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Contact tracing is a fundamental intervention in public health. When systematically applied, it enables the breaking of chains of transmission, which is important for controlling COVID19 transmission. In theoretically perfect contact tracing, all new cases should occur among quarantined individuals, and an epidemic should vanish. However, the availability of resources influences the capacity to perform contact tracing. Therefore, it is necessary to estimate its effectiveness threshold. We propose that this effectiveness threshold may be indirectly estimated using the ratio of COVID19 cases arising from quarantined highrisk contacts, where higher ratios indicate better control and, under a threshold, contact tracing may fail and other restrictions become necessary.
This study assessed the ratio of COVID19 cases in highrisk contacts quarantined through contact tracing and its potential use as an ancillary pandemic control indicator.
We built a 6compartment epidemiological model to emulate COVID19 infection flow according to publicly available data from Portuguese authorities. Our model extended the usual susceptibleexposedinfectedrecovered model by adding a compartment Q with individuals in mandated quarantine who could develop infection or return to the susceptible pool and a compartment P with individuals protected from infection because of vaccination. To model infection dynamics, data on SARSCoV2 infection risk (IR), time until infection, and vaccine efficacy were collected. Estimation was needed for vaccine data to reflect the timing of inoculation and booster efficacy. In total, 2 simulations were built: one adjusting for the presence and absence of variants or vaccination and another maximizing IR in quarantined individuals. Both simulations were based on a set of 100 unique parameterizations. The daily ratio of infected cases arising from highrisk contacts (
An inverse relationship was found between the
We demonstrated the impact of applying an effectiveness threshold for contact tracing on decisionmaking. Although only theoretical thresholds could be provided, their relationship with the number of confirmed cases and the prediction of pandemic phases shows the role as an indirect indicator of the efficacy of contact tracing.
Contact tracing is a fundamental activity in public health and is the process of identifying, triaging, and monitoring individuals exposed to a communicable disease to prevent secondary transmission [
In Portugal, contact tracing and the imposing of quarantine measures are tasks specific to public health units [
Such conditions were more likely to be verified in moments with a lower number of confirmed cases (ie, corresponding to the troughs of the epidemic curve) [
The effectiveness of contact tracing as a mechanism for breaking transmission chains occurs because of the quarantine of contacts of confirmed infected cases. Should these contacts develop the disease, they will not transmit it to other community members [
Hence, the effectiveness threshold of contact tracing is the point at which its utility as a health intervention that includes quarantine in controlling and breaking transmission chains is defined. The proportions of confirmed cases from quarantined individuals above this threshold could be indicative of effective contact tracing, and the pandemic combat strategy could rely mainly on this intervention. In contrast, proportions below the threshold could indicate that contact tracing is not effective and that further interventions may be necessary to stop transmission.
The proportions of confirmed cases from quarantined individuals lower than the effectiveness threshold may have occurred at different stages throughout the pandemic, mainly in periods of case surges, situations in which there was a need to implement more restrictive measures (namely, general or selective confinements). Imposing confinements has been demonstrated to affect pandemic control regarding case numbers, hospital admissions, and deaths because of COVID19 when implemented at least 14 days before the peak of a case surge [
Despite the public availability of several global databases on COVID19, especially concerning the number of new cases, deaths, tests performed, and vaccination data, there is no information on the number of infected cases coming from individuals identified as highrisk contacts through contact tracing. Furthermore, data on quarantined individuals are scarce. Portuguese data were until recently an exception in that the DirectorateGeneral of Health (DGS) reported in a daily bulletin on COVID19 the number of highrisk contacts identified, defined as individuals in quarantine by mandate from health authorities [
Infection dynamics and the effectiveness of contact tracing are not only influenced by the vaccinated population (for which the DSSG also kept curated data). They may also be affected by the prevalence of different SARSCoV2 variants at different points in time. The European Centre for Disease Prevention and Control keeps a data repository on the prevalence of different variants in Europe reported through The European Surveillance System in an openaccess database [
The aim of this study was to identify the ratio of COVID19 cases that occurred in individuals in quarantine mandated by health authorities and the potential use of this ratio as a proof of concept of an indicator for assessing pandemic control in parallel with other established criteria such as the transmissibility index (Rt) and incidence.
In this study, we collected data on COVID19 for Portugal and built an expanded structure to a susceptibleexposedinfected recovered (SEIR) compartmental model to emulate the pandemic. All compartment data came from collected data except for those of protected individuals. The purpose of the model was to estimate the daily number of quarantined individuals who became infected and the daily number of susceptible and vaccinated individuals who were quarantined as data regarding those values were lacking.
We input different values for each known strain of SARSCoV2 (infection risk [IR] and time until infection and until the end of quarantine) and immunity from vaccination and ran the model for 690 days. Consequently, we estimated the daily ratio of cases arising from daily quarantined individuals (
According to the epidemic case curve, we defined 3 pandemic phases in the Portuguese data: interpandemic phase, alert phase, and pandemic phase. As a proof of concept, we estimated the best hypothetical cutoff for our
The compartment transition dynamics were in accordance with the following equations (note that only equations 1 and 3 were estimated, corresponding to compartments S and E, for which real data were not available. All other compartment data were directly collected from official reports [
Data sources and model inputs and outputs.
Parameter  Value  Source 
Susceptible (S)  Base case: 10 million  Estimated 
Protected through vaccination (P)  P (Protection) × V  Estimated 
Quarantined (Q)  Base case: 0  DSSG^{a} [ 
Exposed not traced (E)  Base case: 0  Estimated 
Infected (I)  Base case: 2  DSSG [ 
Recovered from infection (R)  Base case: 0  DSSG [ 
Vaccinated (V)  Base case: 0  DSSG [ 
Inoculation vaccine efficacy (Ve_{0})  10%60% (uniform distribution)  Hall et al [ 
Maximum vaccine efficacy (Ve_{max})  Ve_{0}95% (uniform distribution)  Hall et al [ 
Waned vaccine efficacy (Ve_{waned})  (Ve_{0} + Ve_{max})/2  Estimated 
Inoculation booster dose efficacy (Be_{0})  Ve_{waned}80%  Estimated 
Maximum booster dose efficacy (Be_{max})  Be_{0}95%  Estimated 
Waned booster dose efficacy (Be_{waned})  (Be_{0} + Be_{max})/2  Estimated 
Time until maximum efficacy (∆te_{max})  15 d  Hall et al [ 
Time until waned efficacy (∆te_{waned})  180 d  Hall et al [ 
Variant prevalence  Base case: other=100%; alpha, beta, gamma, delta, and omicron=0%  ECDC^{b} [ 
Maximum time until γ (∆tγ_{max})  214 days (uniform distribution)  DGS^{c} [ 
Average time until γ (∆tγ_{mean})  2∆tγ_{max} days (uniform distribution)  DGS [ 
Maximum time until κ and ς (∆tκς_{max})  214 days (uniform distribution)  Estimated 
Average time until κ and ς (∆tκς_{mean})  2∆tκς_{max} days (uniform distribution)  Estimated 
IR^{d} simulation A  10%50% (uniform distribution)  ECDC [ 
IR simulation B  0.1%2.5%  Calibration 
Quarantined infected  γQ  Estimated 
Ratio of cases from quarantined ( 
γQ/(γQ + ιE)  Estimated 
^{a}DSSG: Data Science for Social Good.
^{b}ECDC: European Centre for Disease Prevention and Control.
^{c}DGS: DirectorateGeneral of Health.
^{d}IR: infection risk.
The sources of data are described in full in
Data were inserted into an epidemiological model based on compartmental models already applied to COVID19 and other epidemiological contexts [
Moreover, 2 transitions have not been reported in the reviewed literature on expanded SEIR models, namely, the direct transition between compartments S and I and the transition between vaccinated or protected compartments and quarantine or exposure compartments. As our model needed to comprise these transitions, several differences from already published models had to be introduced. To keep with the base SEIR structure, compartment E included only individuals who would progress to compartment I without previous contact tracing. The model was extended with compartments Q and P.
Compartment S (susceptible) is the initial compartment of the model (ie, the starting point for all individuals). From compartment S, individuals can progress to compartments P, Q, and E.
Compartment P (protected) refers to the group of individuals who are immune to SARSCoV2 infection because of vaccination. Although the main effect of vaccines is protection against severe disease and not protection against infection [
Compartment Q (quarantined or exposed with tracing) includes all highrisk contacts of SARSCoV2–infected individuals for whom quarantine was mandated by a public health authority according to norms and guidelines [
Compartment E (exposed without tracing) includes exposed individuals who will develop COVID19 and have not been traced by health authorities. Compartment I consists of all the confirmed cases of SARSCoV2 infection.
Compartment R (recovered) includes all individuals who recovered from infection. Compartment R is the terminal compartment as the model is not circular. This is because an individual previously infected with COVID19 cannot be considered a highrisk contact before 180 days have passed from the date of infection or when not presenting symptoms suggesting SARSCoV2 infection [
The daily ratio of cases arising from individuals in quarantine mandated by health authorities was given by the quotient of the daily transitions between compartments Q and I (represented by γ) and the daily total confirmed cases.
Compartmental model without subcompartments (A) and with subcompartments (B). E: individuals exposed to SARSCoV2 who will develop COVID19 and have not been traced by health authorities; I: infected individuals; P: subset of susceptible individuals protected from SARSCoV2 infection through vaccination; Q: quarantined individuals; Q0, Q1, Q2,..., Qn: subcompartments of the quarantine compartment (each number represents the number of days since exposure); R: recovered individuals; S: susceptible individuals.
The compartmental model was run for 690 days (from March 2, 2020, to January 20, 2022). The data collected populated compartments Q, I, and R daily. In the absence of data for compartment S, an initial value of 10 million individuals was assumed, which is a frequently used approximation for the Portuguese population. This value ensured that the model was not circular and obviated the need to consider the possibility of reinfection. Data on compartments P and E were calculated, and expected values for other parameters were defined within each simulation run (
Regarding daily transitions between compartments and subcompartments, the model was built in 3 steps: transitions from subcompartments inside Q (step 1), SEI transitions (step 2), and SQ and PQ transitions (step 3). Despite the existence of data for all compartments and some model parameters, owing to a lack of data in each of these steps regarding the actual number of individuals that transition in each iteration of the model, some parameters were input by defining expected values for the parameters that govern those transitions. Only those parameters were estimated (
In step 1, QS, QP, and QI transitions are the sums of the number of individuals who each day (represented by each subcompartment) transition to compartments S, P, or I, respectively. Each day, transitions from each subcompartment are governed by the probabilities of staying in compartment Q (transition to the following subcompartment Q_{i+1}), making the transition γ to compartment I, or returning to compartment S or P (transitions κ and ς). The probabilities of transition of each subcompartment were defined according to the IR of a highrisk contact of a confirmed COVID19 case and to the mean and maximum periods an individual remains exposed until the person either develops the infection or is considered susceptible again. Any of these 5 parameters can vary with the viral variant to which the individuals were exposed and with the quarantine period. Therefore, in the context of the described model, the probability of transitioning from any subcompartment to compartment S, P, or I is given by the sum of the product of the transition probability of each specific variant and the number of individuals exposed to that variant. Step 1 is concluded after calculating the ratio of new daily cases of infection that came from compartment Q (resulting from transition γ) to the total number of new daily cases (
Steps 2 and 3 aim to keep the model closed (a necessary condition for using compartmental models), allowing the model to simulate the following day. The number of individuals that follow the ε and ι transitions in step 2 is given by the difference between the number of new daily infections and the number of individuals in transition γ. The φ and χ transitions in step 3 are calculated as the difference between the number of exposed individuals (highrisk contacts) on the following simulation day and the number of individuals inside compartment Q after steps 1 and 2. The value of this transition corresponds to the number of individuals inside subcompartment Q_{0} at the beginning of step 1 on the following simulation day.
Finally, compartment P was defined according to 5 parameters, namely, vaccine efficacy after inoculation, maximum efficacy, and efficacy after a waning period as well as the elapsed time between vaccination and maximum efficacy and between maximum efficacy and waned efficacy. This compartment was calculated as the product of the probability of each vaccinated individual being protected from SARSCoV2 infection and the number of vaccinated individuals.
To operationalize the proposed model, several assumptions were made, namely regarding the transitions involving compartments Q and P.
A Markov chain Monte Carlo simulation defined the transitions between the Q subcompartments. Consequently, to compute the transitions between subcompartments, a series of steps were taken (
Regarding compartment P, and particularly vaccine efficacy, a constant rate was assumed between two events: (1) efficacy after inoculation and maximum efficacy and (2) maximum efficacy and waned efficacy. Hence, vaccinated individuals were probabilistically placed in compartment P each day according to the time elapsed since inoculation. After defining new values for booster efficacy, the same procedure was replicated for individuals with booster doses. Individuals in compartment P could transition to compartment Q, but their IR was set at 0. Individuals in compartments S and P would transition to compartment Q according to the ratio between those compartments.
Owing to the inherent variability in the timing of contact tracing and imposing quarantine, it is impossible to accurately define certain parameters in the model for each variant of SARSCoV2 and each vaccine. Therefore, we decided to simulate a set of different parameterizations for these parameters. For each variant, the maximum time elapsed for the transition to compartments I and S was assumed to range from 2 to 14 days. The mean time for these transitions ranged from 2 days to the maximum time previously set for each variant. IR was set to range from 10% to 50% [
Regarding vaccination, we defined an efficacy between 10% and 60% after inoculation and a maximum efficacy of up to 95% [
To calculate the different transition probabilities, we assumed that everyone who transitions to compartment Q has a predetermined probability of becoming infected, called infection risk (IR). IR is specific to each variant.
For each variant included, 2 different Poisson distributions were applied to the individuals who would become infected and those who would return to compartments S and P. Each distribution had a maximum and mean time until transition.
From the combined transition probabilities of each distribution and according to the IR, a transition table was constructed, including the pooled transition probabilities from each subcompartment to the following subcompartment, to compartments S and P, and to compartment I.
A Monte Carlo simulation was then applied to this transition table. Therefore, on each simulation day, an individual in a subcompartment of the Q compartment transitions probabilistically to the following subcompartment, to compartments S and P, or to compartment I.
In total, 2 different simulations were performed, approaching the objective in 2 different ways.
Simulation A aimed to be as approximate as possible to what we could expect in real life by accounting for the potential influence of variants and vaccination. However, it assumed that the risk of infection was fixed for all variants at the beginning of the simulation. For this simulation, 100 parameterizations were established, and these were quadrupled to reflect the presence and absence of the influence of variants and vaccination. Each parameterization was iterated 100 times, totaling 40,000 model iterations with an estimated run time of 8 hours and 20 minutes.
Simulation B ignored the presence of different variants (we assumed that all cases came from the variant “others”) and estimated the maximum value of IR for which the model kept the
Main characteristics of the implemented simulations.

Simulation A  Simulation B 
Used information on vaccines  Yes  Yes 
Used information on variants  Yes  No 
Number of different parameterizations, n  400 (100 × 4)  200 (100 × 2) 
Ceiling value for the ratio of cases ( 
No  Yes 
Parameters included fixed infection risk  Yes  Yes 
Number of iterations per parameterization, n  100  Approximately 1000 
Total model iterations, n  40,000  200,000 
Estimated run time  8 h and 20 min  8 h and 20 min 
The parameters’ characteristics were aggregated in means and medians and described for each simulation. For each simulation, the distribution of the ratio of daily cases resulting from the γ transition (
Since other COVID19 measures for lockdown were presented in a 14day moving average, we applied a 14day moving average to the daily
Local maxima for 14day data on confirmed infected cases were computed, and local minima immediately before and after the computed local maxima were maintained. By using an adaptation of the method used by VázquezSeisdedos et al [
It was assumed that, by definition, a contact tracing effectiveness threshold would have to be surpassed during a pandemic phase. Specifically, the value of the effectiveness threshold would have to occur in an alert phase, defined as the critical period between the wave’s inflection point and up until 14 days before the day when the peak number of cases was registered. In addition, it was assumed that contact tracing as a main strategy for pandemic combat was reset after the end of each pandemic phase (ie, local minimum after case peak) and the model entered an interpandemic phase. For this reason, an effectiveness threshold was calculated considering the interpandemic and alert phases. In other words, we included the entire period analyzed except for periods between the end of an alert phase and the end of each pandemic phase. The data between a local minimum and the next inflection point (interpandemic phase) were considered as controls for determining the threshold. As there was a more considerable period of control data than the period in the alert phase, we oversampled the test data. This classification into interpandemic, alert, and pandemic phases was loosely based on the World Health Organization pandemic phases for influenza [
This threshold estimation from the
Effectiveness threshold values were estimated for each simulation using a receiver operating characteristic model applying the Youden method. The sensitivity, specificity, and positive predictive value (PPV) were calculated for the theoretical effectiveness threshold. Sensitivity represented the proportion of days in the alert phase in which the
Finally, we implemented a 2part sensitivity analysis. First, we estimated a hypothetical threshold for each of the different parameterizations used (both in simulations A and B) and ran a multiple linear regression with the estimated thresholds as the dependent variable. We included 10% increases in IR of a highrisk contact of a confirmed COVID19 case and mean and maximum periods during which an individual remains quarantined until they either develop the infection or are considered susceptible in days, for each strain, and 10% increases in maximum and waned efficacies for complete vaccination and booster doses as independent variables. The second part consisted of fixing the maximum and minimum values for parameters included in simulation B (all but those related to virus variants) and estimating both the thresholds obtained and the Spearman correlation coefficients with the results of simulation B. We presented the range of thresholds and correlations obtained.
The adapted compartmental model was constructed, the simulations were run, and the data were analyzed using the statistical computing and graphics software R (version 4.0.2; R Foundation for Statistical Computing) in the integrated development environment RStudio (version 2022.07.1+554; Posit). Packages
The
Simplifying the presented model to a singleday transition (all quarantined individuals must progress to being infected or return to being susceptible or protected), if on any given day 100 individuals were in compartment Q and the total number of new cases was 20 and…:
...the infection risk (IR) was defined as 10%, the q estimate would be 0.5 (10% × 100/20). In other words, we would expect half the cases to come from quarantined individuals.
...the IR was defined as 20%, the q estimate would be 1.0 (20% × 100/20). In other words, we would expect all cases to come from quarantined individuals.
...the IR was defined as 30%, the q estimate would be 1.5 (30% × 100/20). In other words, we would expect more cases from quarantined individuals than the total number of new cases.
Therefore, the
Our study used publicly available aggregated secondary data with no characteristics that allowed for individual identification. Therefore, the research team considered that there were no relevant data protection and privacy issues to report.
The characteristics of the parameterizations used in the model and for each simulation are described in
Descriptive analysis of simulation parameters.

Simulation A  Simulation B  



Δtγ_{max}^{a} (d), median (IQR)  8 (6)  8 (6) 

Δtγ_{mean}^{b} (d), median (IQR)  4 (4)  4 (4) 

∆tκς_{max}^{c} (d), median (IQR)  8 (7)  9 (7) 

∆tκς_{mean}^{d} (d), median (IQR)  4 (4)  4 (4) 

IR^{e} (%), mean (SD)  28.9 (13.6)  0.8 (0.5) 



Ve_{0}^{f}  33.7 (16.3)  33.7 (16.3) 

Ve_{max}^{g}  64.8 (21.8)  64.8 (21.8) 

Ve_{waned}^{h}  49.2 (14.9)  49.2 (14.9) 



Be_{0}^{i}  65.1 (14.0)  65.1 (14.0) 

Be_{max}^{j}  80.1 (12.3)  80.1 (12.3) 

Be_{waned}^{k}  49.2 (14.9)  49.2 (14.9) 
^{a}Δtγ_{max}: maximum time until γ.
^{b}Δtγ_{mean}: average time until γ.
^{c}∆tκς_{max}: maximum time until κ and ς.
^{d}∆tκς_{mean}: average time until κ and ς.
^{e}IR: infection risk.
^{f}Ve_{0}: vaccine efficacy.
^{g}Ve_{max}: maximum vaccine efficacy.
^{h}Ve_{waned}: waned vaccine efficacy.
^{i}Be_{0}: booster dose efficacy.
^{j}Be_{max}: maximum booster dose efficacy.
^{k}Be_{waned}: waned booster dose efficacy.
The daily value of the
An inverse relationship between the values of the
Despite the different procedures underlying the 2 simulations (A and B), the maximum
The analysis of all iteration results according to the presence or absence of different variants and vaccination’s protective effect (
Results of simulation A in comparison with the daily total number of new COVID19 cases at scale.
Results of simulation A separated according to inclusion or exclusion of vaccines or SARSCoV2 variants.
Results of simulation B in comparison with the daily total number of new COVID19 cases at scale.
Results of simulation B separated according to the inclusion or exclusion of vaccines.
For simulation A, we observed that the value of the
In Portugal, population lockdowns were imposed during all pandemic phases except the fifth one [
Daily COVID19 case number evolution with identification of wave peaks and periods included in the effectiveness threshold for contact tracing. Vertical dotted lines represent the imposing of population lockdown measures by Portuguese authorities in each pandemic phase during the period of analysis.
Testing periods for theoretical effectiveness threshold.
Days  Phase  Classification 
117  Interpandemic  Negative 
1826  Alert  Positive 
82219  Interpandemic  Negative 
220250  Alert  Positive 
302309  Interpandemic  Negative 
310321  Alert  Positive 
400469  Interpandemic  Negative 
470497  Alert  Positive 
588659  Interpandemic  Negative 
660676  Alert  Positive 
Regarding the correlation and threshold analyses (
Multiple linear regression model parameter coefficients for threshold determination.
Variant and model parameter  Coefficient (95% CI)  



Δtγ_{max}^{a}  0.02 (−0.04 to 0.08)  .51  

Δtγ_{mean}^{b}  −0.08 (−0.16 to 0.01)  .07  

∆tκς_{max}^{c}  −0.10 (−0.16 to −0.05)  <.001  

∆tκς_{mean}^{d}  −0.08 (−0.15 to −0.02)  .02  

IR^{e}  0.52 (0.41 to 0.64)  <.001  



Δtγ_{max}  −0.03 (−0.09 to 0.03)  .35  

∆tγ_{mean}  −0.04 (−0.12 to 0.03)  .25  

∆tκς_{max}  0.03 (−0.02 to 0.09)  .26  

∆tκς_{mean}  −0.08 (−0.15 to 0.00)  .04  

IR  0.05 (−0.07 to 0.18)  .41  



Δtγ_{max}  0 (−0.07 to 0.06)  .99  

Δtγ_{mean}  0.01 (−0.07 to 0.09)  .79  

∆tκς_{max}  0.03 (−0.04 to 0.10)  .41  

∆tκς_{mean}  0.01 (−0.07 to 0.09)  .78  

IR  0.01 (−0.11 to 0.13)  .84  



Δtγ_{max}  −0.04 (−0.11 to 0.04)  .31  

Δtγ_{mean}  0.07 (−0.01 to 0.16)  .08  

∆tκς_{max}  0.03 (−0.02 to 0.09)  .23  

∆tκς_{mean}  0.01 (−0.06 to 0.09)  .75  

IR  0.15 (0.03 to 0.28)  .02  



Δtγ_{max}  0.04 (−0.02 to 0.10)  .19  

Δtγ_{mean}  −0.04 (−0.12 to 0.04)  .32  

∆tκς_{max}  0.02 (−0.04 to 0.09)  .47  

∆tκς_{mean}  −0.08 (−0.16 to 0.00)  .048  

IR  0.11 (−0.04 to 0.26)  .16  



Δtγ_{max}  0.02 (−0.05 to 0.08)  .59  

Δtγ_{mean}  0.00 (−0.07 to 0.06)  .89  

∆tκς_{max}  −0.04 (−0.09 to 0.01)  .16  

∆tκς_{mean}  0.08 (0.01 to 0.15)  .02  

IR  0.08 (−0.03 to 0.20)  .16  
Ve_{0}^{f}  0.02 (−0.04 to 0.09)  .48  
Ve_{max}^{g}  0.01 (−0.05 to 0.06)  .85  
Ve_{waned}^{h}  N/A^{i}  N/A  
Be_{0}^{j}  −0.11 (−0.20 to −0.02)  .02  
Be_{max}^{k}  0.01 (−0.09 to 0.11)  .84  
Be_{waned}^{l}  N/A  N/A 
^{a}Δtγ_{max}: maximum time until γ.
^{b}Δtγ_{mean}: average time until γ.
^{c}∆tκς_{max}: maximum time until κ and ς.
^{d}∆tκς_{mean}: average time until κ and ς.
^{e}IR: infection risk.
^{f}Ve_{0}: vaccine efficacy.
^{g}Ve_{max}: maximum vaccine efficacy.
^{h}Ve_{waned}: waned vaccine efficacy.
^{i}N/A: not applicable; coefficients for waned efficacy were not calculated as this parameter was input in the model as the mean of the inoculation and maximum efficacies (
^{j}Be_{0}: booster dose efficacy.
^{k}Be_{max}: maximum booster dose efficacy.
^{l}Be_{waned}: waned booster dose efficacy.
Correlation and threshold estimates for fixed parameters.
Variable and value  Correlation^{a} (range)  Threshold (range)  



2  0.870.88  0.00313.372  

14  0.880.99  0.00211.057  
∆ 


2  0.920.98  0.0372.965  

14  0.890.95  0.0022.696  
∆ 


2  0.910.99  0.01613.372  

14  0.880.99  0.00211.278  
∆ 


2  0.950.99  0.0307.385  

14  0.880.99  0.0023.137  



0.001  0.870.99  0.0020.031  

0.500  0.880.99  1.24813.372  



0.10  0.870.99  0.00213.372  

0.60  0.900.96  0.0460.448  



0.10  0.890.99  0.00213.372  

0.95  0.870.96  0.00210.744  



0.10  0.870.99  0.00213.372  

0.80  0.890.99  0.00213.144  



0.10  0.890.99  0.00213.372  

0.95  0.890.99  0.0308.352 
^{a}Compared with simulation B results.
^{b}Δtγ_{max}: maximum time until γ.
^{c}Δtγ_{mean}: average time until γ.
^{d}∆tκς_{max}: maximum time until κ and ς.
^{e}∆tκς_{mean}: average time until κ and ς.
^{f}IR: infection risk.
^{g}Ve_{0}: vaccine efficacy.
^{h}Ve_{max}: maximum vaccine efficacy.
^{i}Be_{0}: booster dose efficacy.
^{j}Be_{max}: maximum booster dose efficacy.
In this study, we developed a compartmental model to describe the relationship among the official data on COVID19 for Portugal, namely regarding the DGS definition of quarantined individuals. The main aim of this study was to estimate the daily ratio of cases that occurred in individuals in imposed quarantine through contact tracing (
To the best of our knowledge, this study is the first attempt to use COVID19 contact tracing data to define the infection dynamics of SARSCoV2. The main reason for this might be that the DGS was, as far as the authors are aware, the only national health authority with public data regarding individuals under quarantine systematically imposed by health authorities.
We obtained
To demonstrate the impact of applying this theoretical
In addition, a sensitivity analysis of the parameters used to model compartment Q transitions showed that only the IR for any strain and the vaccine efficacy at booster dose inoculation did significantly affect the
Therefore, our study should be regarded as a demonstration of the application of an effectiveness threshold for contact tracing as a measure of pandemic control. Consequently, the
We deem it necessary to draw attention to the importance of open data and data sharing as a way to catalyze research and accelerate innovation and development that shortens the time between the detection of potentially epidemic pathogens and the development of appropriate containment and mitigation measures [
The main limitation of our study arises from implementing a compartmental model and simulation methods to estimate transitions between compartments, potentially leading to results that only partially reflect reality as it occurred at each moment of the period analyzed. Furthermore, it is not possible to exclude the influence that the implementation of confinement measures or other measures implemented during each pandemic wave may have had on the proportion of new cases from exposed individuals. However, the impact of considering different parameterizations of variants or vaccination efficacy, demonstrated by the visual overlap between the results of the different simulations as well as the conducted sensitivity analysis, strengthens the potential robustness of the results and the model.
In addition, the parameters chosen include other limitations as they are based on data collected in studies that were run under controlled conditions [
Few studies have compared the risk of infection per virus variant throughout the quarantine period. Therefore, although we acknowledge that some variants might have been more transmissible than others, it did not allow us to determine whether this variable translated only to a higher IR or a lower time until the transition between exposed and infected (or any other hypotheses, for that matter). To mitigate the impact of this limitation, different values for parameters (which assumed that all variants could have variable IR and time to infection) had the potential to depict variant differences concerning the degree of contagion and infection. The residual impact of different variants, except for their respective IRs, on the simulation results may attenuate this limitation.
It is not possible to exclude the hypothesis that the effect of implementation of confinement measures (or other measures implemented during each pandemic phase) on the ratio of new cases from quarantined individuals may have led the model to present
Finally, given the format in which the data were collected, the analysis could be conducted only at the national level. This lack of data granularity demanded an implicit assumption that any change in the number of cases or individuals under quarantine imposed by health authorities affects national, regional, and local levels simultaneously and proportionally. Greater data detail, both geographical and sectorial, would potentially allow for determining effectiveness thresholds—and the consequent implementation of pandemic containment measures—at those differentiated levels, with a more apparent effortbenefit ratio that could be better understood and better accepted by the population. This would most likely result in higher population adherence to containment measures and finertuned control of both the sanitary and socioeconomic impacts of the pandemic.
As we have stated, this work is a starting point for using data on highrisk contacts of COVID19 cases to define transmission dynamics of SARSCoV2, which could lead to further studies addressing the aforementioned limitations and discussions of alternative hypotheses and perspectives left unexplored in this study. An immediate suggestion could be the case of a scenario in which the variation in IR for compartment Q occurs within each iteration. Such dynamic IR should reflect the implementation of pandemic combat measures that have been proven to change the IR (namely, largescale testing [
Another scenario to tackle in future work—perhaps also resorting to simulation methods—would be to investigate what would happen should contact tracing come to a halt and what would be the impact of fully transferring resources allocated to this public health task to other activities regarding epidemiological dynamics and the pandemic’s impact on a country’s sanitary and economic dimensions.
As mentioned previously, the theoretical definition of contact tracing effectiveness thresholds that would allow for the relief of restrictive measures is beyond the scope of this work. Nonetheless, it might be equally valuable to define thresholds for restrictive measure relief as it could be helpful for communication by public officials during pandemic scenarios as well as to balance the scales between the implicit healtheconomy dichotomy that so often arose in narratives with global reach. There is also a relevant economic component to this work as it is intertwined with the efficiency of these pandemic control measures and their direct and indirect impacts on countries’ health and economy. This work might be very helpful in drawing lessons that better enable us to adapt our society to events such as pandemics.
Finally, and from a health economics standpoint, it would be interesting to understand how the population perceives contact tracing as an instrument used by health authorities to contain pandemics and what the perception would be, for example, in terms of adherence to pandemic control measures should contact tracing cease.
Our work provides important information for policy and decision makers, namely in terms of epidemiology and pandemic/crisis management, as well as for public health professionals. It also constitutes a relevant source of information and an objective acknowledgment of the importance of contact tracing, which was widely used worldwide in all stages of the COVID19 pandemic.
Concretely, this study, although using secondary data, allowed us to (1) provide a proof concept for using estimates of an effectiveness threshold for contact tracing as a primary measure of pandemic containment and (2) consider the potential use of this effectiveness threshold as a decision variable for imposing more restrictive measures, namely, lockdowns, along with indicators that were used to assess the pandemic situation (such as the transmissibility indexes and case incidence). Our results are consistent with this last possibility and, notwithstanding the presented limitations, reveal a path that, we hope, deserves further exploration.
DirectorateGeneral of Health
Data Science for Social Good
infection risk
positive predictive value
susceptibleexposedinfectedrecovered
This paper was based on the scientific efforts of the research team developed upon the execution of the research protocol that won the first edition of the Scholarship Award Amélia Leitão, promoted by the Portuguese National Association of Public Health Doctors and Pfizer Laboratories. This work would have been impossible without the generous contribution of Data Science for Social Good in compiling and curating the data reported in PDF format by the DirectorateGeneral of Health. This is our way to give back to the community as part of the benefits we received through them.
The data sets generated and analyzed during this study are available from the corresponding author upon reasonable request. The code we developed and on which these simulations are based is available on a GitHub repository [
None declared.