CRISP: A Probabilistic Model for Individual-Level COVID-19 Infection Risk Estimation Based on Contact Data
CCRISP: A Probabilistic Model for Individual-LevelCOVID-19 Infection Risk Estimation Based onContact Data
Ralf Herbrich ∗ ZalandoBerlin, Germany [email protected]
Rajeev Rastogi
AmazonBangalore, India [email protected]
Roland Vollgraf
ZalandoBerlin, Germany [email protected]
Abstract
We present CRISP ( C OVID-19 RI sk S core P rediction), a probabilistic graphicalmodel for COVID-19 infection spread through a population based on the SEIRmodel where we assume access to (1) mutual contacts between pairs of individualsacross time across various channels (e.g., Bluetooth contact traces), as well as(2) test outcomes at given times for infection, exposure and immunity tests. Ourmicro-level model keeps track of the infection state for each individual at everypoint in time, ranging from susceptible, exposed, infectious to recovered. Wedevelop a Monte Carlo EM algorithm to infer contact-channel specific infectiontransmission probabilities. Our algorithm uses Gibbs sampling to draw samples ofthe latent infection status of each individual over the entire time period of analysis,given the latent infection status of all contacts and test outcome data. Experimentalresults with simulated data demonstrate our CRISP model can be parametrizedby the reproduction factor R and exhibits population-level infectiousness andrecovery time series similar to those of the classical SEIR model. However, due tothe individual contact data, this model allows fine grained control and inference fora wide range of COVID-19 mitigation and suppression policy measures. Moreover,the algorithm is able to support efficient testing in a test-trace-isolate approach tocontain COVID-19 infection spread. To the best of our knowledge, this is the firstmodel with efficient inference for COVID-19 infection spread based on individual-level contact data; most epidemic models are macro-level models that reason overentire populations. The implementation of CRISP is available in Python and C++at https://github.com/zalandoresearch/CRISP. The COVID-19 pandemic has spread rapidly around the world, with the number of infections anddeaths steadily growing. Most governments around the world have been completely unprepared todeal with the COVID-19 outbreak, which UN Secretary-General Antonio Guterres has referred toas humanity’s worst crisis since World War II. While governments around the world had plans inplace in the event of a pandemic, the peculiarities of COVID-19 (e.g., delayed onset of symptoms,asymptomatic transmission) have challenged these preparations. Governments have reacted byimplementing measures such as nationwide lock-downs, that require people to stay inside theirhomes, enforcing social distancing and therefore breaking the COVID-19 infection chain. However,a blunt mechanism such as a lock-down (over an extended period) can cause severe damage to theeconomy, and so, there is a need to find alternative measures to slow down or stop the spread withoutincremental effects in other areas of society. These alternatives have to be built in a solid foundation ∗ The ordering of authors is alphabetical. All authors contributed equally to the paper.Preprint. Under review. a r X i v : . [ c s . S I] J un uch as widespread testing and the isolation of infected (or potentially infected) individuals viacontact-tracing.Contact tracing technologies [20, 19] have shown promise in tracking the spread of the diseaseacross the population. These mobile apps capture social contact information between users such ascontact duration, distance, etc. using Bluetooth signals on devices. The fine-grained contact data ofindividuals collected by the apps can enable: • Individual risk score prediction.
The contact data, combined with information about COVID-19 positive test cases, can be used to predict the likelihood of infection for each individual.The individual risk scores can be leveraged by governments and organizations to prioritizetesting as well as to identify individuals that need to enter isolation/quarantine. • Hotspot detection.
Tracing technologies can help authorities identify areas with a highdensity of contacts and/or individuals with high infection risk. This can allow policymakersto make more effective decisions, for example, by imposing highly restrictive measuressuch as lock-downs, shelter-at-home, or school closures only in COVID-19 hotspots whileallowing activities to remain closer to normal in unaffected areas. • Insights about infection spread.
Contact tracing can provide insights into the relative im-portance of different modalities of disease transmission (e.g., through intermediate surfacesvs individual contact), risk of infection transmission based on contact characteristics suchas duration and distance, most likely locations (e.g., schools, work, malls) for the spreadof disease, and "super spreaders" who come in close proximity with a large number ofindividuals and so must be frequently tested for infection.To achieve the above-mentioned benefits, we need to devise new models and inference algorithms foranalyzing contact tracing data. This is because existing epidemics models [3, 15, 14, 6, 5] focus onestimating population-level statistics such as percentage of the population infected, number of daysfor the epidemic to peak, etc. as opposed to the infection state of each individual in the population.Other models [16, 17] that use ML-based inference techniques assume complete knowledge of theinfection state of each individual at each time instant. However, in the COVID-19 scenario, (1) theinfection status of individuals is not known until they are tested, and (2) infectious time of individualsare unknown since individuals may infect others while asymptomatic. Finally, governments are usingcontact tracing data [20, 19] to identify and test individuals who have come in direct contact withCOVID-19 positive test cases. However, the fact that asymptomatic individuals may have infected alarge number of people prior to displaying symptoms and being tested, delays the detection of thesenewly infected individuals by only using contact tracing apps.Our main contributions can be summarized as follows: • We propose CRISP ( C OVID-19 RI sk S core P rediction), a probabilistic graphical model forCOVID-19 infection spread through diverse contacts channels between individuals. Ourmodel uses latent variables to represent the epidemiological states of individuals based onthe SEIR model [12] at different points in time, and captures both the transitions betweenstates as well as test outcomes. • We develop a Monte Carlo EM algorithm to infer infection transmission probabilities acrossa range of contact channels. Our algorithm uses block-Gibbs sampling to draw samples ofthe latent infection status of each individual over the entire time period, given data aboutcontacts and test results. • We provide implementation details to accelerate both the block-Gibbs sampling and theforward sampling algorithm. A Python and C++ implementation of CRISP is available athttps://github.com/zalandoresearch/CRISP. • We conduct experiments with simulated data which demonstrate that our CRISP model canbe parametrized by the reproduction factor R and exhibits population-level infectiousnessand recovery time series similar to those of the classical SEIR model. However, due tothe individual contact data, this model allows fine grained control and inference for awide range of COVID-19 mitigation and suppression policy measures. Furthermore, weshow that a testing-and-quarantining policy based on infection risk scores computed by theCRISP algorithm is able to mitigate COVID-19 infection spread while quarantining fewerindividuals compared to other policies based on contact-tracing and symptom-based testing.2o the best of our knowledge, this is the first comprehensive model for COVID-19 infection spreadthat (1) captures the infection states of individuals and transitions between them using the SEIRmodel, and (2) leverages contact tracing and test outcome data to infer model parameters suchas contact-channel specific infection rates using scalable and computationally efficient inferencealgorithms. We classify related work into four broad categories: epidemic models, Machine Learning (ML) basedinference of model parameters, influence maximization in social networks, and contact tracing apps.
In recent years, there has been research on modeling individual dynamics of epidemics [3, 15, 14].However, this work typically resorts to mean-field theory to model virus spread over a network, andthus does not characterize the dynamic infectious state of each individual over time.Ferguson et al. [6, 5] use a compartmental transmission model to simulate the spread of influenzaacross a population, and analyze interventions such as antiviral prophylaxis and social distancing tohalt a pandemic. The authors use a stochastic model of individuals co-located in households that arerandomly distributed across a geographical region, and infection risk from 3 sources – household,place and random contacts in the community. The infection transmission rates for the 3 sources andrecovery rates are based on analysis of historical data. In contrast, we leverage real individual contacttracing data and outcomes of tests on individuals to infer the infection transmission rate for eachcontact and the likelihood of infection for each individual.Lorch et al. [10] propose a spatiotemporal epidemic model that uses marked temporal processesto represent the epidemiological condition of each individual (based on a variation of the SEIRcompartment models), individual mobility patterns, test outcomes, and testing and contact tracingstrategies. The authors design an efficient sampling algorithm for the model using Monte Carloroll-outs that is able to predict the spread of COVID-19 under different testing & tracing strategies,social distancing measures, and business restrictions, given contact histories of individuals. Theyuse Bayesian optimization techniques to infer model parameters (e.g. infection transmission rate)that minimize the difference between the real positive COVID-19 cases and those in the Monte-Carlo simulations. In addition, they demonstrate the efficacy of their model using real COVID-19data and mobility patterns of Tübingen, Germany. Our Monte Carlo EM inference algorithm formodel parameters is computationally much more efficient than the Bayesian optimization techniquesemployed in [10].
In [16], the authors consider the problem of inferring latent social networks based on networkdiffusion or disease propagation data. Given the times when nodes become infected, but not whoinfected them, the authors identify the optimal network that best explains the observed data. Theauthors present a maximum likelihood approach based on convex optimization with a L -like penaltyterm (that encourages sparsity) to estimate the conditional probability of infection transmissionbetween every node pair. A key difference from our work is that [16] assumes complete knowledgeof infected nodes and infection times. In contrast, in the COVID-19 scenario, (1) the infection statusof nodes is not known until they are tested, and (2) infection times of nodes are unknown since nodesmay not show symptoms even though they are infected (and infecting others).Warriyar et al. [17] introduce a novel R statistical software package EpiILM for simulating infec-tious disease spread, and carrying out Bayesian MCMC-based statistical inference for spatial and/or(contact) network-based models in the Deardon et al. [4] individual-level modelling framework. Inindividual-level models (ILMs), the epidemiological state of each individual (e.g., susceptible orinfected) is assumed to be perfectly known at each time instant, which makes it relatively straight-forward to estimate model parameters such as infection transmission probabilities (as a function ofcovariates) using maximum likelihood estimation or Bayesian inference using Metropolis-HastingsMCMC. However, in the COVID-19 scenario, epidemiological states of individuals are hidden untilthey are tested, and this complicates Bayesian inference in our probabilistic model setting.3 z z z z z z z z z z z o Figure 1: Graphical model ofthe CRISP contact infectionspread model for 3 people over4 time steps where individual u = meets both individual u = and u = at time t = and one test outcome ofindividual u = at time t = .Note that this model has no cy-cles as we assume the infec-tion status z u,t only dependson variables z v,t (cid:48) before timestep t , t (cid:48) < t . However, dueto the "memory" that the state z u,t = E and z u,t = I have,we require edges into the en-tire past of an infection trace. The
Influence Maximization problem aims to select k users in a social network that maximizeinfluence spread, and was first modeled as an algorithmic problem by Kempe et al. [8]. [9] presents acomprehensive survey of different diffusion models that capture the information diffusion process andapproximation algorithms to maximize influence. The papers assume that diffusion model parameterssuch as influence probabilities are given and focus on selecting k users to maximize influence spread.In contrast, our paper focuses on the problem of estimating model parameters related to infectiontransmission probabilities for each contact, given social contact information between users andCOVID-19 test results for users.[11] addresses the problem of finding the "backbone" of an influence network. It employs networksparsification to preserve only the links that play an important role in the propagation of information.[7] considers the problem of estimating influence probabilities between users in a social graph. Givena social graph and a log of actions by users, the Maximum Likelihood Estimator (MLE) of influenceprobability of node u on node v is simply the fraction of actions performed by u that are alsoperformed by v . Unlike [7], in our setting, the infection status and times of nodes are latent, and needto be inferred by our algorithms. To combat the spread of COVID-19, governments have launched contact tracing apps[20, 19] thatuse Bluetooth signals on mobile phones to track contacts between users. Users who have come indirect contact with COVID-19 positive test cases are considered to be at high risk of infection, andsubject to tests and quarantine actions. However, a key problem with this approach is that COVID-19infected users are typically tested only after they show symptoms, and typically, infected users showsymptoms 5-6 days post infection. These asymptomatic users may have infected a large number ofusers over multiple hops prior to displaying symptoms and being tested. This delays detection ofinfected users using contact tracing apps, and limits their effectiveness to proactively test and isolateinfected users to contain the spread of COVID-19. In contrast, our probabilistic modeling algorithmCRISP predicts the likelihood of a user getting infected with COVID-19 through a chain of socialcontacts involving asymptomatic users, and identifies infected users early, even though they may bemultiple hops from a user who has tested positive for COVID-19 and even before they begin to showsymptoms. Our inference algorithm also learns infection transmission probabilities for each contactchannel. 4
CRISP Infection Spread Model
Our CRISP model is an SEIR model at the level of every individual (see [12] for an introduction).Note that we consider discrete time steps t implicitly assumed to be at the level of single days. Weassume that we are given the following two datasets for a given set S of individuals: • D contact = { ( u i , v i , t i , x i ) } Ni = ⊆ S × S × N × N J of N quadruples of a pair of twoindividuals ( u i , v i ) who have met at time t i with specific features x i . Here we assume thatthe feature vector x i describes the overall contact between u i and v i via the number x i,j of mutual contacts over channel j (e.g., Bluetooth encounters, queuing together, sharingpublic transportation). We assume D contact to be symmetric so that ( u, v, t, x ) ∈ D contact ↔ ( v, u, t, x ) ∈ D contact . • D test := { ( u i , t i , o i ) } Ki = ⊆ S × N × {
0, 1 } of K triplets of individual u i taking a test at time t i with the test outcome o i where o i = indicates a negative test outcome.We model the T discrete time steps of infection status for each individual. Our model has | S | × T many latent variables Z := { z u,t } u ∈S ,t = ∈ { S, E, I, R } | S | × T that represent the four stages ofinfection : • z u,t = S : individual u has not been infected and is susceptible, • z u,t = E : individual u is infected but not contagious, • z u,t = I : individual u is infected and is contagious, • z u,t = R : individual u has recovered and is not contagious.Let us use the notation Z u,t := { z u,1 , . . . , z u,t } to denote the set of latent states z u,t of individual u up to and including time t and Z t := (cid:83) u Z u,t . In addition to the latent infection status of allindividuals at each time step, we also model K variables o u,t ∈ {
0, 1 } for the test outcomes in D test ,that is, O := { o u,t : ( u, t, o u,t ) ∈ D test } . Then, our graphical model G := ( V , E ) between thevariables V := Z (cid:83) O has the following edges:1. E time = (cid:83) u E u and E u := { ( z u,t , z u,t (cid:48) ) t In order to define f , we assume that an infection occurs from exogenous influenceswith a fixed probability p ∈ [ 0, 1 ] or with probability of p j ∈ [ 0, 1 ] for every instance of a contactthrough the contact channel j if the contact was already in the state I . Thus, the probability that noinfection occurred at time t equals f ( u, t, Z t ) = ( − p ) · (cid:89) ( v,u,t, x ) ∈D contact : z v,t = I J (cid:89) j = ( − p j ) x j . (5) Infection Status Model In order to define g and h , let us assume we have a point density function q E : N + (cid:55) → [ 0, 1 ] and q I : N + (cid:55) → [ 0, 1 ] for the probability q E ( d E ) that the exposure ( z u,t = E ) lastsfor d E time steps (and similarly for the duration of the infectiousness). Examples of functions q E and q I are the probability mass functions of the binomial, negative-binominal or geometric distributions.However, for the case of COVID-19, we will use discrete probabilities established from analysis ofthe population in [1] and [18]. Moreover, let π ( n ; q ) = q ( n ) − (cid:80) n − = q ( i ) = P ( d = n ) P ( d ≥ n ) = P ( d = n | d ≥ n ) , (6)be the conditional probability (according to q ) that the duration is exactly n time steps given that theduration is at least n time steps. Then, g ( u, t, Z u,t ) = π (cid:18) t − max t (cid:48) ≤ t { t (cid:48) : z u,t (cid:48) = S } ; q E (cid:19) (7) h ( u, t, Z u,t ) = π (cid:18) t − max t (cid:48) ≤ t { t (cid:48) : z u,t (cid:48) = E } ; q I (cid:19) (8)Note that the first argument to both g and h is the number of E and I states up to and including time t in the state sequence Z u,t . Test Outcome Model Finally, we need to define the probability of a test outcome o given theinfection status z u,t of individual u at time t . Since there are two types of mistakes of a test, we use P ( o | z u,t ) = α if z u,t = I ∧ o = − α if z u,t = I ∧ o = − β if z u,t ∈ { S, E, R } ∧ o = if z u,t ∈ { S, E, R } ∧ o = (9)We assume both (cid:28) and (cid:28) . It is easy to implement more sophisticated test accuracymodels here, in particular to distinguish between different infection states. Also, we can easily model α and β which are dependent on how many days an individual has been in state I ; this change wouldnot affect the block-Gibbs sampling scheme in Section 4 in an adverse way. Prior Model In order to complete the description of the full probabilistic model, we have tospecify P ( Z ) . For simplicity, we assume these probabilities to be a delta-peak at state S , that is, P ( z u,1 = S ) = for all u ∈ S . 6 Inference in the CRISP Model For inference in the aforementioned model we are interested in computing the infection risk scoreof every individual u at every time step t given the test outcomes O available as well as the hyper-parameters θ := ( p , p , . . . , p J ) which cannot be set by knowledge of the diseases: the J parameters p j represent the probabilities of COVID-19 infection transmission through the contact channel j and p captures the probability that an infection occurs at any time-step from exogenous influences.In order to estimate θ , we will maximize the log-likelihood of the data O , that is θ ∗ = argmax θ log ( P ( O | θ )) (10) = argmax θ log (cid:32) (cid:88) Z T P ( Z T | O , θ ) · P ( O | θ ) (cid:33) (11) = argmax θ log (cid:32) (cid:88) Z T P ( Z T , O | θ ) (cid:33) , (12)where the second decomposition explicitly contains the posterior P ( Z T | O ) . However, this posterioris not analytically tractable and therefore we will approximate it by performing block-Gibbs samplingof an infection trace z u := ( z u,1 , . . . , z u,T ) of individual u keeping all other infection traces { z v : v (cid:54) = u } fixed. This requires a computationally efficient procedure to sample from the conditional probabilitydistribution P ( z u | { z v : v (cid:54) = u } , O , θ ) which we describe in the following subsection. Since we assume that the total number of days of the model, T , is not large , we will enumer-ate all possible sequences of infection traces z u and compute the un-normalized probability of P ( z u | { z v : v (cid:54) = u } , O , θ ) for all terms that depend on elements of the trace z u in order to re-normalizeand draw from this distribution. Also, as our model is an SEIR model, we know that each sample z u can be uniquely represented by a triple ω = ( t , d E , d I ) ∈ N × N + × N + of time steps with t being time steps individual u is in state S , d E being time steps in state E , d I being time steps in state I and the remaining T − t − d E − d I being time steps in state R .There are three groups of factors that (might) involve z u in the (un-normalized) conditional probabilitydistribution P ( z u | { z v : v (cid:54) = u } , O , θ ) : T − (cid:89) t = P ( z u,t + | Z t ) (cid:124) (cid:123)(cid:122) (cid:125) A ( z u ) · (cid:89) v (cid:54) = u T − (cid:89) t = P ( z v,t + | Z t ) (cid:124) (cid:123)(cid:122) (cid:125) B ( z u ) · (cid:89) ( t,o ) ∈T u P ( o | z u,t ) (cid:124) (cid:123)(cid:122) (cid:125) C ( z u ) . (13)The first set of factors, A ( z u ) , captures the temporal evolution of the infection state changes of z u directly and can be reduced to three factors based on ω and all the contacts v that could have infectedindividual u . The second set of factors, B ( z u ) , captures the factors where the infectiousness of u might impact other individuals v . Finally, the third set of factors, C ( z u ) , captures the outcome oftests on individual u . Factors A ( z u ) In order to derive a compact representation of A ( z u ) , we assume that it can bewritten in terms of A ( z u ) = l ( t ) · l E ( d E ) · l I ( d I ) · l infected (14) As of today, the COVID-19 pandemic is active for 90 days. z v,t = I of other individuals v that had contact with u only affect u in thesusceptible state, we can derive l ( t ) and l infected from (4) by collecting the f terms (see (5)): t − (cid:89) t = f ( u, t, Z t ) · ( − f ( u, t , Z t )) (15) = (cid:32) t − (cid:89) t = p u,t (cid:33) · ( − p ) t − · ( − ( − p ) p u,t ) (16) = (cid:32) t − (cid:89) t = p u,t (cid:33) · (cid:18) − ( − p ) p u,t p (cid:19) (cid:124) (cid:123)(cid:122) (cid:125) l infected · ( − p ) t − p (cid:124) (cid:123)(cid:122) (cid:125) l ( t ) , (17)where p u,t := (cid:81) ( v,u,t, x ) ∈D contact : z v,t = I (cid:81) j ( − p j ) x j . Note that l ( t ) is the density function of thegeometric distribution. Similarly, given (4) and (7) we can derive l E ( d E ) as l E ( d E ) = d E − (cid:89) d = ( − g ( u, t + d, Z u,t + d )) · g ( u, t + d E , Z u,t + d E ))= d E − (cid:89) d = (cid:32) − q E ( d ) − (cid:80) d − = q E ( i ) (cid:33) · q E ( d E ) − (cid:80) d E − = q E ( i ) (18) = d E − (cid:89) d = (cid:32) − (cid:80) di = q E ( i ) − (cid:80) d − = q E ( i ) (cid:33) · q E ( d E ) − (cid:80) d E − = q E ( i ) (19) = q E ( d E ) . (20)A similar derivation shows that l I ( d I ) = q I ( d I ) which proves that the computational complexity ofcomputing A ( z u ) has been reduced to one factor for each contact during the S states of u and threeadditional factors corresponding to the compact representation ω for z u . The number of factors donot directly scale up with T . Factors B ( z u ) In order to derive a compact representation of B ( z u ) , we note that only the cases of z v,t = S potentially contain the value of z u,t for v (cid:54) = u (see the value range of the function g and h in (4)). In fact, looking at (5) it becomes evident that it requires z u,t = I . Thus, B ( z u ) is defined by T (cid:89) t = (cid:89) v ∈C S ( u,t ) f ( v, t, Z t ) (cid:89) v ∈C E ( u,t ) ( − f ( v, t, Z t )) , (21)where C S ( u, t ) and C E ( u, t ) are the individuals that u met at time t who were susceptible and haveeither stayed susceptible or got exposed, respectively: C S ( u, t ) := { v : ( u, v, t, x ) ∈ D contact ∧ z v,t = S ∧ z v,t + = S } , C E ( u, t ) := { v : ( u, v, t, x ) ∈ D contact ∧ z v,t = S ∧ z v,t + = E } . In this subsection, we describe how we can speed up the block-Gibbs sampling step for drawing asample infection trace z u for individual u . Constant terms A key observation is that the term C ( z u ) = (cid:81) ( t,o ) ∈T u P ( o | z u,t ) in (13)—corresponding to test outcomes for individual u —is a constant for each infection trace z u irrespectiveof the values of other infection traces { z v : v (cid:54) = u } . Thus, C ( z u ) can be pre-computed at the start ofthe block-Gibbs sampling algorithm for individual u and then reused every time we evaluate thelikelihood of an infection trace z u . Similarly, the terms l ( t ) = ( − p ) t − p , l E ( d E ) = q E ( d E ) and l I ( d I ) = q I ( d I ) in A ( z u ) in (14) are constant for each infection trace z u irrespective of theinfection traces of other individuals. Hence, these terms can also be pre-computed at the start of theblock-Gibbs sampling algorithm for individual u .8 ontacts into u The remaining term l infected in A ( z u ) (see (14)) captures the contribution due tocontacts into individual u from other (infectious) individuals v who are in state z v,t = I prior to u herself getting infected at t . As a result, l infected depends on the values of other infection traces { z v : v (cid:54) = u } and needs to be recomputed during each block-Gibbs sampling step to draw sample z u . Letus define l infected ( t ) := p u,t , (22) l (cid:48) infected ( t ) := − ( − p ) p u,t p , (23)where p u,t := (cid:81) ( v,u,t, x ) ∈D contact : z v,t = I (cid:81) j ( − p j ) x j (see also (17)). Then, we have l infected = t − (cid:89) t = l infected ( t ) · l (cid:48) infected ( t ) (24)Thus, at the start of each block-Gibbs sampling step, we pre-compute (22) and (23) for each timestep t , and then use (24) to compute l infected for a particular infection trace z u . Note that this involvesonly t multiplications (or additions in the log-domain). Contacts out from u We next turn our attention to computing B ( z u ) for each infection trace z u that captures the contribution due to contacts out from u . We introduce two states Z It and Z ¬ It whichare identical to Z t except for the value of infection state z u,t which is I is Z It and one of { S, E, R } in Z ¬ It . Now, let B ( z u , t, Z t ) := (cid:89) v ∈C S ( u,t ) f ( v, t, Z t ) (cid:89) v ∈C E ( u,t ) ( − f ( v, t, Z t )) (25)be the inner terms in (21). Note that for all contacts ( u, v, t, x ) ∈ D contact , the terms B ( z u , t, Z It ) and B ( z u , t, Z ¬ I ) differ only in the factor (cid:81) j ( − p j ) x j that is in f ( v, t, Z It ) but not in f ( v, t, Z ¬ It ) since u is infectious at this time t in Z It but not in Z ¬ It . Also, note that values of z u,t (cid:48) for t (cid:48) < t donot affect B ( z u , t, Z t ) . We can then obtain B ( z u ) for each infection trace z u value as B ( z u ) := Constant · t + d E + d I (cid:89) t = t + d E B ( z u , t, Z It ) B ( z u , t, Z ¬ It ) , (26)where Constant is the product of B ( z u , t, Z ¬ It ) over all time steps t and can be ignored due tonormalization of the sampling distribution. Again, the ratio B ( z u , t, Z It ) /B ( z u , t, Z ¬ It ) can be pre-computed for all time steps t at the start of the block-Gibbs sampling step for z u , and then used tocompute B ( z u ) for each infection trace z u as in (26). Putting it all together Note that the quantities l infected ( t ) , l (cid:48) infected ( t ) and B ( z u , t, Z It ) /B ( z u , t, Z ¬ It ) only depend on the infection status of individual u at time t be-cause the infection traces z v of all other individuals v are fixed when we are drawing a block-Gibbssample for z u . Thus, the (un-normalized) conditional probability for each z u value is obtained bytaking the product of A ( z u ) , B ( z u ) and C ( z u ) , which in turn are computed efficiently as describedabove from pre-computed values of l ( t ) , l E ( d E ) , l I ( d I ) and C ( z u ) at the start of the algorithm,and l infected ( t ) , l (cid:48) infected ( t ) and B ( z u , t, Z It ) /B ( z u , t, Z ¬ It ) for all time steps t at the start of theblock-Gibbs sampling step. Additional implementation optimizations We use two additional ideas to accelerate the imple-mentation of the block-Gibbs sampling algorithm: • We never materialize the infection trace z u because it is uniquely described by the triple ω = ( t , d E , d I ) ; each value z u,t can be computed by no more than three comparisons of t with t , t + d E and t + d E + d I . Thus, the whole state of the latent variable model isrepresented by × | S | integers. • We carry out all computations of probabilities in the log-domain so all functions becomesums and products instead of products and powers.9 lgorithm 1: Block-Gibbs sampling algorithm for CRISP model /* Initialization */ Initialize each z u = S · /* Precomputations independent of contact data */ forall ( t , d E , d I ) ∈ N + × N + × N + such that t + d E + d I ≤ T do Pre-compute l ( t ) , l E ( d E ) and l I ( d I ) Construct the sequence z u with t states S , d E states E , d I states I and T − t − d E − d I states R Pre-compute C ( z u ) according to (13) repeat Pick a random index u /* Precomputations dependent on contact data */ forall time steps t do Pre-compute l infected ( t ) using (22) and l (cid:48) infected ( t ) using (23) Pre-compute ratio B ( z u , t, Z It ) /B ( z u , t, Z ¬ It ) using (25) forall ( t , d E , d I ) ∈ N + × N + × N + such that t + d E + d I ≤ T do /* Infection trace specific computations */ Construct the sequence z u with t states S , d E states E , d I states I and T − t − d E − d I states R Compute log ( A ( z u )) = log l ( t ) + log l E ( d E ) + log l I ( d I ) + log ( l infected ) using (24) Compute log ( B ( z u )) using (26) Set l t ,d E ,d I = log ( A ( z u )) + log ( B ( z u )) + log ( C ( z u )) /* Block-Gibbs sampling step */ Sample ( t ∗ , d ∗ E , d ∗ I ) with probability ∝ exp ( l t ∗ ,d ∗ E ,d ∗ I − max t ,d E ,d I ( l t ,d E ,d I )) Set z u with ( S , E , I , R ) states corresponding to ( t ∗ , d ∗ E , d ∗ I ) return Z i = Z until convergence Block Glibbs Sampling Algorithm Algorithm 1 is block-Gibbs sampling algorithm for sampling Z iT from our CRISP model. It cycles through (random) individuals u , sampling the vector of latentvariables z u from the conditional distribution P ( z u |{ z v : v (cid:54) = u } , O , θ ) until convergence. We can use thesamples Z , . . . , Z mT drawn by this algorithm to compute the infection risk score for an individual u at time t by taking the fraction of samples Z iT in which the latent infection state z u,t ∈ { E, I } . In order to estimate the hyper-parameters θ of the CRISP model, would like to find θ ∗ that maximizesthe log-likelihood log (12). However, since this is intractable, we propose to use the Monte CarloExpectation-Maximization (EM) algorithm [2]. We will use EM to refine θ in successive iterations.Let θ old be the value of θ computed in the previous iteration. Then, in the E step of the currentiteration, we will estimate the expected complete-data log-likelihood (cid:88) Z T P ( Z T | O , θ old ) · log ( P ( Z T , O | θ )) . (27)We will use the block-Gibbs sampling procedure described in Algorithm 1 to approximate theposterior distribution P ( Z T | O , θ old ) over the latent infection status of individuals u . If the samplesdrawn from the posterior P ( Z T | O , θ old ) are Z , . . . , Z mT , then in the M step, we will compute θ thatmaximizes the expected complete-data log-likelihood θ next = argmax θ m (cid:88) i = log (cid:0) P (cid:0) Z iT , O | θ (cid:1)(cid:1) (28) = argmax θ m (cid:88) i = − (cid:88) t = (cid:88) u log (cid:0) P (cid:0) z iu,t + , O | Z it , θ (cid:1)(cid:1) , (29)10here z iu,t + is the infection state for individual u at time t + in sample Z iT . If t i0 denotes thenumber of initial S states in the sample infection trace z iu , we note that by virtue of (4) only the first t i0 terms depend on θ which reduces the above maximization term to only m (cid:88) i = (cid:88) u t i0 − (cid:88) t = log (cid:0) f (cid:0) u i , t, Z it | θ (cid:1)(cid:1) + log (cid:0) − f (cid:0) u i , t i0 , Z it | θ (cid:1)(cid:1) . We use stochastic gradient descent to compute the θ values that maximize the above expression. Wealso note that for numerical stability, we re-parameterize p j via w j as p j = exp ( w j ) / ( + exp ( w j )) which allows for an unconstrained optimization over w . We can extend the block-Gibbs sampling algorithm in CRISP to a federated learning setting [13]where local contact and test outcome data for an individual u are utilized to compute the block-Gibbssample z u on the individual’s mobile device without ever needing to be shared with anyone else. Thishas two benefits: (1) We distribute the block-Gibbs sampling algorithm across hundreds of millionsof mobile devices in the world and thereby utilize their distributed computational power, and (2)Contact and test outcome data for an individual are stored only on the individual’s mobile device andnot shared with other mobile devices—-this preserves a user’s privacy. In the federated setting, thecontact data is never centralized—instead for each individual u , her device executes the block-Gibbssampling step to draw sample z u only using the locally available contacts and test outcome datafor u , as well as additional “minimal statistics” sent to u by the devices of its past contacts. In thefollowing two paragraphs, we explain how to compute the factors A ( z u ) , B ( z u ) and C ( z u ) in (13) ina federated setting (see Algorithm 2 for the pseudo-code which runs on every mobile device). Factors A ( z u ) and C ( z u ) A key observation is that the terms l ( t ) , l E ( d E ) , l I ( d I ) in A ( z u ) aswell as the factor C ( z u ) can all be computed locally on the device with the contact and test outcomeinformation available on the device. In order to compute the remaining term l infected in A ( z u ) , weonly require information on the individuals v who had a contact with u at each time step t andthe infection status z v,t of v at the time of the contact. Individual u ’s mobile device already hasthe contact information for u ; thus all that is required to compute l infected are the current infectiontraces z v for all individuals v who have had contacts with u . Since each infection trace is uniquelycharacterized by a ( t , d I , d E ) triple, we require the mobile devices of all individuals v who havehad a contact with u to send u ’s device the ( t , d I , d E ) triple corresponding to z v . Factor B ( z u ) In order to compute B ( z u ) as defined in (21), we require the term f ( v, t, Z t ) foreach individual v who has had a contact with u at time t and whose infection state z v,t = S . Let f − u ( v, t, Z t ) be defined as in (5) over all contacts of v at time t except for individual u . Then, thedevice for each individual v who has had a contact with u at time t and whose infection state z v,t = S sends to u ’s device the quantity f − u ( v, t, Z t ) computed based on v ’s view of the infection traces ofits contacts. These terms are used by u ’s device to compute B ( z u ) as defined in (21). In this section, we present two types of experimental evaluations:1. Population Level COVID-19 Infection Spread . In the first set of experiments, we willdemonstrate that CRISP is capable of modelling infection spread across an entire population.We will relate our individual-level parameters θ to more classical measures of infectionspread such as reproduction factor R and demonstrate that the structure of the contactpatterns allow more fine grained control of the infection spread which can be used foralternative containment measures of the COVID-19 pandemic.2. Test and Quarantine Efficacy of CRISP Model . In the second set of experiments, we willassess the test and quarantine efficacy of the CRISP model by comparing the populationhealth after 5 months under three testing and quarantining policies: (1) symptom-based, (2)contact-tracing-based, and (3) CRISP model-based.11 lgorithm 2: Federated block-Gibbs Sampling algorithm for CRISP model /* Initialization */ Initialize z u = S · Initialize N v = ( ∞ , 0, 0, ∅ ) for all v // Stores minimal statistics from contacts /* Precomputations independent of contact data */ repeat /* Update minimal statistic received in the incoming queue */ forall { ( t v0 , d vE , d vI , { f − u ( v, t, Z t ) } ) } v in the incoming message queue do N v ← ( t v0 , d vE , d vI , { f − u ( v, t, Z t ) } ) /* Precomputations of test outcomes */ forall ( t , d E , d I ) ∈ N + × N + × N + such that t + d E + d I ≤ T do Pre-compute l ( t ) , l E ( d E ) and l I ( d I ) Pre-compute C ( z u ) for this sequence according to (13) /* Precomputations dependent on contact data */ forall time steps t do Pre-compute l infected ( t ) using (22) and l (cid:48) infected ( t ) using (23) and ( t v0 , d vE , d vI ) in N v for allpast contacts v Pre-compute ratio B ( z u , t, Z It ) /B ( z u , t, Z ¬ It ) using (25) and { f − u ( v, t, Z t ) } in N v for allpast contacts v forall ( t , d E , d I ) ∈ N + × N + × N + such that t + d E + d I ≤ T do /* Infection trace specific computations */ Construct the sequence z u with t states S , d E states E , d I states I and T − t − d E − d I states R Compute log ( A ( z u )) = log l ( t ) + log l E ( d E ) + log l I ( d I ) + log ( l infected ) using (24) Compute log ( B ( z u )) using (26) Set l t ,d E ,d I = log ( A ( z u )) + log ( B ( z u )) + log ( C ( z u )) /* Block-Gibbs sampling step */ Sample ( t ∗ , d ∗ E , d ∗ I ) with probability ∝ exp ( l t ∗ ,d ∗ E ,d ∗ I − max t ,d E ,d I ( l t ,d E ,d I )) Set z u with ( S , E , I , R ) states corresponding to ( t ∗ , d ∗ E , d ∗ I ) /* Send minimal statistic to all contacts */ forall v in past contact list do F = { f − v ( u, t, Z t ) : ( u, v, t, x ) ∈ D contact ∧ t ≤ t ∗ } Send message ( t ∗ , d ∗ E , d ∗ I , F ) to v until forever In all these experiments, we use the parameters α = and β = in (9) and match thedistribution q E and q I of exposure and infectiousness duration to the empirical distributions providedin the medical literature [1, 18]. This is both used in the generation of the simulated test outcomedata as well as for the CRISP inference algorithms as these parameters are publicly known. We willuse the notation q I for the expectation of the empirical distributions q I .In order to simulate realistic epidemiological spread, we need to translate a reproduction factor R at t = into contact data. By definition, R is the average number of individuals that an infected personwill infect over the entire period of being infectious. Thus, for a reproduction factor R and a contactchannel j with transmission probability p j ∈ [ 0, 1 ] , we need to generate C ( R , p j ) := R / ( q I · p j ) many connections on average for all individuals in each time step. Conversely, for any process thatgenerates η j connections to unique and distinct individuals over channel j in each time step, theeffective R over contact channel j with 100% transmission probability equals q I (cid:80) j η j . The actualnumber of contacts is drawn form a binomial distribution with n = S − and a rate p = C ( R ,p j ) ( S − ) .Note that the rate is one half of the target contact rate because all contacts are symmetrically mirrored.12 .1 Population Level COVID-19 Infection Spread In order to assess if the CRISP model is able to provide realistic population-level statistics forCOVID-19 infection spread, we simulate a population of | S | = 10, 000 individuals over a period of274 days (9 months). We single out an individual u for whom we set p = so that she will getinfected with probability 100% at t = ("patient 0"); for all other people we assume a p = − tomodel a miniscule chance of infection spread from exogenous sources. We assume a single contactchannel with a 1% chance of transmission, p = . We simulate five scenarios: • No Mitigation . Since R of COVID-19 is estimated to be 2.5, at any time t we generate C ( ) random connections for every individual at every time step. • Social Distancing After 60 Days . Intuitively, the "locality" of the contact patterns shouldplay a role in the infection spread of COVID-19: if an individual is in contact with a broadrange of other individuals, the spread should be faster than if unique number of people incontact over time is small. In order to demonstrate that this concept has indeed an effecton the infection spread, we performed an additional simulation where we kept the unique number of people that every individual meets in every time step at C ( ) but introducedthe concept of "social bubbles" where all individuals form groups of 20 who have a largenumber of interactions with each other (i.e., equivalent to C ( 2, p ) but only rare interactionswith people from other bubbles equivalent to C ( ) (see Figure 3 for a picture of thecontact matrix with random connections and with "social bubbles"). • Mitigation After 60 Days . For t ≤ , we generate C ( ) random connections forevery individual at every time step. Afterwards, we assume that mitigation measures aretaken which reduce the reproduction rate to . Thus, we generate only C ( ) randomconnections for every individual at every time step t > 60 . • Suppression After 60 Days . For t ≤ , we generate C ( ) connections for everyindividual at every time step. Afterwards, we assume that lock-down measures are taken tosuppress the pandemic which reduce the reproduction rate to . • Suppression After 60 Days and Release of Lock-down after 120 Days . This scenario issimilar to the previous scenario but we assume that due to very low infection numbers, thelock-down is released after 60 days. Thus, we generate C ( ) random connections forevery individual for t > 120 . Figure 3: Snapshot of the contactmatrix of the first 200 individualsfor random connections and with"social bubbles".In Figure 2, we show the plot of (cid:80) u P ( z u,t = z ) over t = 1, . . . , 274 days for z ∈ { E, I, R } (orange = E , red = I ,blue = R ) from forward samples of the CRISP model forthese scenarios. As one can see, with no mitigation there is ahigh peak around day t ∗ = and eventually herd-immunityis achieved at 85% of infected population. Even though thenumber of unique contacts in each time step is the same, "so-cial bubbles" flatten the curve, thus slowing down the infectionbut growth rates of infected people are still super linear untillarge parts of the population had been in contact with the dis-ease. Note that a similar mitigation policy is currently usedin Belgium. In case of mitigation to R = , growth ratesare pushed to sub linear but the pandemic is still continuouslygoing on after 9 months. Not surprisingly, suppression is mosteffective at bringing the infections back to nearly 0% after 120 days. However, if the lock-down islifted after days, a second wave of infections will cause an exponential increase in infectiousnessafter only two weeks (dashed lines). Note that all these effects were computable by simply forwardsampling our individual-level CRISP model. In order to assess the test and quarantine efficacy of the CRISP model, we consider a populationof | S | = 1, 000 individuals for days (5 months) with a uniformly random contact pattern of C ( ) = contacts on average per individual and day. We simulate the actual infectionspread by applying the following sequence in each time step (i.e., day): At the beginning of each time13 o mitigation days after patient 0 got infected mitigation with localized contact pattern days after patient 0 got infected mitigation after 60 days days after patient 0 got infected release after 60 days lock-down days after patient 0 got infected Figure 2: Population level COVID-19 infection spread for three different scenarios: (top-left) Nomitigation ( R = ). (top-right) No mitigation until day 60 and then using "social bubbles". Notethat R remains at 2.5 the entire time. (bottom-left) Mitigation after 60 days by reducing R to . (bottom-right) Lock-down at day 60 and reduction of R to (solid lines). In dashed lines we showthe effect of a subsequent re-opening of a subsequent contact rate increase to R of 2.5 starting at day120.step, we query the testing-and-quarantining policy for a list of individuals which need to be testedand need to be in quarantine during this step (this will only be done after t ∗ = to simulate anundetected initial outbreak). Each policy is constrained to select no more than test candidates perday (1% of the total population). Given the quarantined individuals on that day, we remove contactsfrom and to the quarantined individuals for that day and then use the CRISP forward model (4) andCRISP test outcome model (9) to draw one sample of the next simulated infection state of everyindividual as well as the actual test outcomes of the requested test candidates. If the infection stateof an individual changes from E to I in this sampling step, we assume that with 50% probability,the individual generates symptoms. Finally, at the end of the time step, the testing-and-quarantiningstrategy is revealed the test outcomes as well as the list of symptomatic individuals (again, provided t ≥ t ∗ ). We single out an individual u for whom we set p = so that she will get infected withprobability 100% at t = ("patient 0"); for all other people we assume a p = − to model asmall chance of infection spread from exogenous sources.1. Symptom-Based Policy . For every time step t ≥ t ∗ , we will request testing for up to 10symptomatic individuals from the previous time step. For all individuals with a positive testoutcome on the previous day, we will institute a quarantine for ρ time steps where ρ rangesfrom to days in our evaluation.2. Contact-Tracing Policy . For every time step t ≥ t ∗ , we will request testing for up to 10symptomatic individuals from the previous time step. If there are less than 10 symptomaticindividuals, then we will request the remaining tests for individuals in quarantine sorted indescending order of the number of contacts they have had in the past 7 days with peoplewho have tested positive. For every individual with a positive test outcome, we will not onlyquarantine her but also all the contacts she had in the past 7 days for ρ time steps where ρ ranges from to days in our evaluation; for every individual with a negative test outcome,we will remove her from quarantine.3. CRISP Model-Based Policy . For every time step t ≥ t ∗ , we will use block-Gibbs samplingof infection traces z u to estimate P ( z u,t ) for every individual u at the current timestep t based on the contacts and test outcomes prior to time step t . We will request testing14igure 4: Effect of different mit-igation policies on the infectionpercentage and quarantine days af-ter T = days (5 months).The y -axis shows the percentageof population that got infected withCOVID-19 during the 150 days.The x -axis shows the total numberof days that individuals were quar-antined. The error-bars are com-puted as the standard deviation over20 random initializations of the for-ward model simulating the T = days while not affecting the ran-domization of the contact matrices.for up to 10 symptomatic individuals from the previous time step. If there are less than10 symptomatic individuals, then we will request the remaining tests for individuals (whohave not tested positive before) in descending order of ^ P ( z u,t = I ) . We will quarantineany individual who is not yet quarantined but whose estimated probability ^ P ( z u,t ∈ { E, I } ) exceeds a given policy threshold τ EI ; we will release an individual from quarantine oncetheir estimated probability ^ P ( z u,t ∈ { S, R } ) exceeds a given policy threshold τ SR . Note thatwe increase p in the block-Gibbs sampling by a factor of to account for "patient 0".In order to gauge the efficacy of each policy, we measure two quantities at the end of the simulation( t = ): (1) Percentage of population that got infected during the 150 days, and (2) total number ofdays that individuals were quarantined (e.g., if a policy locks down for the entire 150 days, this wouldresult in 150,000 quarantine days). Varying the policy parameters ρ , τ EI and τ SR results in curves onthe two dimensions of infection percentage and quarantine days. The closer a curve is to the origin,the more effective is the policy in terms of "health" (infection) and "economic" (quarantining) cost.In Figure 4, we plot curves for the three policies with ρ ∈ { 2, 7, 14, 21 } , τ EI ∈ { } , τ SR = . For comparison, we also show the two extreme points corresponding to "no mitigation"(i.e., zero quarantine days but the largest infection percentage of 90%) and "full lock-down" (i.e.,largest quarantine days of 120,000 and near-zero infection percentage). All three curves exhibita negative slope where a higher percentage of quarantine days corresponds to a more effectivemitigation of infection spread. Of the three policies, our CRISP-based policy achieves the bestperformance in terms of the smallest number of quarantine days for a given infection percentage(i.e., Pareto frontier). This is because our CRISP model is able to accurately identify infectioususers (even though they may be asymptomatic) and test/quarantine them proactively– this helps toprevent infection from spreading across the population while at the same time quarantining fewerindividuals with a high likelihood of getting infected. In contrast, the symptom-based policy onlytests individuals with symptoms and then quarantines the individuals who have tested positive. As aresult, since 50% of the infected individuals are asymptomatic, they never get tested and quarantined,thus resulting in a spread of infection to 60% of the population. Similarly, the contact-tracing policy,by isolating all contacts of positive tested individuals (many of whom may have low likelihoodsof getting infected), is able to achieve the absolute smallest infection percentage but at the cost ofmassive quarantining (30% of the population). Figure 5 shows a visualization of these effects in oneof the simulation runs for ρ = , τ EI = , and τ SR = . In this paper, we proposed a probabilistic graphical model for COVID-19 infection spread throughindividual contacts that captures the epidemiological state of each individual based on the SEIRmodel. We developed a computationally efficient block-Gibbs sampling-based algorithm to infer15igure 5: Infection trace and quarantining statistics for symptom-based (left), contact-tracing (middle),and CRISP ( τ EI = , τ SR = ) model-based (right) testing-and-quarantining policy over theduration of 150 simulated days (blue = S , orange = E , green = I , red = R ). In the bottom plots, weshow a stacked bar chart of quarantined individuals per day grouped by actual infection status. Whilethe number of quarantined individuals for the symptom-based policy is small, the infection spreadis not contained and the quarantining keeps growing exponentially. In contrast, the contact-tracingpolicy effectively suppresses infection spread while regularly quarantining more than 25% of thepopulation. The CRISP model-based policy is initially picking a large number of individuals forquarantining but is then able to keep it at a low-level, in particular of susceptible individuals.the COVID-19 infection risk score of all individuals at any time, given test outcome and mutualcontact information between individuals. An efficient C++-based Python implementation of ourinference algorithm is available at https://github.com/zalandoresearch/CRISP. Through experimentswith simulated data, we showed that the CRISP model is able to model macro-level characteristics ofthe COVID-19 infection at county level ( ≈ 10, 000 individuals) and effectively mitigate COVID-19spread by pro-actively quarantining and testing individuals with high risk of infections.As part of future work, we would like to further accelerate our inference procedure using otherapproximation techniques such as Variational Bayes and Expectation Propagation [2]. Our inferencealgorithm can also be speeded up by exploiting the parallelism inherent in our block-Gibbs Samplingalgorithm. For example, it is possible to concurrently sample infection traces of two individualswith no contacts in common. It is also known that the hyper-parameters of the SEIR model varywith demographic attributes such as age, socio-economic status, or location (see, for example [10]who present a location-varying infection spread model). We would like to extend our model withgroup-level hyper-parameters to account for this variation. We would also like to explore the causalimpact of mitigation or suppression policy measures (e.g., school closures, shop closures, small groupgatherings) on COVID-19 infection spread when using contact-level data. Finally, we would like toconsider more sophisticated models of COVID-19 transmission through different modalities, andcontacts with varying duration and distance characteristics. Acknowledgments We would like to thank Sebastian Munoz, Christopher Gandrud, JasvinderKandola and Peter Herbrich for all their valuable input and feedback on earlier drafts of this paper.We are also indebted to Christoph Thöns for his support with Python and C++ programming. References [1] Jantien A Backer, Don Klinkenberg, and Jacco Wallinga. Incubation period of 2019 novelcoronavirus (2019-nCoV) infections among travellers from Wuhan, China, 20-28 January 2020. 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