COVID-19 until vaccine supply catches up with demand.

Acknowledgements

We acknowledge support from the National Sciences andEngineering Research Council of Canada (NSERC), NSERCCOVID-19 Alliance Grant (MDP, MA, CTB) and OntarioMinistry of Colleges and Universities COVID-19 RapidResponse Grant (MA, CTB).

Patent pending, United States Provisional Applica-tion No. 63/122,587.

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[32] M. E. J. Newman, S. H. Strogatz, and D. J. Watts.Random graphs with arbitrary degree distributionsand their applications. Phys. Rev. E, 64:026118, Jul2001.

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[41] Ashleigh R Tuite, David N Fisman, and Amy LGreer. Mathematical modelling of COVID-19 trans-mission and mitigation strategies in the population

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of Ontario, Canada. CMAJ, 192(19):E497–E505,2020.

[42] Jeroen JA van Kampen, David AMC van de Vi-jver, Pieter LA Fraaij, Bart L Haagmans, Mart MLamers, Nisreen Okba, Johannes PC van den Akker,Henrik Endeman, Diederik AMPJ Gommers, Jan JCornelissen, et al. Shedding of infectious virusin hospitalized patients with coronavirus disease-2019 (COVID-19): duration and key determinants.MedRxiv, 2020.

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Methods

We model the spread of infectious disease in a popula-tion as a percolation process on the network of contacts.Each vertex in this network represents an individual per-son and each edge links two people who come into contact.Our particular approach is a generalization of Newman’sprobability generating function framework [33] to weightednetworks.

Percolation Theory

The key mathematical object in the original formalism isthe probability generating function of the degree1 distri-bution,

G0(x) =∞∑k=0

pkxk ,

where pk is the fraction of vertices in the network hav-ing degree k. This function is useful for writing analyticexpressions for moments of the distribution; for example,the average degree is given by µ = G′0(1).

An infectious disease spreads in the network throughoccupied edges. In the original model, each edge has auniform probability T for being occupied. The basic re-production number2 R0 corresponds to the expected num-ber of other occupied edges attached to a vertex at the endof a random occupied edge. This number is given by

R0 = TG′′0(1)G′0(1)

.

In terms of the mean µ and the standard deviation σ ofthe degree distribution, this equation is equivalent to

R0 = T [µ(1 + σ2

µ2) − 1] .

Weighted Percolation

In order to model vaccine prioritisation strategies basedon exposure notification apps it is necessary to incorpo-rate contact duration into the percolation model. To dothis we consider the contact network to be weighted: eachedge is further equipped with a weight w. The transmis-sion probability along an edge for the percolation processis assumed to depend on that edge’s weight, so that thesingle parameter T is replaced with a distinct Tw for eachweight w.

For our purposes in this paper the weight representsthe number of time-steps over which the contact took

1The degree of a vertex is the number of edges attached to it.2In heterogeneous populations, R0 is defined as “the expected

number of secondary cases produced, in a completely susceptiblepopulation, by a typical infected individual during its entire periodof infectiousness” [15].

place. However, the weights could represent any factorwhich influences the transmission probability along anedge, such as the nature of the contact, the setting, orthe presence or absence of PPE. Unless stated otherwise,our analytical results hold for this general interpretationof weights. Given that this is quite a flexible formalismit can be applied to a broad range of modelling tasks.

For weights representing contact duration there is anatural candidate for the transmission probabilities Tw.If one assumes that in each time-step there is an inde-pendent probability T1 of transmission, the transmissionprobability after w time-steps is

Tw = 1 − (1 − T1)w .

Once again, Tw is the probability for a weight-w edgeto be occupied. The infection spreads through occupiededges.

In a weighted network we represent the configurationof edges around each vertex by a generalized degree, de-noted k. This is a vector having an entry for each of thepossible weights appearing in the network. For a vertex ofgeneralized degree k, the entry corresponding to weightw, denoted kw, is the number of contacts having weight w.There are two important quantities we can extract fromk: First, the degree of a vertex, that is, the number ofdistinct contacts, is simply ∑w kw. Second, the weighteddegree of a vertex, that is, the sum of the weights on eachedge, is ∑w wkw. When w represents contact duration,the weighted degree is the total exposure time of thatindividual to others.

To generalize Newman’s results to the weighted set-ting we introduce a multivariable generating function forthe distribution of generalized degrees. There is one vari-able yw for each weight w appearing in the network, andwe write y for the vector of these variables. Let qk denotethe fraction of vertices in the network having generalizeddegree k. Then the multivariable generating function ofthe network is

Q(y) =∑k

qkyk , yk =∏

w

ykww .

The generating function G0(x) for the degree distri-bution in a weighted network is related to this functionby G0(x) = Q(1x), where 1 is a vector with each entrybeing 1.

In order to derive the basic reproduction number, weneed the distribution of occupied degree, i.e., the numberof occupied edges attached to a vertex. This distributionis given by

G0(x;T ) = Q(1 − T + Tx) ,

where T is the vector composed of the transmission prob-

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abilities Tw. Furthermore, we introduce

G1(x;T ) = G′0(x;T )

G′0(1;T )

as the probability distribution of the number of other oc-cupied edges attached to a vertex reached by following arandom occupied edge (cf. [33]). By definition, the ba-sic reproduction number R0 is precisely the mean of thisfinal distribution, i.e. R0 = G′1(1;T ).

One can express R0 directly in terms of the generatingfunction Q(y). This is accomplished by introducing thedifferential operator ∇T , defined as

∇TQ(y) =∑w

Tw∂Q(y)∂yw

.

Then, we find

R0 =∇2

TQ(1)∇TQ(1) .

Alternatively, this result can be rewritten in a formwhich is more amenable to computation,

R0 =Av [(∑i Tw(i))

2] −Av [∑i T2w(i)]

Av [∑i Tw(i)].

In this equation, the sums ∑i are taken over the contactsof each vertex, with w(i) being the weight of that contact,and the averages Av [⋯] are taken over all vertices in thenetwork. Note that this formula holds regardless of theinterpretation of the weights as contact duration.

Total Exposure Time

Let us now focus on the case when the weights representcontact duration and Tw = 1 − (1 − T1)w. In this case,the weighted degree of a vertex is the total exposure timeof that individual to others, measured in discrete timesteps. To better understand the impact of the total ex-posure time on the reproduction number, consider theapproximation Tw ≈ wT1, which is valid when wT1 << 1for all w. Under this approximation, we find the heuristicformula

R0 ≈ T1⎡⎢⎢⎢⎣µ(1 + σ

2

µ2) −

Av [∑iw(i)2]µ

⎤⎥⎥⎥⎦.

Here, µ and σ are the mean and the standard deviationof the total exposure time. Our model therefore predictsthat the total exposure time has a direct influence on thespread of disease.

Vaccination on Contact Networks

We model vaccination as a stochastic process wherein avertex of generalized degree k has a probability v(k) of

being vaccinated. Vaccination modifies the original con-tact network by removing vaccinated vertices, as we as-sume that a vaccinated individual can neither becomeinfected nor infect others.

We introduce the post-vaccination reproduction num-ber Rv as the reproduction number in the non-vaccinatedsub-population after a vaccination process. Since vacci-nation is modelled as a stochastic process, Rv is a randomvariable.

The formulas for the basic reproduction number R0

can be adapted to give the expected post-vaccination re-production number. This adaptation requires two changes:Firstly, some vertices are removed themselves. Each ver-tex of generalized degree k has a probability (1 − v(k)) ofremaining in the residual network. Secondly, some con-tacts of the remaining vertices are removed. A weight-wcontact has the probability

ϕw = 1 − Av [kwv(k)]Av [kw]

of remaining non-vaccinated. Hence, the expected post-vaccination reproduction rate is given by

E [Rv] =Av [(1 − v(k)) ((∑i Tw(i))

2 −∑i T2w(i))]

Av [(1 − v(k))∑i Tw(i)],

where Tw = Twϕw, and the averages Av [⋯] on the right-hand side run through all vertices in the network.

Efficiency of Vaccine Prioritisation

In general, one expects that prioritising the vaccinationof individuals with greater exposure to others achieves agreater reduction in Rv than vaccinating the same num-ber of less-connected individuals. This translates to slow-ing the infection more effectively and a reduction in thenumber of infections. Vaccine prioritisation becomes par-ticularly important when only a small supply of vaccinesis available. In that situation, adopting a strategy thatprioritises highly connected individuals for vaccinationcan save the lives of more people.

To compare different prioritisation strategies we areconcerned with not only how much they reduce the spreadof disease, but also how many vaccines it takes for thatoutcome. To that end, we introduce a novel measure ofprioritisation efficiency,

Ev = 1 −Rv/R0

V,

where V is the fraction of the population which receivesthe vaccine.

In a stochastic process of vaccination, V , Rv and Ev

are all random variables. In particular, the vaccine cov-erage rate V has the expected value E [V ] = Av [v(k)]

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and the variance Var[V ] = 1N

Av [v(k) (1 − v(k))], whereN is the size of the population. Therefore, for computingthe expectation value of the efficiency Ev in a large pop-ulation, we may neglect the variance in V and treat it asa fixed number at its mean value. Hence, we can write

E [Ev] = 1 − E [Rv] /R0

E [V ] .

For a uniform vaccination probability v without anyprioritisation, one would have E [Rv] = (1 − v)R0 andE [Ev] = 1. Through a realistic vaccine prioritisationstrategy, it is possible to achieve a significantly higherefficiency, while the exact numbers depend on the under-lying network.

The Hot-spotting Strategy

We propose a hot-spotting strategy that can be imple-mented simply on the existing Bluetooth based exposurenotification apps. This strategy is based on a Bernoulliprocess for each exposure registered in the app with achance of β for success. The app users with at least onesuccess are given the priority for vaccination. Hence, ifan app user has a total exposure number n in the app,they will have a chance of 1 − (1 − β)n for being selectedfor prioritised vaccination in this strategy.

In order to estimate the impact and efficiency of ourhot-spotting strategy on the whole population, we musttake into account that not every person in the populationuses the app in question. To this end, we assume thatthere is an app usage rate of U and that the app usersare homogeneously distributed in the population. We alsoassume that there is no preferential attachment betweenapp users. Then, for a random individual with the contactstructure k, which is not necessarily registered in the appnetwork, the probability for them being an app user andalso being selected for prioritised vaccination is given by

v(k;β,U) = U (1 −∏w

γkww ) ,

whereγw = 1 −U +U (1 − β)w .

Model Network

The well-known POLYMOD study surveyed over 7000 in-dividuals across 8 European countries [30]. Respondentskept a log of all contacts made on a single day noting,among other features, how long the contact lasted andhow frequently that contact is made. In the language oftime-weighted networks, the inclusion of duration datameans that the survey responses sample from the gener-alized degree distribution of the daily contact network.

To model COVID-19, the network should capture thecontacts made over the typical infectious period of thedisease. This varies significantly between patients, withone study estimating a median of 8 days after the onsetof symptoms [42]. For simplicity we choose a period of14 days, as it aligns well with the frequency responses inthe survey.

Using the daily contact data we must generate samplesfrom the generalized degree distribution of the fortnightlycontact network. We accomplish this by a bootstrappingtechnique. More precisely, for each contact recorded, re-spondents chose between 5 options concerning both theduration and the frequency of that contact as shown inTable 2. We can therefore represent each respondent’scontacts on that day in a 5 × 5 matrix whose (i, j) entryis the number of contacts recorded with frequency keyi and duration key j. For respondent n we denote thismatrix by Dn.

Keys Frequency Duration(1) daily < 5 mins(2) 1 − 2 times per week 5 − 15 mins(3) 1 − 2 times per month 15 − 60 mins(4) less than once a month 1 − 4 hrs(5) first time > 4 hrs

Table 2: Possible responses on POLYMOD survey concern-ing frequency and duration of contact.

Our goal is to extrapolate fortnightly contact matri-ces Fn from the daily contact matrices Dn. In our boot-strapping procedure, it is important that we distinguishbetween daily repeating contacts, which have frequencykey (1), and infrequent contacts, which have frequencykeys (2)-(5). We denote by In the matrix of infrequentcontacts for respondent n. The matrix In is obtainedfrom the same respondent’s Dn by setting the first row ofdaily repeating contacts to 0.

The matrices Fn are created by sampling from thedaily contact matrices. Each sample is generated as fol-lows:

1. Sample 14 respondents, n1, . . . , n14.

2. Set F ′n =Dn1 + In2 + . . . + In14 .

3. Produce Fn from F ′n by dividing each entry in thesecond row by 3 and rounding it to the nearest in-teger.

In words, we sample 14 daily contact logs from the sur-vey. Then, we add together the 1-day duration-frequencymatrices for each of the samples, excluding the daily re-peating contacts from all but the first sample. Then, as-suming that a frequency-key-(2) contact is seen 3 times in

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a fortnight, we divide the second row by three to accountfor repeated counting of the same contact.

Finally, we assign to each type of contact in the matrixFn a certain weight. Each unit of weight is approximately10 minutes of contact time, which corresponds to the to-ken exchange rate of the exposure notification apps. Wechose these weights to be given as in the following weightmatrix:

W =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 12 36 120 4800 3 9 30 1200 1 3 10 400 1 3 10 400 1 3 10 40

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Hence, we obtain a list of generalized degrees sampledfrom a virtual weighted contact network, in which the setof possible weights is {1,3,9,10,12,30,36,40,120,480}.

Simulation

The simulations presented in the main text are basedon 500 samples of fortnightly generalized degree distri-butions generated through the bootstrapping proceduredescribed above. We interpret these 500 samples as anobserved cohort of 500 individuals within a much largerpopulation. As such, the recorded contacts are assumedto lie outside the observed cohort.

For the chosen data of 500 samples, we fix the unittransmission probability at T1 = 0.000375 to get the ba-sic reproduction number R0 = 1.501. We select a setof parameters β for the vaccination probability functionv(k;β,U) of the hot-spotting strategy (and a set of pa-rameters v for the uniform strategy) to target a homoge-neous distribution of the vaccinate rate V in the outcomeover its range of possible values. 10,000 simulations areperformed for each selected parameter. This accountsto a total of 990,000 simulations for the strategies withU = 100%. For the strategies with U = 20%, 40%, and60%, we ran a total of 190,000, 390,000, and 590,000 sim-ulations, respectively.

Each simulation consists of four steps:

1. The vaccination probabilities v(k;β,U) for each in-dividual, and the probabilities ϕw for each weightare calculated for the given parameters.

2. For each component kw of the generalized degreeof each individual, a binomial process is performedwith the probability ϕw. The outcome of this pro-cess replaces kw as the residual contact number.

3. For each individual, a Bernoulli trial is performedwith the probability v(k;β,U). If the outcome is asuccess, the individual is removed from the list withall their contacts, otherwise they remain.

4. The base reproduction number is computed in theresidual list and saved as the post-vaccination re-production number Rv. The vaccine coverage V isdeduced from the length of the residual list. Theefficiency Ev is calculated from Rv and V .

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Supplementary Information

Uniform vacc.

Hotsp., U=20%

Hotsp., U=40%

Hotsp., U=60%

Hotsp., U=100%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

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Uniform vacc.

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0% 10% 20% 30% 40%5% 15% 25% 35%

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Vaccine coverage V

Figure 4: The efficiency for the hot-spotting strategy forR0 = 2.2 with various app usage rates are compared with theuniform vaccination in relation to vaccine coverage. The dotsrepresent the mean efficiency obtained from 10,000 simula-tions for each dot. The shaded regions show the intervals thatcapture 90% of all simulation results. The dashed lines in thebottom figures show the expected efficiency obtained from theformula in Methods.

Uniform vacc.

Hotsp., U=20%

Hotsp., U=40%

Hotsp., U=60%

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10% 20% 30% 40% 50% 60% 70% 80% 90% 100%0

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Post-vacc.reproductionnumberRv Uniform vacc.

Hotsp., U=40%

10% 20% 30% 40%5% 15% 25% 35%1.

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Uniform vacc.

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Figure 5: The post-vaccination reproduction numbers forthe hot-spotting strategy for R0 = 2.2 with various app usagerates are compared with the uniform vaccination in relationto vaccine coverage. The dots represent the mean efficiencyobtained from 10,000 simulations for each dot. The shadedregions show the intervals that capture 90% of all simulationresults. The dashed lines in the bottom figures show the ex-pected post-vaccination reproduction numbers obtained fromthe formula in Methods.

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