Abstract
Background:
The value of frequent, rapid testing to reduce community transmission of SARS-CoV-2 is poorly understood.
Objective:
To define performance standards and predict the clinical, epidemiologic, and economic outcomes of nationwide, home-based antigen testing.
Design:
A simple compartmental epidemic model that estimated viral transmission, portrayed disease progression, and forecast resource use, with and without testing.
Data Sources:
Parameter values and ranges as informed by Centers for Disease Control and Prevention guidance and published literature.
Target Population:
U.S. population.
Time Horizon:
60 days.
Perspective:
Societal; costs included testing, inpatient care, and lost workdays.
Intervention:
Home-based SARS-CoV-2 antigen testing.
Outcome Measures:
Cumulative infections and deaths, number of persons isolated and hospitalized, and total costs.
Results of Base-Case Analysis:
Without a testing intervention, the model anticipates 11.6 million infections, 119 000 deaths, and $10.1 billion in costs ($6.5 billion in inpatient care and $3.5 billion in lost productivity) over a 60-day horizon. Weekly availability of testing would avert 2.8 million infections and 15 700 deaths, increasing costs by $22.3 billion. Lower inpatient outlays ($5.9 billion) would partially offset additional testing expenditures ($12.5 billion) and workdays lost ($14.0 billion), yielding incremental cost-effectiveness ratios of $7890 per infection averted and $1 430 000 per death averted.
Results of Sensitivity Analysis:
Outcome estimates vary widely under different behavioral assumptions and testing frequencies. However, key findings persist across all scenarios, with large reductions in infections, mortality, and hospitalizations. Costs per death averted are roughly an order of magnitude lower than commonly accepted willingness-to-pay values per statistical life saved ($5 to $17 million).
Limitations:
Analysis was restricted to at-home testing. There are uncertainties concerning test performance.
Conclusion:
High-frequency home testing for SARS-CoV-2 with an inexpensive, imperfect test could contribute to pandemic control at justifiable cost and warrants consideration as part of a national containment strategy.
Primary Funding Source:
National Institutes of Health.
Previous research has identified social distancing (masks, de-densification, lockdowns), large-scale diagnostic testing, and vaccination as essential elements of a coordinated plan to contain the COVID-19 pandemic. One strategy receiving less formal attention to date is the high-frequency use of low-cost, rapid, home-based antigen testing for SARS-CoV-2 combined with self-enforced isolation for those who have a positive result (1–5). Advocates of this approach point to several potential advantages: prevention by isolation, focusing on “infectiousness” rather than “infection,” and reducing strain on laboratories that must conduct viral polymerase chain reaction (PCR)–based diagnostic testing (6–8). Critics argue that such an approach risks poor uptake and adherence; frequent false-negative results, leading to unfounded reassurance; and frequent false-positive findings, resulting in needless isolation and lost work productivity (9).
We sought to formalize this discussion by capturing both the promise of and concerns about home-based antigen testing in the structure of a mathematical model. We aim to identify the circumstances that would have to prevail—about the accuracy of antigen testing, the cost of both initial and confirmatory tests, lost workdays arising from isolation (based on both true- and false-positive results) and disease, and individual behavior—for such an intervention to be of clinical, economic, epidemiologic, and policy interest.
Methods
Study Design
We adapted a simple compartmental epidemic model (Supplement Figure, available at Annals.org) to capture the essential elements of a U.S. population–wide, home-based testing program to detect, isolate, and contain the people with SARS-CoV-2 infection who are most likely to transmit the virus to others. This strategy acknowledges that this testing will be negative in a proportion of people with positive PCR results, especially those in the recovered phase, who have high cycle thresholds and may no longer harbor replication-competent virus. Features we sought to portray included the epidemiology of infection; the natural history of COVID-19 illness; the behavioral response to test availability, test results, and isolation; and the financial consequences of testing, hospitalization, and lost workdays arising from illness, isolation, and (potentially incorrect) test findings. A spreadsheet implementation of the model permitted us to vary critical input data parameters and to examine how different test performance attributes (such as frequency, sensitivity, specificity, cost), behavioral responses (such as acceptance of antigen testing, willingness to self-isolate, propensity to abandon isolation), and epidemiologic scenarios would translate into both health outcomes (such as tests administered, true- and false-positive results, new infections, person-days requiring isolation, hospitalizations, and deaths) and economic performance (such as testing costs, inpatient costs, lost productivity, and cost-effectiveness). Given the rapid spread of infection and the speed of new advances, we adopted a short, 60-day planning horizon.
Input data (Supplement Table 1, available at Annals.org) were obtained from published sources; whenever possible, we adhered to planning scenarios and data guidance for modelers provided by the Office of the Assistant Secretary for Preparedness and Response and the Centers for Disease Control and Prevention (10–46). Because our aim was to identify the circumstances under which widespread, rapid home testing might warrant inclusion as part of a national containment strategy, we deliberately “tipped the scales” to portray the intervention in a less favorable light, choosing false-positive and false-negative rates at the upper end of the plausible range, inflating costs, and exaggerating levels of nonadherence with recommended protocols. To provide context for our estimates of the incremental cost per death averted, we followed the guidance of the Office of the Assistant Secretary for Planning and Evaluation and applied the value of a statistical life, a benchmark of the societal willingness to pay for reductions in mortality risks (46). Here again, we erred on the side of conservatism, adopting the lower bound value ($5.3 million per statistical life saved) from the recommended range.
Compartmental Model
We made 2 notable changes to the traditional susceptible-exposed-infected-recovered compartmental modeling framework (Supplement Figure). First, we separated the single “infected” compartment into 4 subcompartments: asymptomatic, mild/moderate, severe, and critical. This permitted us to capture more fully the natural history, epidemiology, and resource use associated with progressive COVID-19. Second, we introduced a parallel set of states to distinguish between epidemiologically “susceptible or infectious” individuals and persons no longer susceptible or infectious due to isolation or death. In “susceptible or infectious” compartments, we assumed that individuals interact in ways that permit infectious contact and transmission of SARS-CoV-2; in “isolation” compartments, no transmission was possible.
We provided 2 pathways into the isolation compartments. First, individuals with severe or critical infection were isolated by virtue of their advanced COVID-19 symptoms and hospitalization. A user-defined variable also permitted some proportion of individuals with mild/moderate symptoms to self-quarantine. Testing offered a second avenue to isolation.
We considered different background epidemic severities. We assumed a baseline effective reproduction number (Rt) of 1.3 based on public health interventions and explored values ranging from 0.9 to 2.1 in sensitivity analysis. As detailed in the Supplement (available at Annals.org), we assumed that 60% of infections would produce symptoms, that 10% of persons with symptoms would advance to severe disease, and that 15% of those with severe disease would advance to critical illness. Base occupancy times for persons progressing to more advanced illness were 3, 10, 6, and 4 days in the exposed, asymptomatic, mild/moderate, and severe disease states, respectively. We used mortalities of 1%, 5%, and 20% in the mild/moderate, severe, and critical disease states.
Performance of Testing
Regular opportunities to be tested for SARS-CoV-2 contagion were offered to persons in the active uninfected, exposed, asymptomatic, and mild/moderate compartments. We reviewed the data on the performance of antigen testing and found sensitivity estimates ranging from a low of 41.2% to a high of 100% and specificities from 97% to 99.9% (7, 29–36). We then focused our estimates on cases when testing was accompanied by either a low cycle threshold (high titers of virus) or the ability to isolate replication-competent virus, because these cases represented people in the most transmissible stage of COVID-19 and excluded those in the recovery phase who might still have positive PCR results (17, 37, 38). Although some studies described antigen sensitivity as exceeding 90% under these circumstances, we deliberately lowered our base-case sensitivity assumption to 80% to reflect less-than-optimal testing characteristics in the home setting. We did not consider how repeat testing might increase this sensitivity. Similar motivations led us to adopt a lower-bound value of 95% for test specificity, and we explored even lower values (90%) in sensitivity analysis (7). We examined weekly testing in the base-case analysis but considered frequencies ranging from daily to once every 15 days in sensitivity analysis.
To capture both the costs and the potential delays of confirmatory testing, we assumed that the shipment of testing kits to households would include a swab for obtaining a PCR test, to be self-collected at home and sent to a central laboratory in the event of a positive rapid test result. Individuals whose initially false-positive result had led them to adhere to recommended isolation protocols were assumed to return to the active population after a 3-day delay (47). This assumption ignored the small possibility of repeatedly false-positive confirmatory test findings.
Finally, we assumed that in the absence of disease progression, successful isolation of infected persons for 10 days would lead to recovery, and that uninfected persons would return to the active population.
Behavioral Response
To account for concerns about individual willingness to adhere to testing and isolation protocols, we adopted a spectrum of highly pessimistic assumptions regarding the behavioral response to the testing intervention (Table 1). Specifically, we assumed that a large proportion of the population (50% in the base case; 75% in the worst case) would elect not to use the tests provided to them. Furthermore, we assumed that even among the minority who did perform the test, a large proportion (50% in the base case; 75% in the worst case) would elect to ignore a positive test finding and refuse to self-isolate. This means that in the worst case, only 6.25% (25% of 25%) of persons would be assumed to adhere to the recommended testing and isolation protocols. Finally, we assumed that even among that small proportion of persons who might elect to self-isolate, 20% each day would abandon isolation and return to the active population, against recommended guidance.
Table 1.
Testing Behavior Scenario | |||
---|---|---|---|
Best Case | Base Case | Worst Case | |
% of individuals who make use of tests offered | 80 | 50 | 25 |
% of individuals who self-isolate after a positive test result | 75 | 50 | 25 |
Daily % of individuals who abandon isolation | 5 | 20 | 33 |
Economic Outcomes
We assigned 3 categories of cost: testing, inpatient, and lost productivity. Here again, we chose values that would deliberately bias the analysis against the intervention: exaggerating the costs of testing and lost workdays, and understating the costs of hospitalization. Testing costs included both the initial testing kit ($5 [range, $1 to $10]) and the confirmatory test ($20 [range, $5 to $50]). The current price of a single over-the-counter antigen test for SARS-CoV-2 is roughly $25 or more, a figure that includes a substantial markup for both the manufacturer and the retailer and that offers no quantity discount (39). Experience with other infectious diseases (malaria, for example) suggests that high-volume, paper strip–based antigen tests can be obtained at wholesale prices as low as $0.20 per unit for government-based purchase orders of the magnitude being considered here (40). The initial test costs were assigned on an “intention-to-treat” basis: Mailing test kits to someone’s home incurred the cost, regardless of whether the individual chose to use the test. Inpatient costs were assigned per day with severe illness ($1000) or critical illness ($2500) (16, 41–43). Every day spent in the hospital or in isolation—regardless of whether isolation was the result of a true- or a false-positive finding—was treated as a day of lost productivity, which we assigned a cost of $180, based on the daily per capita gross domestic product (44, 45).
Valuing Outcomes
To assist in the interpretation of our estimates of the incremental cost per death averted, we adhered to the guidance of the Office of the Assistant Secretary for Planning and Evaluation and applied the value of a statistical life (46). This is a widely accepted means of formalizing the tradeoff that individuals make in exchanging wealth for small reductions in the risk for death over a defined period (48, 49). By way of illustration, a willingness to pay $500 for a 1 in 10 000 reduction in the risk for dying in the current year translates into a statistical life value of $5 million ($500/0.0001). The value of a statistical life imposes fewer constraints on individual preferences than the quality-adjusted life-year, another widely accepted benchmark measure of the societal willingness to pay for health benefits. This makes the value of a statistical life a particularly pragmatic and attractive choice for an analysis such as ours, where the outcome of interest is a broad measure of deaths averted across the U.S. population.
Estimates of the value of a statistical life vary widely. In the spirit of biasing the analysis against the testing intervention, we adopted the lower bound ($5.3 million) from the recommended range (central estimate, $11.3 million; upper bound, $17.2 million) as our base value (Supplement).
Role of the Funding Source
This work was supported by the National Institute on Drug Abuse and the National Institute of Allergy and Infectious Diseases, National Institutes of Health. The funding sources had no role in the design, analysis, or interpretation of the study; the writing of the manuscript; or the decision to submit the manuscript for publication.
Results
Base Case
In the absence of a testing intervention, the model anticipates 11.6 million infections, 119 000 deaths, and $10.1 billion in total costs ($6.5 billion in inpatient costs and $3.5 billion in lost productivity) over a 60-day horizon (Table 2). Weekly home testing under base-case assumptions could reduce infections to 8.8 million and deaths to 103 000. Lower inpatient costs ($5.9 billion) would partially offset additional outlays for testing ($12.5 billion) and greater lost workdays ($14.0 billion). Although testing and isolation would increase the total number of workdays lost (from 19.6 million to 78.0 million days), individuals could expect to be isolated for 0.17 day (1.1 days under the most pessimistic possible data assumptions) owing to false-positive findings. Compared with the status quo, the testing intervention would produce a cost per infection averted of $7890 and a cost per death averted of $1.43 million. Applying the lowest available estimate from the recommended range for the benchmark value of a statistical life ($5.3 million), this suggests that the intervention would be exceptionally good value.
Table 2.
No Testing | Testing: Behavioral Worst Case | Testing: Behavioral Base Case | Testing: Behavioral Best Case | |
---|---|---|---|---|
Total infections (millions) | 14.9 | 14.0 | 11.0 | 5.5 |
Total Deaths (thousands) | 125 | 120 | 106 | 77 |
Workdays Lost (total; millions) | 21.4 | 33.1 | 77.4 | 197.6 |
Workdays Lost (per person) | 0.07 | 0.11 | 0.26 | 0.66 |
False-Positive Isolation Days (total; millions) | 0 | 10.5 | 52.4 | 172.6 |
False-Positive Isolation Days (per person) | 0 | 0.04 | 0.17 | 0.58 |
Total Costs ($ billions) | 10.35 | 23.88 | 31.86 | 53.37 |
Infections averted (thousands) * | 902 | 3,969 | 9,474 | |
Deaths averted * | 4,205 | 18,945 | 48,130 | |
Incremental costs ($ billions) * | 13.52 | 21.50 | 43.01 | |
ICER† ($/Infections averted) | 15,000 | 5,400 | 4,500 | |
ICER† ($/Deaths averted) | 3,200,000 | 1,100,000 | 890,000 | |
Cumulative Cost Breakdown ($ billions) | ||||
Base test | - | 11.40 | 11.39 | 11.35 |
Confirmation test | - | 0.16 | 0.61 | 1.41 |
Inpatient (severe disease) | 3.04 | 2.94 | 2.59 | 1.88 |
Inpatient (critical disease) | 3.45 | 3.42 | 3.34 | 3.16 |
Productivity | 3.86 | 5.95 | 13.93 | 35.56 |
Total | 10.35 | 23.88 | 31.86 | 53.37 |
Per Person Cost Breakdown ($) | ||||
Base test | - | 38.00 | 37.97 | 37.84 |
Confirmation test | - | 0.52 | 2.03 | 4.70 |
Inpatient (severe disease) | 10.15 | 9.81 | 8.63 | 6.26 |
Inpatient (critical disease) | 11.49 | 11.41 | 11.13 | 10.54 |
Productivity | 12.87 | 19.85 | 46.43 | 118.55 |
Total | 35.41 | 79.59 | 106.19 | 177.89 |
Compared to “No Test” scenario
ICER = incremental cost-effectiveness ratio
Sensitivity to Behavioral Factors
Test acceptance and adherence to isolation protocols would greatly influence the magnitude of all estimated outcomes (Figure 1 and Table 2). Under best-case behavioral assumptions, the testing intervention alone could begin to contain the epidemic; under less favorable behavioral scenarios, the effect of testing on the spread of infection would be less pronounced. However, even under the worst-case behavioral scenario (25% participation; 25% isolation of persons with positive results; 33% rate of daily abandonment), more than 600 000 infections could be prevented over 60 days.
Costs too would vary widely depending on behavioral assumptions (Table 2). Greater adherence to program protocols would invariably produce greater overall costs, comprising higher costs of testing (initial and confirmatory) and higher costs of lost workdays, partially offset by lower costs of inpatient care. But cost-effectiveness ratios (such as costs per infection and per death averted, compared with no testing) would behave more stably. Even under the most pessimistic behavioral assumptions, the incremental cost per death averted ($4.1 million) would remain below the most stringent recommended benchmark value of a statistical life ($5.3 million).
Sensitivity to Test Frequency
Figure 2 shows the critical role played by test frequency. A test that elicits a poor behavioral response will still prevent a large number of infections, if offered with sufficient frequency. Even under worst-case behavioral assumptions, for example, cumulative infections could be cut more than 30% via the daily offer of testing.
Sensitivity to Rt
Under all scenarios and testing assumptions, greater epidemic severity (as measured by the reproductive number, Rt) would produce more infections, more deaths, and higher costs. It would also engender more favorable results for the testing intervention, as measured by infections averted, deaths averted, and cost-effectiveness ratios (Table 3).
Table 3.
Total Cost ($) | Total infections | Total deaths | Incremental cost ($)* | Infections averted* | Deaths averted* | ICER† ($/Infections averted) | ICER† ($/Deaths averted) | |
---|---|---|---|---|---|---|---|---|
Rt = 0.9 | ||||||||
No Test | 6,423,408,708 | 3,856,513 | 65,380 | |||||
Base Case | 28,265,635,826 | 3,084,105 | 60,406 | 21,842,227,118 | 772,408 | 4,974 | 28,278 | 4,390,952 |
Rt = 1.3 | ||||||||
No Test | 10,351,589,938 | 14,930,587 | 124,637 | |||||
Base Case | 31,856,176,855 | 10,961,643 | 105,692 | 21,504,586,917 | 3,968,943 | 18,945 | 5,418 | 1,135,079 |
Rt = 1.7 | ||||||||
No Test | 21,094,569,851 | 53,002,937 | 286,342 | |||||
Base Case | 41,662,061,788 | 38,203,510 | 227,026 | 20,567,491,937 | 14,799,426 | 59,317 | 1,390 | 346,740 |
Rt = 2.1 | ||||||||
No Test | 44,402,386,231 | 131,798,989 | 638,154 | |||||
Base Case | 64,250,192,494 | 103,668,349 | 504,479 | 19,847,806,264 | 28,130,639 | 133,675 | 706 | 148,478 |
Compared to “No Test” scenario
ICER = incremental cost-effectiveness ratio
Sensitivity to the Costs of Testing
With initial and confirmatory test costs set to their base-case values ($5 and $20), the intervention had an incremental cost-effectiveness ratio of $1.4 million per death averted. Setting testing costs to the lowest ($1 and $10) and highest ($10 and $50) values in our estimated range yielded incremental cost-effectiveness ratios per death averted of $802 000 and $2.2 million, respectively.
Size and Composition of the Isolated Population
Regardless of the testing protocol, large numbers of people will be required to isolate. In the absence of testing, we expect to observe 20 million person-days spent in isolation, all of them attributable to symptoms and hospitalization (Table 2). Notably, this means that the average individual can expect to lose 0.06 workday to isolation. Under base-case assumptions, testing will increase total days spent in isolation to 78 million (or 0.24 day per person). However, only 20% of those days will be attributable to hospitalization, the remainder being the result of testing (10% true-positive results and 70% false-positive results). Under the most pessimistic possible assumptions (that is, best-case adherence to isolation protocols coupled with worst-case test performance), the average member of the population can expect to spend 1.1 days in unnecessary isolation for a false-positive result over the course of the 60-day horizon.
Discussion
Our model-based analysis found that a nationwide program of frequent, antigen-based home testing and self-isolation could greatly reduce total infections and mortality at a justifiable cost. We arrived at this conclusion by using methods of cost-effectiveness analysis and assumptions that were specifically chosen to portray all aspects of the intervention—performance of antigen testing, the behavioral response of individuals to testing and isolation protocols, and societal willingness to pay to avert untimely deaths—in an unfavorable light. Use of more middle-of-the-road data assumptions would only serve to strengthen our policy conclusion. Our bottom-line message is: Do not let the perfect be the enemy of the good; even a highly imperfect home-based testing program could confer enormous benefit.
With an analysis that uses the entire U.S. population as its target, the numerical results reported here are staggering in their magnitude, sensitive to small changes in the input values, and difficult to digest. Although we have attempted to address this by reporting values on a per capita basis and by conducting extensive sensitivity analysis, we nevertheless urge the reader to focus less on our numerical point estimates and more on the remarkable robustness of our qualitative policy finding—namely, that a nationwide rollout of frequent, home-based testing and self-isolation is justified on both epidemiologic and economic grounds.
Our study has limitations. In addition to the many limitations of any model-based evaluation, our analysis does not account for the potential benefits of regular, inexpensive, rapid testing in other, more targeted settings. Schools, factories, air travel, concerts, recreational sports, large family celebrations, and places of worship might choose to make such testing a prerequisite of participation. Because a small number of people with COVID-19 account for a high proportion of secondary cases, such a strategy would remove some of the most contagious individuals from crowded settings, eliminating case clusters and preventing super-spreader events (50).
In addition our analysis also does not stratify the at-risk population by age. Although our assumed mortality rates from severe and critical disease (5% and 15%) mirror those of people aged 65 years or older—and we generally expect older individuals to be the ones to experience severe and critical illness from COVID-19—we are unable to report either the years of life that might be gained or the age-adjusted deaths and productivity losses that might be averted via a program of mass testing (51–53). This is an important next step that could further address distributive equity concerns regarding mass testing. Some observers have questioned the ability of frequent, rapid, antigen testing to reduce transmission, raising several concerns (9). These include the lower sensitivity of antigen testing compared with PCR testing (increasing the risk that infectious people will remain in public owing to the erroneous belief that they are not infectious); the high number of false-positive test results, leading to unnecessary isolation; poor adherence to the recommended testing and isolation recommendations; and the massive expense if testing is broadly applied. We acknowledge these concerns and, wherever possible, we have tried to give them voice by adopting modeling assumptions and input data values that tip the scales against nationwide antigen-based home testing.
The strategy of frequent rapid testing to reduce SARS-CoV-2 transmission and decrease COVD-19 cases began generating widespread attention in the popular press and on social media many months ago (54, 55). Implementation on a population-based level, however, has been limited to date. A program of massive, rapid antigen testing in Slovakia on consecutive weekends appears to have contributed to a reduction in COVID-19 cases beyond what would have been expected through standard infection control measures (56). Initiation of a community-based testing pilot in Liverpool was associated with a decline in cases, but it is not clear whether this was the result of testing or increased infection-control measures and other restrictions (57). Current obstacles to broader use of these tests in the United States include the requirement for a physician’s order for certain tests, limited availability, and continued high prices. The U.S. Food and Drug Administration has granted emergency use authorization to 3 highly effective vaccines for COVID-19 and plans to review other future candidates. However, most public health officials agree the availability of these vaccines does not obviate the need for increased testing, at least in the short term (58). There are several reasons why broad testing will remain important during the vaccine rollout. These include limited vaccine supply, vaccine hesitancy and refusal, and the emergence of SARS-CoV-2 variants that are more contagious and potentially less susceptible to existing vaccines (59, 60). All of these factors strongly suggest that COVID-19 as a health threat will be with us for the foreseeable future, even if the vaccines help bring case numbers under control.
A recent study by Du and colleagues (61) concluded that screening once a week, coupled with strict enforcement of a 2-week isolation period for confirmed cases, could be justified on cost-effectiveness grounds. Although both that study and ours adopt similar underlying modeling frameworks and cost structures, we believe that we make a unique contribution by directly confronting the principal policy objections to mass screening: imperfect test performance and poor adherence to the testing and isolation strategies. By building a high degree of skepticism into our modeling of the performance of the intervention, we increase the robustness of the finding that this strategy can prevent transmission and save lives at a reasonable cost. As a result, we believe that our approach strengthens the findings first reported by Du and colleagues.
Our findings also confirm those of other investigators that home-based rapid testing can have a dramatic effect in reducing transmission and COVID-19 mortality at a justifiable cost (1, 32), with testing frequency a key component of why this broad testing strategy is so effective. Factors currently in play that could worsen the pandemic—a slow start to the rollout of widespread vaccination, “pandemic fatigue” regarding social distancing and mask wearing, and emergence of variants that are more easily transmissible—only underscore the importance of frequent rapid testing as a strategy to reduce the number of new cases.
Supplementary Material
Funding:
By grant R37DA015612 from the National Institute on Drug Abuse (Dr. Paltiel) and grant R01AI042006 from the National Institute of Allergy and Infectious Diseases (Dr. Sax), National Institutes of Health.
Footnotes
Disclosures: Disclosures can be viewed at www.acponline.org/authors/icmje/ConflictOfInterestForms.do?msNum=M21-0510.
Reproducible Research Statement: Study protocol, statistical code, and data set: Available in the Supplement (available at Annals.org).
Previous Posting: This manuscript was posted as a preprint on medRxiv on 8 February 2021. doi:10.1101/2021.02.06.21251270
Contributor Information
A. David Paltiel, Public Health Modeling Unit, Yale School of Public Health, New Haven, Connecticut.
Amy Zheng, Harvard Medical School, Boston, Massachusetts.
Paul E. Sax, Brigham and Women’s Hospital and Harvard Medical School, Boston, Massachusetts.
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