COVIsim: an Agent-based Model for Evaluating Methods of Digital Contact Tracing
Abstract
Background: The rapid global spread of COVID-19 has led to an unprecedented demand for eﬀective methods to mitigate the spread of the disease, and various digital contact tracing (DCT) methods have emerged as a component of the solution.
In order to make informed public health choices, there is a need for tools which allow evaluation and comparison of DCT methods.
Methods: We introduce an agent-based compartmental simulator we call COVIsim, integrating detailed consideration of virology, disease progression, social contact networks, and behaviour/mobility patterns, based on parameters derived from empirical research.
We verify by comparing to real data that COVIsim is able to reproduce realistic COVID-19 spread dynamics, and perform a sensitivity analysis to verify that the relative performance of contact tracing methods are consistent across a range of settings.
We use COVIsim to perform cost-beneﬁt analyses comparing no DCT to: 1) standard binary contact tracing (BCT) that assigns binary recommendations based on binary test results; and 2) a rule-based method for feature-based contact tracing (FCT) that assigns a graded level of recommendation based on diverse individual features.
Findings:
We ﬁnd all DCT methods consistently reduce the spread of the disease, and that the advantage of FCT over BCT is maintained over a wide range of adoption rates.
Feature-based methods of contact tracing avert more disability-adjusted life years (DALYs) per socioeconomic cost (measured by productive hours lost).
Interpretation : This research provides a useful testbed to compare and optimize real-world implementations of contact tracing (CT) schemes, a ﬁrst step in responsible and informed use of CT as an epidemic intervention tool.
Our results suggest any DCT method can help save lives, support re-opening of economies, and prevent second-wave outbreaks, and that FCT methods are a promising direction for enriching BCT using self-reported symptoms, yielding earlier warning signals and a signiﬁcantly reduced spread of the virus per socioeconomic cost.
1. Introduction
While vaccine development is underway, countries facing the COVID19 pandemic are confronted with a limited choice of non-pharmaceutical interventions to control virus propagation.
Mathematical modelling and econometric studies show that lockdown has been eﬀective in containing the
COVID-19 pandemic in many settings [1, 2].
However, sustained limitation of human contact is likely not viable with regards to individuals’ mental health [3] and ﬁnancial security [4], children’s development [5], or national and global economies [6].
Improving eﬀectiveness of non-pharmaceutical public health strategies is therefore of paramount importance to mitigate the spread of the disease, at least in the short and medium term.
Contact tracing (CT) is a public health strategy for mitigating the spread of infectious diseases, and consists of alerting individuals that they have recently been in close contact with an infected person (hereafter, an ‘index case’) and taking preventative measures.
In manual contact tracing (MCT ), this process is undertaken by public health workers who telephone, email, and/or interview conﬁrmed index cases and subsequently do the same for all persons the index case has been in contact with during their suspected window of infectiousness (hereafter, ‘contacts’).
While MCT has been used successfully to contain previous respiratory infection epidemics (e.g. SARS in 2003), the massive scale of the COVID-19 pandemic has overwhelmed the capacity of many public health departments undertaking MCT [7].
The eﬃcacy of MCT is also limited by peoples’ recall of their contacts, which is problematic especially when asymptomatic or presymptomatic transmission plays a large role, as appears to be the case for COVID-19 [8, 9].
Digital contact tracing (DCT) , which uses smartphone apps to send messages between contacts with little to no delay, has been successfully deployed in some settings (e.g. Singapore and South Korea [7]) to supplement the limited manpower available for MCT.
By enabling large-scale processing of individual-level data (e.g. on testing, presence of symptoms), facilitating communication with potentially-exposed individuals (e.g. through phone alerts), and allowing for non-identifying location-based tracing, DCT has the potential to both maximize eﬃciency of CT, as well as provide real-time data to inform understanding of COVID-19 and public health decision-making.
DCT, however, is not free of disadvantages.
Due to a wide range of privacy concerns about smartphone communications, DCT suﬀers from poor adoption by the public [10].
Additionally, most countries using DCT have adopted a simple form which informs and recommends quarantine to all digitally-recorded contacts of cases conﬁrmed through testing.
We call these systems Binary Contact
Tracing (BCT) because they recommend users either to quarantine or not (binary decisions) based on positive or negative test results and a binary decision as to whether or not a dangerous contact took place (binary input features).
COVID-19 is a challenging disease to mitigate with BCT for two primary reasons: BCT currently relies on reverse transcriptase PCR (RT-PCR) tests which have high disease phase-dependent false negative rates, are expensive, and may require a long time to obtain results [11, 12]; and the majority of transmissions of SARS-CoV-2 take place before the infector shows any symptoms, thereby reducing the likelihood that a potential infector would have been tested before transmission [13].
We observe that there are a wide variety of clues potentially available to a contact tracing app on a smartphone which are not leveraged by BCT, and further, that non-binary, individualized recommendations would be possible by leveraging such information.
We call these methods feature-based contact tracing (FCT) , and hypothesize they could provide an important and eﬀective means of reducing the spread of the disease, perhaps even more eﬀectively than BCT at relatively lower adoption rates.
Recognizing this potential, we propose a software testbed 1 we call
COVIsim in which diﬀerent DCT methods can be proposed and benchmarked via cost-beneﬁt analysis in terms of lives saved, reduction in eﬀective reproductive number ( R t ) of the virus, disability-adjusted life years (DALYs) averted, and productive hours lost.
By using an agent-based model as the foundation of this testbed, we are able to simulate a rich set of individuallevel input features, which can be calibrated to match published data for any region of interest by supplying corresponding demographics and contact pattern information for that region.
We verify that COVIsim reproduces COVID-19 case and hospitalization data for the region of Montreal, Canada.
In order to ensure the simulator is a fair and reliable testbed, we also check that the relative ordering of methods is preserved across wide ranges of simulator parameters and over several metrics.
We propose a simple rule-based FCT method which leverages individual-level features to make non-binary recommendations, and compare this approach to BCT and compare both to no-DCT via cost-beneﬁt analyses.
We ﬁnd that both BCT and FCT methods are able to reduce spread of the disease, and our results echo those of recent research [14] suggesting that
DCT methods can still save lives even at low adoption rates.
We ﬁnd evidence that FCT approaches, which leverage rich individual-level features to make graded recommendations, are promising for improving DCT even further.
Additionally, by stratifying DALYs (disability-adjusted life years) over age groups, we observe the highest reduction in DALYs for those over 80 years of age, even with extremely low adoption rates in that age group, showing the protective eﬀects of younger people using DCT.
We note that these results are conservative in estimating the beneﬁts for the most vulnerable populations, since we randomly assign DCT app usage proportional to smartphone usage, while in reality it is likely that more vulnerable people (or those close to more vulnerable people) would be more likely to use DCT.
Our results thus strongly support the usage of DCT methods as a component of eﬀective public health strategies, and we hope COVIsim will be a useful resource for development, benchmarking, evaluation, and improvement of DCT methods.
2. Related work 2.1. Agent-based epidemic modeling
Agent-based models are frequently used to study geospatial and other patterns of disease which vary at an individual level (e.g. [15, 16]).
They are thus often useful for studying diﬀerential eﬀects of policy decisions and interventions on diﬀerent subgroups of the population; for instance [17] use an ABM to study which post-lockdown measures most eﬀectively protect the most vulnerable, in terms of disease incidence, mortality, and ICU occupancy, and [18] and [19] study patterns of COVID-19 spread in diﬀerent representative locations (university, workplace, and highschools), and the impact of diﬀerent intervention scenarios in each of these locations.
One of the interventions studied by these works is digital contact tracing, and both ﬁnd evidence that it can reduce ICU admissions and help curb the spread of the disease.
ABMs are also useful for modeling the impact of outlier individuals or events such as super-spreading, e.g. [20], which are not easily captured by mathematical models of population-level spread.
2.2. Comparing contact tracing methods
Several works have studied the use of smartphone apps in epidemic management, e.g. [21, 22], and some work has begun speciﬁcally on COVID-19.
For instance, Ferretti et al [23] propose a mathematical model of infectiousness based on early epidemic data in China and compare binary contact tracing to manual tracing.
They quantify the contribution of diﬀerent transmission patterns (contamination through symptomatic individuals, presymptomatic individuals, and from the environment) and the requirements for eﬀective contact tracing.
Assuming a 3-day delay in notiﬁcation (and thereby quarantine of the individual), the authors demonstrate manual tracing could not bring
R t below 1 and hence, not control the epidemic, while instantaneous contact tracing by a digital tracing application on smartphones could do so ( R t < 1).
Shamil et al.
[24] follow a similar approach, but with an ABM taking into account realistic contact patterns, studying the potential eﬃcacy of BCT in controlling the spread of the disease.
They ﬁnd strong dependence of the eﬃcacy of digital contact tracing on app adoption, suggesting that BCT alone is insuﬃcient to control a pandemic unless over 60% of the population is using the app.
[14] ﬁnd similar results, emphasizing however that even at very low adoption rates DCT is able to save lives.
This suggests that DCT should be considered an important component of public health strategy for mitigating COVID-19.
We ﬁnd similarly that BCT and FCT are unlikely to control a pandemic on their own at low adoption rates - see 5, showing that even at 60% adoption rate these methods must be combined with other strategies to suﬃciently reduce the number of contacts in the population to bring R t below 1.
However, even a relatively small reduction in reproductive number has exponential eﬀects on cases, and DCT methods can thus save lives.
Perhaps most similar to our work is Hinch et al.
[25], who like us propose an open-source ABM which allows manual and digital contact tracing methods to be compared, with beneﬁts stratiﬁed across age.
Developed concurrently to our simulator, similarities between the two approaches highlight the importance of several design decisions made independently but converging to the same solutions, e.g. the use of agent-based models, a python interface, and the need for empirical testbeds of this nature.
A key diﬀerence in our simulator is the rich set of individual-level features (including e.g. pre-existing medical conditions), which allow us to benchmark feature-based contact tracing methods, and also allow for stratiﬁcation over a larger variety of subgroups.
The cost of this level of detail is computation; our simulator models smaller populations at higher ﬁdelity for the same computational budget.
We perform a scaling analysis (Appendix J) and veriﬁcation of ﬁt to real data (Section 5) in order to ensure the dynamics we produce on these smaller populations are representative of larger populations.
However the simulator of [25] may be preferable for studying binary-only contact tracing methods, or when faster computation is needed relative to individual-level detail.
To our knowledge, our simulator also oﬀers a more detailed cost-beneﬁt analysis to compare methods.
3. Simulator Overview
The simulator is an agent-based compartmental model [26] implemented in Python [27] and C [28], using Simpy [29], a process-based discrete-event simulation framework.
For each agent the simulator tracks transitions through
Susceptible, Exposed, Infectious, and Recovered (SEIR) states, as well as a variety of individual characteristics, including pre-existing medical conditions, self-reported symptoms, and test results.
This rich set of individual features enable the simulation of contact tracing apps which make use of such features.
At the same time, we parameterize our simulator using real-world data when available, and when no data is available we make weak assumptions and investigate the sensitivity of our results with respect to these assumptions (see Section 6.5).
3.1. Demographics
We focus our analysis on the region of Montreal for the period between
March and June, 2020, however, this ABM can be conﬁgured for other regions; see Appendix F for details.
Each agent i in the simulation has individual characteristics (including age, sex, pre-existing medical conditions) denoted
X i .
Dwelling characteristics, workplace association, and contact patterns are derived from age-stratiﬁed surveys and empirical studies; see Appendix A for a comprehensive description of this data and our demographics model.
3.2. Mobility Patterns and Disease Spread
A proportion α of the individuals are initialized as infected, chosen at random to simulate community spread.
Agents move around the city transitioning between locations like households, workplaces and other locations.
The pattern of each individual’s mobility (i.e., which locations they visit, how often they visit them, and how they interact with other individuals at these locations) is set according to [30, 31].
While at a location, agents sample contacts according to age-stratiﬁed contact matrices derived from [32].
We use contact matrices speciﬁed in [33] to simulate contact patterns during lockdown.
A detailed discussion of agents’ mobility patterns is provided in
Appendix E.
3.3. Transmission Characteristics
Virus transmission takes place anytime an infectious and a susceptible individual are within 2 meters of each other for at least 15 minutes, thereby possibly transmitting viral load to a newly infected agent.
We model the probability of disease transmission P according to [23].
Borrowing notation from the authors, brieﬂy, this probability is proportional to age-dependent susceptibility S a of the susceptible agent with age a , location-dependent multiplicative factor B n (for location n ), symptom status (asymptomatic, mild, severe) dependent ratio
A s of the infectious agent, and a surrogate for cumulative viral load ( EVL ) transmitted from the infectious agent for duration δt .
We discuss EVL in the next section.
A proportionality factor r is used to calibrate the reproductive number of the disease spread.
For the sake of completeness, a mathematical form of this transmission model is presented in Eq 1 and Eq 2.
[Equation]
where P ( δt, S a , A s , n ) is the probability of the contagion event.
We use the same values for constants as used by authors in their open-sourced code 2 .
3.4. Individual Disease Trajectory
After such a contagion event, we sample for the infected agent the variables controlling the course of the disease, including symptoms and severity.
Additionally, we sample a time-series of a quantity proportional to viral load as measured by [34, 35], which we call eﬀective viral load EV L ; this quantity represents the interaction of the virus with the host’s immune system.
We further discuss ( EV L ), its dependence on age and preexisting conditions, and how it is sampled in Appendix B.
3.5. RT-PCR Testing
In COVIsim, we model RT-PCR testing and its relation to contact tracing applications.
As discussed earlier, the disease trajectory for each agent depends on their age and preexisting conditions.
Thus, there are agents who experience symptoms on a spectrum from none (asymptomatic) to severe: agents who experience more severe symptoms are more likely to seek an RT-PCR test.
We model age-based COVID-related hospitalization of agents after publicly available data [36].
We calibrate the time-series of hospitalized cases as discussed in section 5.
Given the limited testing capacity at the onset of the pandemic, we model a testing facility with a ﬁxed maximum capacity.
For an infectious agent, the outcome of a test is modeled according to disease phase dependent false negative rates as per [37].
As an example, 4 days after a SARS-CoV-2 infection, the infected agent will have a false negative rate of 67%.
Upon receiving a positive test result, the agent is put to quarantine for d max days.
3.6. Agent Behavior
Unlike existing COVID-19 ABMs, our agents are designed to follow varying levels of mobility restrictions.
The mobility level of an agent corresponds to a fraction ( γ ) reduction in contacts with respect to the distribution of contacts in the post-pandemic mobility situation in Montreal.
Thus, if there are n + 1 such levels, an agent in level n will draw contacts from the post-pandemic activity distribution ( γ = n ), and an agent in levelis in quarantine i.e. no contacts are drawn ( γ = 0).
Levels 1 , ..., n − 1 are represented as a linear scaling of mobility between levels 0 and n .
Our choice is motivated by the desire to have a simple model lumping together the eﬀect of choices individuals can make to reduce their likelihood of becoming infected like washing hands, wearing a mask, and physical distancing.
Each of these levels are further associated with a dropout parameter that represents the fraction of time an agent in level k will drop to leveland return to a higher level of activity (speciﬁcally, post-pandemic numbers of contacts).
ABMs can be conﬁgured to quarantine individuals due to any of the following triggers (a) conﬁrmed positive test (b) self-reporting of symptoms, (c) recommended by the app (d) household member of any of the above cases.
Additionally, in order to model population-wide mobility restrictions we use a Bernoulli distribution with parameter β to sample contacts.
Thus, an agent that usually draws 12 contacts will now draw β ∗ 12 contacts on average.
This modeling choice is a simpliﬁcation of varying degrees of government imposed mobility restrictions, controlled by a population-wide β .
3.7. Communication between Agents
DCT methods rely on capturing high-risk contacts and communicating risk information.
The population-level app adoption rate is a very important parameter because the fraction of contacts captured by this system will be proportional to the square of the app-using fraction of the population.
To distribute apps throughout our simulated population, we use an agedependent distribution across smartphone owners.
We use an age-based breakdown of smartphone users as in [23], and use an UP T AKE parameter to vary the population-level adoption rate.
If an agent is assigned an app, there is a provision to report individual characteristics like age and preexisting conditions as well as daily symptoms.
We further model dropout rates for reporting symptoms as well as a ”drop-in” rate for falsely reporting symptoms to account for malicious users and other confounding factors that produce symptom inputs, such as colds or ﬂu.
When app users i and j have a signiﬁcant encounter on day d (which can lead to contagion) the CT method can exchange messages between them, on the same day or later.
We have considered two kinds of messages: a contact message exchanged on day d allows i and j to keep track of the encounter event (such as when it happened) as well as a handle allowing them to send each other messages later (potentially via some server), while minimizing breaches in privacy.
A contact message could also contain information about the current degree of contagiousness risk they pose to each other on day d .
Later, if i observes new clues suggesting their contagiousness on day d was actually higher, i can send an update message to j , with information which can help j update their own evaluation of contagiousness.
We denote M di,j ( t ) the risk message sent on day t by i to j regarding their encounter on day d .
If t = d this is the initial (contact) message, whereas for t > d this is used to update the information j will have at day t about their encounter with i on day d .
Note that i may send multiple update messages to j , as they acquire more information about having been infectious on day d .
Additionally, we restrict the frequency of communication: if t (cid:48) is the time of the last risk message sent by i about some encounter, the next message is sent at time t (cid:48)(cid:48) only if M di,j ( t (cid:48)(cid:48) ) (cid:54) =
M di,j ( t (cid:48) ).
The clues which agent i may use to come up with its risk messages include symptoms, test results, pre-existing conditions and received risk messages.
The way to come up with these risk messages, as well as how to adjust behavior based on estimated risk, is speciﬁed by the CT method, such as those described below.
4. Digital Contact Tracing methods
In digital contact tracing we have two goals: to reduce the individual’s spread of disease (by recommending a reduction in mobility to risky individuals), and to inform contacts of additional risk.
To control the mobility of agent i on day d , we use apply a recommendation level γ id , where γ id =recommends post-lockdown contact patterns reported in [33] and γ id =recommends full quarantine.
Intermediate values of γ id linearly scale down the number of contacts.
In addition, we predict a vector risk history r d where | r d | = d max .
We use this risk history determine which risk messages M di,j to send, thereby determining what information is available to our contacts such that they may modify their own behavior or to propagate risk further.
4.1. Binary Contact Tracing (BCT)
The most common class of digital tracing methods, Binary Contact
Tracing (BCT), can be viewed as a binary classiﬁer with a class for quarantine and a class for no quarantine.
Most often, the decision boundary is simple: did the individual recently contact someone who received a positive RT-PCR test.
We refer to these methods as Test-based BCT , or simply BCT (since
BCT is rarely implemented with any other input feature).
We next provide a formal description of BCT, then propose FCT in Section 4.2.
Agent i may receive and record a positive test result test =
P ositive with test ∈ { P ositive, Negative, None } and where None indicates no test result was received that day.
As an input to the BCT method, we construct the vector T d which contains up to d max elements of type test representing the tests over the last week.
We can refer on day d to the test results received on day d (cid:48)
(where d − d max < d (cid:48) ≤ d ) as T d ( d (cid:48) ).
The BCT rule is as follows: if any element of P ositive ∈ T d , return a risk history where each element of r d is equal to R max and then send a message M i,j ( d ) to each of the digitally recorded contacts of i .
At the same time, each agent j receiving such M i,j ( d ) will hold onto this message for d max days and quarantine themselves as discussed in the section 3.
In our experiments, we used d max = 14 days.
We describe this formally with Algorithm 1.
We call this particular method BCT1 because it only aﬀects the ﬁrst order contacts of individuals with positive test results; in section 6 we also show results for another baseline, BCT2, where second order contacts may also be quarantined.
Algorithm
1: function ComputeRisk ( T d , r d − 1 , γ d − 1 ) 2: if T d = positive then 3: r d = R max for all d in d max 4:
5: else 6: return
4.2. Feature-based Contact Tracing (FCT)
We describe a class of methods we call Feature-based Contact Tracing (FCT), which leverage the potentially rich set of features available on a smartphone to make graded recommendations.
We also propose a member of this class, a rule-based method for FCT which we call Heuristic-FCT .
Heuristic-FCT uses several input features: self-reported symptoms S d , test results T d , risk messages from contacts M d , and previous risk history r d − 1 , where each bolded variable is a vector containing d max elements (though in the case of S d , each element S d ( d (cid:48) ) includes information about which symptoms and what severity was experienced on that day d (cid:48) ).
Similar to
BCT, the output of this procedure is a vector risk history r d and a scalar mobility recommendation γ d .
As mentioned, the ﬁrst rule of Heuristic-FCT is the same as in Test-based
BCT1 : an index case’s positive test result is propagated to ﬁrst degree contacts who are recommended to quarantine.
We also add a rule stating that F alse ∈ T d = ⇒ r d (cid:48)
=
for all d (cid:48) ∈ { d, ..., d − 7 } .
This negative test result rule overrides the intermediate variables computed by other rules.
The intermediate risk variable s d is produced by grouping symptoms
S d that were reported during the last week into three groups based on how indicative they are of COVID-19.
We grouped symptoms empirically by analyzing the proportion of the time that these symptoms were caused by
COVID-19 in the simulator.
The presence of a highly informative symptom in S d results in a high symptoms risk s d ; a moderately informative symptom leads to results in a moderate symptoms risk; and so on.
The intermediate risk variable m d is produced by received risk messages is produced by ﬁnding the maximal message in M d (cid:48)
ji where d − 14 < d (cid:48) ≤ d .
We then subtract a constant from this message’s risk level and call this new value r (cid:48) d (cid:48) .
We then take this most risky message and place it into one of several categories which determine the intermediate risk level m d .
We then compute the duration of time when agent i could have been infectious if this contact had caused their infection as d (cid:48)
+ 1 < d (cid:48)(cid:48) <
d .
Thus, we conclude m d = r (cid:48) d (cid:48) while d (cid:48)
+ 1 < d (cid:48)(cid:48) <
d .
Finally, when computing r d , we take the maximum risk value for each day across the intermediate risk histories m d and s d , as well as from r d − 1 .
In addition, we have a recovery rule which activates if no symptoms were recorded in the past 7 days, and there were no high risk messages within the last few days.
In this case, we set every element of r d to 0 and set γ = 1.
5. Calibration of the Simulator
In this section we seek to provide evidence (1) that our simulator is producing a reasonable approximation of real-world COVID-19 dynamics; and (2) that it is a reliable testbed for comparing contact tracing methods.
We address (1) by checking the output epidemiological characteristics match published literature (see Figure 1 and Table 1), and by checking that the hospitalization and mortality statistics are well-aligned with those found in real world data (Figure 2).
We address (2) by performing a sensitivity analysis of the simulator to a wide range of parameters, and checking that the ranking between contact-tracing methods is preserved over these settings, for diﬀerent metrics (see Section 6.3).
5.1. Validation of epidemiological characteristics
We calibrate the simulator so that the observed statistics in the simulator are similar to what is observed for COVID-19, plotting SEIR curves in 1 and comparing statistics to published data in 1.
5.2. Comparison with real hospitalization and mortality data
We run 10 simulations using random seeds and plot the mean and standard deviation of the hospitalization and mortality statistics over 100 days.
We compare these results to the real data reported in Montreal from the same time period.
Our simulations use 30,000 people and we report results as a
[Table]
proportion of population.
We see that under these settings the proportion of the population that is hospitalized or deceased aligns with the data from
Montreal (See Figure 2).
6. Experiments 6.1. Setup
We focus our analysis on the region of Montreal and sample agents from publicly available census data [40].
Given a memory intensive infrastructure required for managing risk messages, we run simulations on a smaller population size of 3000 people.
We initialize each simulation with 0.2% of the population as infected (6 infections), and run 10 diﬀerent seeds for each value of β ∈ { 0 . 25 , 0 . 275 , 0 . 30 , ..., 0 . 85 } resulting in a total of 420 runs.
These values are chosen so that we can get estimates ∆ ˆ R of the change in the reproduction number of the virus, as discussed in Section 6.3.
6.2. Baselines
We compare the Heuristic-FCT method proposed in Section 4.2 with the Test-based BCT method discussed in Section 4.1, and a No Tracing scenario.
The scenario of No Tracing corresponds to agents initialized at level 1 (post-lockdown contacts with some restriction advised) instead of level 0 (pre-pandemic contacts), in order to compare our methods in the context of the scenario where economies are gradually reopening.
We use Test-based
BCT1 and Test-based BCT2 to distinguish between the two variants when necessary, otherwise we use Test-based BCT to imply Test-based BCT1 which is the method being considered by most of the countries.
6.3. Evaluation
We compute the following metrics to evaluate various scenarios:
• Average number of contacts per day per human ( C ): We empirically compute the average number of daily contacts per agent in simulation.
• Proxy R
ˆ R ): We use an empirical calculation to estimate the reproductive number R , we call this estimate ˆ R .
At any timepoint in the simulation we may approximate R t by ˆ R t , by computing the infection tree and taking the ratio of number of children number of parents , where parents are recovered agents.
This ratio gives an approximate rate of growth of the tree.
We use ˆ R t at the end of the simulation as our ˆ R .
• Daily cases (%): Percentage of population exposed, i.e., new cases on any given day of a simulation.
• Cumulative cases (%): Percentage of agents in exposed, infectious, or recovered state up until a particular day of a simulation.
• Prevalence (%): Percentage of population in infectious or exposed state on any given day of a simulation • Incidence (%): Number of agents exposed per 1000 susceptible agents on any given day of a simulation.
It is also referred to as attack rate.
As the contact tracing methods proposed in this paper change agents’ behaviour to varying degrees, it is crucial to compare diﬀerent methods across similar social mobility restrictions.
For example, BCT can only use levels 1 and n while FCT can use all levels from 1 to n .
Therefore, at the same value of β , BCT will likely under perform as it can not reduce infections by using intermediate levels.
Thus, for a fair comparison of BCT with FCT, we run simulations with varying values of β ∈ (0 , 1) and match them for average number of contacts.
To compare the performance of diﬀerent methods for the same mobility restriction we empirically compute pairwise diﬀerence in mean ˆ R for a ﬁxed number of contacts C .
This is achieved by obtaining the performance in terms of ˆ R of a method across a spectrum of values of C (by varying β ), and ﬁtting a Gaussian Process (GP) regression [41] to obtain a functional dependence for each method between C and ˆ R .
We denote this ﬁtted regression for method A by ˆ R GPA .
To compute the advantage of method A over B , we ﬁnd
C = x for which A yields ˆ R GPA = 1.
Next we compute the advantage ∆ ˆ R AB of A over B as ˆ R GPB ( x ) − 1.
[Equation]
, where, x is such that ˆ R GPA ( x ) = 1 .
[Equation]
Note that R = 1 is the threshold between exponential growth and exponential decay of the virus and serves as a good point of interest when comparing methods since the objective is to choose a method which brings R below 1, all else being equal (e.g., general restrictions on social mobility).
6.4. Comparison of methods
Figure 3 shows a comparison of the GP regression ﬁts between aforementioned DCT methods at 60% adoption rate, and a No Tracing scenario, across a range of values of C .
We observe that Heuristic-FCT signiﬁcantly improves over Test-based BCT by reducing ˆ R by 6.7%.
Of note is the region around C = 5 .
61 ± 0 .
5 where ˆ R = 1 . 2 for No Tracing .
We compare the performance of DCT methods in this region to set our comparisons in the context of current scenarios (partial lockdown) where government-imposed restrictions keep R under control.
To further investigate the reason for the improvement of Heuristic-FCT over Test-based BCT , we peek into the simulations in the concerned region of
C .
Figure 4 shows that on average, Heuristic-FCT exhibits lower incidence as well as prevalence on any given simulation day.
This is expected as
Heuristic-FCT makes use of far richer features as compared to binary test results used by Test-based BCT to evaluate individual’s risk of infecting others.
Finally, due to network eﬀects, it is likely that adoption rate plays an important role in the performance of DCT methods.
Thus, we evaluate
DCT methods at various adoption rates.
To do this, we run simulations of DCT methods at diﬀerent adoption rates in a similar way as explained in Section 6.3, and concern our analysis on simulations chosen according to the selection criterion described above.
Figure 5 shows the performance of the considered DCT methods at diﬀerent adoption rates.
As expected, the performance diminishes with lower adoption rates, for all the methods.
However, we observe that Heuristic-FCT retains its advantage over BCT methods even at the lower adoption rates.
At the same time, we also note that poor adoption brings both the DCT methods close to No Tracing , thereby making it more diﬃcult to measure signiﬁcant advantages over the partial lockdown scenario.
Thus, we argue that by continuity while at any adoption rate DCT methods can save lives, adoption rate is crucial for their eﬃcacy.
6.5. Sensitivity Analysis
A primary contribution of this work is the creation of an ABM which can act as a testbed for comparing COVID-19 contact tracing methods.
While the majority of the parameters in our ABM are chosen according to published literature, much about COVID-19 is still unknown or uncertain.
For this reason, we conduct a sensitivity analysis of some key parameters which exhibit high variance across diﬀerent studies.
Speciﬁcally, we study the impact of the asymptomatic proportion of the population (see Figure 6), diﬃcult to measure without widespread serological testing, and the initially infected proportion of the population (Figure ?? ).
We observe that the relative eﬃcacy of diﬀerent methods (i e. the ranking of methods) holds across a wide range of settings, a desirable characteristic for a comparison testbed.
7. Cost-beneﬁt analysis
COVID-19 has been as much a challenge for health agencies as it has been for economic decision makers.
Countries around the world have taken unprecedented measures to prevent the collapse of health and economic welfare of people.
These two objectives have come to be at odds with each other in an eﬀort to contain the pandemic.
Therefore, we acknowledge the fact a more useful evaluation of DCT methods should also stand on sound assumptions as it pertains to utility maximization in economic theory.
Therefore, in this section, we attempt to relate simulation dynamics to quantities that policy makers might ﬁnd useful in their decision making process.
To assess the socio-economic burden of COVID-19, we examine the following metrics (a) Disability-Adjusted Life Years (DALYs)[42], a measure of lost years of healthy life, and (b) Temporary Productivity Loss (TPL), a measure of economic cost to society of restrictive measures.
Of particular interest is the trade-oﬀ between these two measures.
We use TPL per DALY averted to compare the cost-eﬀectiveness of DCT methods with respect to the No Tracing baseline.
This can be thought of as an Incremental CostEﬀectiveness Ratio (ICER), which answers the following question: how much more does each unit of additional beneﬁt (in averted DALYs) cost with respect to the No Tracing baseline?
A breakdown of the methodology used to calculate DALYs and TPL can be found in Appendix K.
Of note is a particular assumption involved in computing TPL which is about computing TPL: total foregone work hours due to quarantine, are scaled by a factor of 0.49[43] to account for the proportion of agents able to work from home.
7.1. Setup
To assess the impact of aforementioned CT methods on DALYs and TPL, we run 10 simulations of 3000 agents for each method.
For a reliable economic analysis, it’s necessary to have data on the full trajectory of simulations which reach a post-epidemic steady state consisting of agents that are either susceptible or recovered.
Such a trajectory would help assess whether the
DCT methods actually avert the loss of life years, or simply delay them.
Since CT methods are used to stop individuals from spreading the disease, it is important to know whether the CT methods have successfully reduced the overall proportion of individuals, or if they have simply delayed these infections such that the epidemic takes a same trajectory as in the No Tracing scenario, but on a longer timeline.
If the simulations are only run for 60 days with 0.2% infections seeded at the start, then the epidemic has still not reached a post-pandemic steady state, resulting in a possible overestimation of the beneﬁts of the aforementioned
CT methods.
Thus, we run longer simulations of 90 days with 5% infections seeded at the start of simulations.
This particular choice lets us draw preliminary insights into the cost-eﬀectiveness of CT methods without having to scale up the simulations which are extremely compute intensive.
7.2. Results
As a ﬁrst step towards understanding simulation dynamics in terms of healthcare and economical costs, we compute DALYs and TPL for each scenario considered in Section 6 under the conditions described above.
We use No Tracing as a reference scenario to independently evaluate diﬀerential beneﬁt of DCT methods over doing nothing.
As can be seen in Figure 8, under No Tracing, the total DALYs is 129.42, and the TPL is $1.370M. Test-based BCT slightly aﬀects the health and economic outcomes: the TPL is $1.591M and the total DALYs is 119.42.
However, Heuristic-FCT has a comparatively massive eﬀect on the total
DALYs, which drops to 71.31.
However, this is contrasted by a rise in the TPL to $2.122M: health outcomes are drastically improved at the cost of a greater drop in productivity. Of note is the diﬀerence in impact of both tracing methods: whereas Test-based BCT has very little eﬀect on the health and economic outcomes, Heuristic-FCT reduces total DALYs by 44.90% at the expense of an increase in TPL by 54.89%. Uncer the assumptions discussed at the onset of this section, our experiments (results in table 2) suggest that Heuristic-FCT saves almost six times more DALYs than Test-based BCT does, a signiﬁcant number of lives years saved.
However, we note that TPL of implementing Heuristic-FCT is three times that of Test-based BCT due to higher proportions of agents being quarantine.
Finally, we note that the cost per DALY of Heuristic-FCT is 58% of that of Test-based BCT , as evident from the ratio of TPL per DALYs averted of the two methods.
Thus, the cost per healthy year of life saved by Heuristic-FCT is lower than the Test-based BCT method. Test-based BCT Heuristic-FCT
[Table]
∆ DALYs (years) 10.01 ± 6.
58.12 ± 6.
∆ TPL (hours) $223K ±
$755K ± 3.
∆TPL∆DALYs $22.3K ± 15.5K $13.0K ± 0.
DALYs per age group Since there is evidence in the literature that COVID-19 disproportionately aﬀects the elderly[44][], we stratify the DALYs per person by age in Figure 9.
It is highest among people aged 80-89 years for both CT methods, which is consistent with the literature. It is worth noting that the mean of Heuristic-FCT performs at least as well as Test-based BCT across all age ranges, and notably better for agents aged 60 and over.
By aggregating across age groups weighted by the size of each age group, we ﬁnd that the mean DALY per person rate is 0 . 0425 for No Tracing .
It improves to 0 .
for Test-based BCT and to 0 .
0236 for Heuristic-FCT , meaning that DALYs per person are lower when using Heuristic-FCT .
8. Conclusion
This work presents COVIsim, a simulation testbed for evaluation of DCT methods.
COVIsim is an agent-based compartmental model that is initialized with a synthetic population sampled from the census data.
Daily activities as well as interactions for each agent are sampled according to the empirically derived contact matrices.
We calibrate COVIsim to approximate the spread of COVID-19 virus to the region of Montreal, however, the simulator can be easily conﬁgured for other regions via change of an appropriate conﬁguration ﬁle.
Finally, we propose the FCT class of contact tracing methods that utilize a richer set of input features as compared to BCT methods (which rely on binary signals like presence or absence of a positive test result).
In doing this, we aim to provide infected individuals with a warning signal earlier than BCT methods.
To put FCT in practice, we designed Heuristic-FCT which uses hand-designed rules to inform an individual’s risk of infection and infectiousness to others.
Our empirical results show that Heuristic-FCT results in 6.5% improvement in R t over BCT methods, and both the methods themselves provide a signiﬁcant improvement over a partial lockdown scenario.
Experiments with varying adoption rates suggest that the eﬃcacy of DCT methods is heavily dependent upon adoption rates.
It is, however, observed that Heuristic-FCT retains its beneﬁt over Test-based BCT method across the adoption rate spectrum, but this advantage was not statistically signiﬁcant in the face of very low adoption rates, at the scale of our simulations.
Using an agent-based compartmental model as the foundation of this testbed allows us to simulate a rich set of individual-level features, which we show can potentially be leveraged by DCT methods to improve over the existing BCT methods.
We hope that the baselines established in this work will encourage and enable the informed development of DCT methods as a ﬁrst step in their responsible deployment as an epidemic intervention tool, potentially saving lives at lower economic cost during deconﬁnement and/or second-wave prevention.
Finally, this work joins a growing body of work in considering novel methodologies for rigorous evaluation of interdisciplinary technologies.
Epidemiology is fundamentally an intersectional science, touching sociology, biology, behavioural psychology, geography, political science, ecology, mathematical and computational modeling, and many other ﬁelds, in a society which is increasingly digitized and globalized.
By working together across ﬁelds, with careful empirical study, we have hope in dealing with the impor- tant issues we face.
Future Work A major direction for future work is to benchmark a wider variety of CT methods, including probabilistic and machine-learning based methods which could make even better use of the features our simulator provides than the Heuristic-FCT proposed here.
The data generated by our simulator are potentially of interest for training such models to estimate individual-level characteristics which are predictive of the spread of the disease.
Our cost-beneﬁt analysis using DALYs and TPL is a ﬁrst step towards an integrated framework to help policy makers in their decision process.
We see imbuing economically sound decisions in our simulated agents as a step in creating an integrated framework for richer evaluation of DCT methods.
Although COVIsim is designed to evaluate DCT methods, we foresee directions for the simulator to investigate the impact of a gamut of nonpharmaceutical interventions on containing COVID-19.
For example, with some work and expertise in manual CT methods, one can compare and evaluate various variants thereof.
Another example is to analyse various
COVID-19 testing strategies in conjunction with DCT methods.
Studies of this nature could potentially help public health agencies as well as policy makers in their decision process.
A far cry, though worth mentioning, is the fact that COVIsim has essential components for simulating an outbreak, thereby enabling adaptation to other infectious diseases like inﬂuenza or tuberculosis.
Appendix A. ABM Demographics
Information on the Montreal population regarding age, sex and occupation distribution was retrieved from Canadian Census data [40].
Prevalences of selected medical conditions considered as COVID-19 infection and prognosis risk factors were determined based on prevalence estimates from nationally representative surveys or medical surveillance programs in Canada: heart disease [46, 47, 48, 49], stroke [48], asthma [47, 48], chronic obstructive pulmonary disease (COPD)
[47, 48, 49], cancer [47, 48, 50], diabetes [47, 48, 49, 51], obesity [47, 48, 49, 52, 53], chronic kidney disease (CKD) [47, 48, 49, 54], immuno-suppressed conditions [48] and smoking [55, 56].
National prevalence estimates were extracted based on age group ( < 10, 11-20, 21-30, 31-40, 41-50, 51-60, 61-70, 71-80 and > 80 years of age) and sex.
We determined conditional probability of developing COVID-19 in ABM based on symptoms and risk factors associated with COVID-19 in published literature.
A mathematical modelling study of the epidemic with Canadianspeciﬁc estimates [57] was used to model COVID-19 susceptibility in the pediatric population of the simulator.
B. Inoculum & Eﬀective Viral Load
We model inoculum, the amount of virus transmitted during an exposure event, as a random variable uniformly distributed between 0 .
and 1 . .
The magnitude of inoculum is used to determine the type and severity of symptoms.
We sample parameters for a piece-wise linear model of what we call eﬀective viral load (EVL) 3 .
We think of EVL as a piecewise linear function, attributes of which are sampled for each individual separately.
Figure B. is the mean of sampled eﬀective viral load curve.
C. Testing & Quarantine
We model RT-PCR test allocation as a priority queue.
When an agent experiences symptoms, or is advised by a contact tracing application to seek
E ﬀ ec t
i v e
V i r a l
L oa d
Time
Incubation
Onset
Plateau
Post plateau
Post plateau a test, this agent is added to the queue.
For diﬀerent symptom severities, we assign a diﬀerent probability of seeking a test.
At some points during the pandemic, Montreal experienced a restricted testing capacity.
In order to model that setting, we limit the daily testing capacity to a proportion of the population (generally set to 0.1%).
To determine which people get tested under such a restriction, we allocate tests to people with more severe symptoms and app-based recommendations.
After a test has been conducted, there is a delay before results are returned (2 days in our experiments).
During this period, the individual who is being tested is isolated, and if the test returns positive then an additional 12 days of isolation is recommended.
Any quarantine which is applied to the person receiving the test is also applied to other members of their household.
D. Hospitalization
We adopt a simple model of hospitalization and interaction within them.
To simulate post-lockdown scenarios, we assumed no infections in hospitals.
We adopt a probabilistic model of hospitalization where likelihood of being admitted to the hospital or ICU depends on symptom severity and age.
Mortality rates are conditioned on age-group following data from Quebec public health (date?)
The duration of a hospitalization, likelihood of requiring critical care, and mortality rate given critical care by age follow nationally conducted surveys available publicly 4 .
The number of hospitals is deﬁned in relation to the population, using the same ratio of hospitals to people as are found in Quebec (1.99 hospitals per 100,000 people).
The number of hospital beds per capita, and occupancy ratios are taken from 5 , and the number of icu beds per capita and occupancy ratios are taken from 6 .
Hospitals are staﬀed by doctors and nurses, who are modelled as people with a profession that requires they work at the hospital and have protected interactions with patients.
Literature on the link between underlying medical conditions and COVID19 encompasses risk of being hospitalized due to COVID-19, risk of severe complications (e.g. mechanical ventilation) and risk of mortality.
Preexisting medical conditions were used to model: 1) hospitalizations and deaths outcomes and 2) conditional probability of symptoms.
To model hospitalizations and deaths in the population, we used risk ratio estimates from studies focusing on risk of hospitalisation and risk of death as outcomes.
Risk ratios adjusted for other individual characteristics were preferred over crude estimates.
To inform the hospitalisation and death outcome model from the simulator, we selected the following risks ratios: diagnosis of heart disease, 1.17 [48], stroke history, 2.16 [48], asthma with recent oral corticosteroid use, 1.13 [48],
COPD, 1.08 [47], cancer (excluding haematological malignancy), 1.72 [48], diabetes, 2.24 [47], obesity stages 1 and 2 (body mass index (BMI) = 30-39.
kg/m 2 ), 1.8, obesity stage 3 (BMI > 40 kg/m 2 ), 2.45 [47], CKD, 2.60 [47], immuno-suppressed because of asplenia, 1.34 and because of immunosuppressive conditions (excluding asplenia and haematological malignancy), 1.70 [48].
Given the uncertainty on the association between smoking and COVIDprognosis [56], we did not consider this risk factor in the simulator.
E. Contact Patterns
In this section we describe contact patterns in pre-pandemic situation.
Scenarios of lockdown and contact patterns in intermediate behavior levels are a modiﬁcation by a factor γ n as discussed in the section 3.
E.1. Empirical Matrices
We use empirically derived matrices in 2017 from [32] for Canada that we further project on to Montreal’s demographical structure.
Projection of country-wide matrix to a regional matrix is done via method described in [59].
However, ABM can be conﬁgured to bypass the step of regional projection of contact matrices.
Given a discrepancy between population wide mean daily contacts inferred from projected matrices and Montreal’s number of contacts reported in a 2020 survey [33], we scale the projected matrices appropriately.
We ran 12 simulations with no infected agent i.e. α =to observe the pre-pandemic contact patterns to yield simulated contact patterns in this section.
Additionally, the simulated contact patterns shown in this section are descaled with the same multiplicative factor that is used to scale the projected matrices.
E.2. Dwelling Characteristics
As discussed in the main section, agents are grouped into houses according to census data [60].
Thus, we simulate dwelling characteristics of the city of
Montreal.
However, it can be conﬁgured easily for any other city by using appropriate parameter values.
We consider house of sizes ranging from 1 to
Age distribution of agents living solo also follows census data.
Further, house sizes ranging from 2 to 5 consider three broad categories of dwelling characteristics - (a) couple with x kids, (b) single parent with x kids, (c) random allocation, where x represents number of kids required to complete the house size.
For example, for a house with single parent and of size 5, x = 4.
The distribution of these characteristics also follow from census data.
Finally, we also consider senior residencies where a proportion of agents above age 65 live.
We inform this proportion from the census data as well 7 .
As a result of housing allocation discussed above, we yield a contact pattern as shown in Figure E.12.
We make two observations (a) there is an oversampling of contacts towards the older age groups: It is because older agents grouped in collectives like senior residencies are modelled explicitly.
This choice was motivated from [61] which suggests inclusion of collectives in proper response to the COVID-19 pandemic.
(b) a slight discrepancy we observe in the intensity of the main diagonal is due to insuﬃcient social gatherings at households.
E.3. Workplace Allocation
We consider an age-dependent workplace allocation such that agents in each age group have a probability of attending a school or a workplace.
We consider schools for the following age groups (a ∗ ) 2-4 years old (y.o) (b) 4y.o (c) 5-12 y.o (d) 12-17 y.o (e ∗ ) 17-19 y.o: (f ∗ ) 19-24 y.o (g ∗ ) 25-29 y.o, where ∗ marks the age group in which only a fraction (informed by census data) of agent population was allocated schools.
Further, we assume 100% employment so that all agents older than 17 y.o and younger than 65 y.o. were allocated a workplace.
Agents in senior residencies are allocated a common room as their workplace where they get together during working hours.
Such allocation of younger agents to schools give a contact pattern as shown in Figure E.14, while a workplace allocation of adults yield a contact pattern shown in Figure E.13.
E.4. Random (Other) Locations
To model infections at locations other than workplace and house, we consider locations where agents remain for a relatively shorter duration as compared to house and workplace.
Speciﬁcally, we model interactions at locations like restaurants, grocery stores, and parks.
Note that this category of locations is also termed as “other” in [32].
Further, as the mean number of contacts at house are greater than the number of residents, it was important to consider socializing activities organized at houses.
To do this, we maintain a pool of agents that an agent interacts with, and bring them together for a social activity at either a restaurant or at the agent’s house.
We discuss scheduling of these activities next.
A contact pattern resulting at these random locations is shown in Figure E.15.
E.5. Activity Scheduling & Mobility Pattern
We consider adult supervision for agents below 14 y.o i.e. except for when agent goes to school, at least one adult agent (older than 14 y.o) has to be present all the time.
Thus, we pre-plan the schedule of agents older thany.o at the start of the simulation, and plan the schedule of agents younger than 14 y.o during the simulation.
Planning the schedule takes into account workplace opening hours as well as regularly scheduled activities like social gatherings 8 , exercising, and grocery shopping 9 .
Thus, an agent who has gone to a grocery store or a restaurant on one day will be less likely to go again during that same week, and so on.
The schedule additionally depends on the day of the week.
For instance, agents with school as workplaces are scheduled to be at school on weekdays, whereas most of their time will be spent at home on weekends.
On the day of activity, however, these activities might stand cancel due to sickness, quarantining requirements, or hospitalization.
In these situations, location of activity is appropriately changed for a required duration.
At the same time, if an agent requiring adult supervision is sick, has to quarantine, or is hospitalized, an adult from the same house has to cancel the activity to stay with the agent.
Of note, agent’s mobility i.e. presence in locations other than house is reduced when they are experiencing symptoms (to a degree proportional to symptom severity).
Note that we do not change the schedule unless the agent is quarantining i.e. normal mobility is maintained all the time unless the agent is put in the level n .
Figure ??
show a breakdown of contacts at each location on weekdays and weekends.
E.6. Sampling of Contacts
We implement age-stratiﬁed contact sampling as informed from empirically derived contact matrices as described in Appendix E.1.
Speciﬁcally, for each agent we draw number of contacts as per the location-speciﬁc agestratiﬁed number of contacts obtained from the contact matrices.
We use a negative binomial distribution [63] to draw number of contacts.
Further, we use these matrices to infer probability of interaction with other agents in each age group, thereby, implementing location dependent assortativity in interactions.
We call these interactions as encounter .
Finally, we also draw amount of time spent in each encounter as per the survey conducted in California [64] standardized to the demographics of Montreal.
Thus, we obtain an aggregated contact pattern as shown in Figure E.17.
F. Simulator Customization
We brieﬂy describe the required steps for customization of the simulator.
Location-speciﬁc demographic and contact data may be modiﬁed simply by adding a new conﬁguration ﬁle to the conﬁgs folder, and specifying it either through the command-line while running the simulator or modifying the default behavior of the simulator through the core conﬁguration ﬁle.
Conﬁguration ﬁles are written using YAML, a human friendly data serialization standard.
Essentially, these ﬁles contain key-value pairs, where the values for a new region must be speciﬁed and contained in the new conﬁguration ﬁle.
Examples of modiﬁcations required to model a new region include: population-level distribution of age, how often people go out to stores, socialize, and contact others.
G. Privacy, Security, and Phone technology
We provide details of the messaging protocol that uses Bluetooth signals to exchange tokens.
The privacy protocol which provides anonymous message exchange between phones which have exchanged these tokens is covered in more depth in another paper 10 .
In the context of the current work, we focus on the description of how we model the Bluetooth communication range of phones and on the description of the clustering algorithm we use to prepare received messages for input to a risk prediction model.
Bluetooth Details
Given a ground-truth distance and duration for our encounter, we want to determine if two app-users should exchange encounter messages.
We only exchange encounter messages if the perceived distance is below 2 meters and the duration greater than 15 minutes.
To compute this, we use a naive model of the Bluetooth noise which uses a per-person noise and a per-encounter noise.
Each smartphone has a diﬀerent noise sampled uniformly betweenand 1.
Each encounter takes the mean of this value across both users.
We then apply a relative oﬀset to the real encounter distance by multiplying the combined user noise and a uniform random variable centered at 0 with range [0 . 5 , − 0 .
The magnitude of the distance oﬀset is up to 2 metres when the real distance is 2 metres, and up to 0.5 metres when the ground truth distance is 1 metre.
Encounter Messages
An encounter message m enc is composed of the day the encounter occurred d and the sender’s quantized risk at the time of the encounter r d .
The formal deﬁnition of a risk message is:
[Equation]
An app user (say “Alice”) receives new encounter messages four times per day in batches; we call the set of new encounter messages on day d , M new .
The risk messages which Alice has already clustered are noted M enc (which is an empty set when Alice ﬁrst installs the app).
It is useful to think of encounter messages as database records.
Alice inserts records into her database whenever she encounters other app users (“Bobs”) with their current risk estimate.
Update messages are like database update statements, and are used to change the risk values in old encounter messages.
Risk Update Messages
If a user (say “Bob”) had many encounters with Alice, and Bob subsequently receives a positive test result, becomes sick (reporting symptoms), or otherwise re-evaluates his risk, then Bob will send risk update messages to Alice.
Similar to encounter messages, these risk update messages may be sent through the server up to four times per day.
More speciﬁcally, if Bob updates his estimate of his risk on some previous day d old , then Bob will construct a risk update message for every encounter message that he had on d old and these messages are sent to the users with whom Bob had a contact.
If this update message is sent on day d , it is composed of the current day d , his new risk r new , his previous risk estimate for that day r old and the day of the encounter d old .
Therefore, the formal deﬁnition of a risk update message is:
[Equation]
Encounter and Update Message Clustering
By virtue of the strong privacy protocol in place, we are not able to create message clusters that correspond the true chains of contacts for the encountered users.
The only information we have to create clusters is the day and risk level of the encounters.
Our hypothesis is that a user’s contact patterns can provide a rough idea on the number of individual people they encounter on each day.
All encounter messages received on a given day with a similar risk level are put into the same cluster.
Given that there are 16 risk levels, it is only possible to create 16 new clusters on a given day, unless the user receives update messages.
Every update messages is created for exactly one encounter message.
If there are fewer update messages for a given day / risk combination than there are encounter messages for that day / risk, then we split the existing cluster for that day / risk into two, with one cluster containing newly updated encounter messages, and another containing the rest of the messages.
We do not claim that this clustering algorithm is optimal, that the input format to the neural network is optimal, or that this messaging protocol is optimal.
There exists an interesting trade-oﬀ between privacy and risk precision which we hope will be explored in future work.
H. Transmission
Agent-Agent Transmission All encounters are assumed to be at a distance equal or less than two meters.
Transmission events can only take place between infectious and susceptible individuals.
Although the ABM models all encounters taking place between individuals, a susceptible individual is only considered exposed (i.e. the model considers that an eﬀective contact has taken place) when the encounter lasts a minimum of 15 minutes and bernoulli distribution with probability determined via Equation 2 gives 1.
The likelihood of viral transmission during a given eﬀective contact is proportional to the time duration of the encounter, and depends broadly on characteristics of both the infected and susceptible individual.
We use the transmission model as explained in [23].
Thus, encounters taking place in certain locations (i.e. senior residencies and households) are also inherently more prone to result in a transmission event, all other factors being equal.
This is modeled via B n .
At the same time, characteristics aﬀecting infectiousness (of the infected individual) include progression of the disease (i.e. eﬀective viral load) ( EV L ), whether (and to what extent) they are symptomatic ( A s ).
Susceptibility of the exposed individual depends notably on age ( S a ).
Environment-Agent Transmission Please note, that the probability of environmental transmission is not supported by any published study, and we consider 0 environmental transmissions in the experiments in this paper.
However, ABM can be conﬁgured to simulate environmental transmissions.
Empirical estimates of such transmissions stands at 10% as per [23], therefore, we consider a transmission model that models environmental infections in pre-pandemic scenario.
Given the lack of such estimation in post-lockdown scenario, we do not consider any such transmissions.
We model these transmissions by considering a linearly time-decaying probability of location being contaminated which is triggered by the presence of an infectious individual.
Initial magnitude of contamination is dependent on the agent’s current phase of the disease.
Further, the duration of such contamination is informed from experimental study [65], which lists surfaces and the duration for which virus survives on them.
We consider an experimental environment transmission model that estimates probability of infecting agent as proportional to (a) contamination strength of the location, and (b) susceptibility of an agent.
The proportionality factor being the environmental transmission control knob which lets us model the disease spread as per the observed data.
I. Symptoms
COVID-19 and other disease symptoms.
Symptoms in the model include:
Fever, Chills, Gastroenteritis (Gastro), Diarrhea, Nausea/vomiting, Sneezing, Cough, Fatigue, Hard time waking up, Headache, Confused, Loss of consciousness, Runny nose, Sore throat, Chest pain, Trouble breathing, Loss of taste/smell, and Aches, as shown in Figure I.18.
We represent symptoms as a set of diseases that are sampled on each day.
[Figure]
Sneezing
[Figure]
In each of the following sections we detail the probability of a given symptom in each of 5 phases of the disease.
I.1. Fever, Chills
Disease/Phase
Fever
Chills
COVIDIncubation 0% 0%
Onset 20% 80%
Plateau 80% 50%
Post-plateau0% 0% Post-plateau0% 0%
Flu
First day 70% 0%
Main 70% 0%
Last day 30% 0%
I.2. Gastroenteritis, Diarrhea, Nausea / Vomiting
If innoculum> 0 .
6 then the probability of this person having Gastrologic symptoms on any day is ( innoculum − 0 . 15) / 4, and if on any day they have
Gastrologic symptoms, then for each following day they will also have this type of symptom.
[Table]
I.3. Fatigue, Hard Time Waking Up, Unusual, Headache, Confused, Lost Consciousness
Let A represent Age/200, C represent Carefulness/2, and let I represent
Inoculum.
Let F represent A+(I*0.6)-C. Let HW represent the symptom,
Hard time waking up, and LC represent the symptom, Lost consciousness.
The probability of a person having fatigue as a symptom is given in the ﬁrst column, while the probabilities of having the other symptoms given that this person has fatigue are given in the columns following the Fatigue column.
In order to have unusual symptoms, the person must also be over 75 years of age; that is, the probabilities in the table given for unusual symptoms are the probabilities of having unusual symptoms given this person is overyears of age and is experiencing fatigue.
[Table]
I.4. Trouble Breathing, Sneezing, Cough, Runny Nose, Sore Throat, and Severe Chest Pain
Let I represent Inoculum, and let C represent Carefulness/4.
Let TB represent the symptom trouble breathing, ST represent the symptom sore throat, and SCP represent Severe chest pain.
The probability of having trouble breathing as a symptom is given in the
TB column for the corresponding COVID-19 phases.
For an individual to experience severe chest pain (SCP), the individual must also be extremely sick; that is, the probabilities given for having SCP as a symtpom are the probabilities of having SCP given an individual has trouble breathing and is extremely sick.
I.5. Light/Moderate/Heavy Trouble Breathing
The probability of having light/moderate/heavy trouble breathing is given by the probability of an individual having light/moderate/severe symptoms of COVID-19 and the probability of having trouble breathing as a symptom.
I.6. Loss of Taste
The probability of having loss of taste as a symptom of COVID-19 is 25% during the onset phase, 35% during the plateau phase, and 0% for all other phases of COVID-19.
I.7. Aches
Aches are not caused by COVID-19 in this simulator, but are caused by the ﬂu.
The probabilities of having aches on the ﬁrst and last days of the ﬂu are 30% and 80% respectively, while the probability of having aches for all other days with the ﬂu is 50%.
J. Scaling Analysis
We wish to verify that the dynamics of the simulator at the population sizes we model are representative of larger populations.
Because of the computational demands of an agent-based model, particularly with messaging between agents, it is more eﬃcient to model smaller populations, as long as the dynamics remain reprsentative.
As shown in Figure J.19, we ﬁnd that population sizes above 2k to be representative of the dynamics of larger populations across a range of metrics.
We thus ensure all experiments are run with populations of 2k and over.
K. Cost-Eﬀectiveness Calculations K.1. DALYs
Disability-Adjusted Life Years (DALYs) are a summary measure of the public health burden associated to a speciﬁc cause’s premature mortality and morbidity.
To calculate DALYs, we individually compute the years of life lost due to premature mortality (YLLs)[66] for agents that died during the simulation, as well as the years of life lost due to disability (YLDs)[67] for agents that were infected and symptomatic.
Disability weights (DW) are taken from the 2017 Global Burden of Disease Study[68]: they represent health preferences such that a DW of 0 is perfect health and a DW of 1 is equivalent to death.
Hence, the higher the amount of DALYs, the worse health outcomes are.
DALYs are calculated without discounting or age-weighting, following the WHO methodology[69].
[Table]
For each agent i ,
[Equation]
DALY
[Equation]
where:
L i := Life expectancy of agent i
DW s :=
Disability weight of state s l i,s :=
duration of state s for agent
i
i died := (cid:40)if agent i died during the simulation otherwise
K.2. TPL
TPL Temporary Productivity Loss (TPL) is the loss in productivity due to absenteeism from work.
To calculate TPL, we extract from the simulator the number of work hours that agents aged 25 to 65 years had to forego due to quarantine, taking care of a dependent (supervision), or illness to the point of being unable to work.
We then multiply this quantity of foregone work hours by the 2019 average hourly wage in Montreal[70] to obtain total
TPL.
We follow the methodology for calculating TPL presented in [44].
Aggregated across all agents,
[Equation]
where: n a 25:65 := number of hours foregone due to a w average := average hourly wage for Montreal workforce
[Table]
K.3. Assumptions and Limitations
Disability weights Covid-19 is a novel disease, and there are no published disability weights for diﬀerent levels of severity of the disease.
Therefore, we use similar conditions as proxies for diﬀerent health states of the agents.
Agent hospitalization status is used as a proxy for actual health status, and can be divided into three categories: agents that are symptomatic and not hospitalized, agents that are hospitalized but not in critical are, and agents that are in critical care.
The disability weights, as well as their equivalent causes in the GBD 2017 study can be found in Table K.
Full trajectory Due to the computational strain of the message-passing within the simulations, observing the full trajectory under binary contact tracing and feature-based contact tracing is currently unfeasible.
Future work will consider longer simulations that reach a post-pandemic steady state.
PPL In addition to Temporary Productivity Loss (TPL) due to absenteeism from work, cost-beneﬁt analyses following the Human Capital Model (HCA) [44] typically include a Permanent Productivity Loss (PPL) due to premature mortality component.
When longer simulations become possible, future work will take into account PPL.
Delaying vs Preventing To properly evaluate the impact of diﬀerent tracing strategies on the socio-economic burden of a disease, it is important to evaluate whether the proposed strategies avert DALYs, or simply delay them[71].
Such evaluations require a longer trajectory to arrive at a postpandemic steady state, which we will consider in future work.
Sensitivity Analysis
The focus of this paper is not the cost-beneﬁt analysis of the outcomes of the simulator, but rather the simulator itself, a sensitivity analysis of the cost-beneﬁt results has been relegated to future work.
[Table]