Estimating protection afforded by prior infection in preventing reinfection: applying the test-negative study design

Abstract The COVID-19 pandemic has highlighted the need to use infection testing databases to rapidly estimate effectiveness of prior infection in preventing reinfection (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document}) by novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variants. Mathematical modeling was used to demonstrate a theoretical foundation for applicability of the test-negative, case–control study design to derive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document}. Apart from the very early phase of an epidemic, the difference between the test-negative estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document} and true value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document} was minimal and became negligible as the epidemic progressed. The test-negative design provided robust estimation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document} and its waning. Assuming that only 25% of prior infections are documented, misclassification of prior infection status underestimated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document}, but the underestimate was considerable only when > 50% of the population was ever infected. Misclassification of latent infection, misclassification of current active infection, and scale-up of vaccination all resulted in negligible bias in estimated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document}. The test-negative design was applied to national-level testing data in Qatar to estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document} for SARS-CoV-2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P{E}_S$\end{document} against SARS-CoV-2 Alpha and Beta variants was estimated at 97.0% (95% CI, 93.6-98.6) and 85.5% (95% CI, 82.4-88.1), respectively. These estimates were validated using a cohort study design. The test-negative design offers a feasible, robust method to estimate protection from prior infection in preventing reinfection.


Introduction
Estimating effectiveness of prior infection in preventing reinfection (PE S ) is essential to understanding the epidemiology of a given infection.2][3][4][5][6][7][8][9] However, there are challenges in estimating PE S using conventional epidemiologic study designs.Such designs require extensive, complete electronic health records to be feasible.Vaccination scale-up makes it difficult to disentangle immunity induced by prior infection from that induced by vaccination.
1][12] Effects of this bias increase with increased cumulative infection exposure in the population. 13Emergence of the Omicron 14 (B.1.1.529)variant and its subsequent subvariants emphasized the need to estimate PE S rapidly once a new variant/subvariant emerges.
Here, we demonstrate a robust, practical method to estimate PE S using a test-negative, case-control study design.This is, to our knowledge, the first use of this method to estimate PE S .While it has been used to study vaccine effectiveness, [15][16][17][18][19][20][21][22] it does not appear to have been used to estimate PE S , perhaps because of a perception that it is not applicable, as most prior and current infections are undocumented, unlike vaccinations, which are typically documented and tracked in health systems.We also provide an application of this method by estimating PE S for SARS-CoV-2 infection in Qatar, at a time when the Alpha 14 (B.1.1.7)and Beta 14 (B.1.351)][23][24][25][26] This article includes two components.The first is a parsimonious mathematical modeling component whose purpose is to motivate the test-negative design and to demonstrate that theoretically it can be applied to provide credible estimates for PE S despite specific sources of bias.This modeling exercise is not intended to provide a simulation of a specific empirical study or discuss all sources of potential bias, but to provide a theoretical foundation of the applicability of such design to estimate PE S .The second component is a real-world application of the testnegative design to actual routine data to generate estimates for PE S .This specific application was conducted because there are already published estimates for PE S using a cohort study design applied to the same data, population, and duration of study. 4herefore, the cohort study design provides a validation for the test-negative design, as both the cohort and test-negative designs yielded the same results when applied to the same data source.

Test-negative case-control study design
1][32] Matching is done to control for differences in the risk of exposure to the infection. 21,22,33Vaccine effectiveness is then derived as 1 minus the ratio of the odds of vaccination in subjects testing positive to the odds of vaccination in subjects testing negative. 15,16[29][30][31][32]

Mathematical modeling and simulation of the test-negative design
Mathematical modeling was used to demonstrate a theoretical foundation for the applicability of the test-negative, case-control study design for deriving effectiveness of prior infection in preventing reinfection (PE S ), that is, the proportional reduction in susceptibility to infection among those with prior infection versus those without. 2Modeling was also used to investigate effects of biases on estimated PE S .While this demonstration was done for SARS-CoV-2 infection, the approach is generic and should be broadly applicable to a range of infections.Moreover, while this demonstration was done for any SARS-CoV-2 infection, regardless of symptoms, the same approach can be applied to other outcomes such as symptomatic infection, asymptomatic infection, severe or critical COVID-19, 34 or COVID-19 death, 35 as long as these outcomes are defined as specific subsets of the broad anyinfection outcome or its direct disease progression.
Several models were devised to simulate SARS-CoV-2 infection transmission in the population and to investigate applicability of the test-negative design.7][38][39][40][41][42] To keep only the essential details for the investigations of this study, the models were parsimonious and not structured by age, nor by infection type and severity.The instantaneous prevalence at each time point, for each population compartment, was used in the analyses of these models.
The first model was the classic susceptible-exposed-infectiousrecovered (SEIR) model, but extended to allow for reinfections (baseline model; Figure 1A).This model was used to demonstrate applicability of the test-negative design and to investigate sources of bias.In this model and its analysis all controls were either susceptible or recovered individuals, and all cases were either infected or reinfected individuals.
Building on previous modeling studies of vaccine effectiveness and its waning, 13,[43][44][45][46][47] the second model was an extension of the baseline model to incorporate scale-up of vaccination in the population (vaccination model; Figure 1B).This model was used to investigate whether vaccination could affect applicability of this method to estimate PE S .Vaccine effectiveness (VE S ) was defined as the proportional reduction in susceptibility to infection among those vaccinated versus those unvaccinated. 40,41VE S was set at 75%, a representative value for the range of coronavirus disease 2019 (COVID-19) vaccines available during times in which incidence was due to pre-Omicron variants. 21,33,48,49][50][51][52] The third model was also an extension of the baseline model, incorporating gradual (linear) waning in protection offered by prior infection against reinfection (waning-of-immunity model; Figure 1C).Time after recovery from infection was modeled as an aging process whereby the recovered population transitions from one population compartment to the next with the average duration spent in each compartment being one month.Each 1-month recovered-population compartment had a set PE S value.PE S was modeled to decline linearly month by month.Accordingly, the recovered population is tracked month by month after recovery to allow for test-negative-study estimation of waning of natural immunity, as is described in the literature for waning of vaccine immunity after the second or booster doses. 25,52,53hese models consisted of coupled nonlinear differential equations that stratified the population into compartments (groups) based on infection status (infected, reinfected, or uninfected) and vaccination status (vaccinated, unvaccinated).Susceptible individuals (vaccinated or unvaccinated) were assumed at risk of acquiring the infection at a force of infection that varied throughout the epidemic due to variation in the contact rate.Recovered individuals (vaccinated or unvaccinated) were also assumed at risk of acquiring the infection, but the force of infection was reduced by the effect of PE S .

Effectiveness of prior infection against reinfection and impact of bias
Applying the test-negative, case-control study design, PE S was derived as 1 minus the ratio of the odds of prior infection in subjects testing positive (such as by polymerase chain reaction [PCR] testing) to the odds of prior infection in subjects testing negative for the infection.The 2-by-2 table used to derive the odds ratio is shown in Figure 2A, as expressed in terms of the baseline model's population variables.The mathematical expression for PE S is also shown in Figure 2A, assuming no form of bias.An underlying assumption is that those being tested are a specific fixed proportion (random sample) of all population variables; that is, the same sampling proportion is applied for each population compartment in the model.We also assumed that those latently infected (E compartment) are as diagnosable as those in acute infection (I compartment), given the wide application of the highly sensitive PCR testing for SARS-CoV-2 infection, and because of existence of large-scale routine testing in many countries, in addition to testing for symptomatic cases.A departure of the latter assumption has been investigated in a sensitivity analysis.

80%
Informed by evidence from existing studies [1][2][3][4][5][6][7][8][9] Proportion of prior infections that are undocumented g p 75% Informed by evidence from existing studies [10][11][12]38 Proportion of latent infections that are undocumented g E 75% Informed by evidence from existing studies [10][11][12]38 Proportion of current active infections that are undocumented g I 75% Informed by evidence from existing studies [10][11][12]38 Several forms of bias may affect estimation of PE S using the test-negative method. Thest critical is misclassification of prior infection status.A proportion g P of those previously infected may not have been diagnosed and may have been unaware of their infections.] Here, we assumed that 75% of prior infections are undocumented, that is, an ascertainment rate of only 25% (Table 1).This ascertainment rate was based on fitting epidemic models to national seroprevalence survey data in Qatar, 12,38,[56][57][58][59] and is consistent with the ascertainment rate estimated for the United States, also, using serological surveys. 10 Unlikaccine effectiveness studies, in which records are typically available to track vaccinations, [15][16][17][18][19][20][21][22]33 most persons with prior infection could be misclassified as persons with no prior infection.Similarly, most currently active infections may not be documented.The 2-by-2 table is thus modified for this bias along with the expression for PE S (Figure 2B).It was assumed that this bias affects both cases and controls similarly, a valid assumption considering that both cases and controls are seeking health care because of symptoms.This assumption is central to the testnegative design strategy.15,16,27,28,[30][31][32] A second source of bias is misclassification of latent infection status.A proportion g E of those with latent infections are asymptomatic, thereby remaining untested and undiagnosed.These cases would be misclassified as controls.The 2-by-2 table is thus modified to accommodate this bias along with the expression for PE S (Figure 2C).We assumed that g E = 75% (Table 1).We also assumed that this bias similarly affects those with and without prior infection.This is a valid assumption considering that both are seeking health care for the same reason, another assumption central to the test-negative design strategy.15,16,27,28,[30][31][32] A proportion g I of cases (current active infections) could be misclassified as controls, because of lack of testing or due to imperfect sensitivity of the testing method, thereby introducing bias.The 2-by-2 table is thus modified for this bias along with the expression for PE S (Figure 2D). We asumed that g I = 75% (Table 1).][32] Estimation of PE S may occur at a time when vaccination is being scaled up, as in the current COVID-19 pandemic.This could introduce bias as vaccination is another form of immune protection.Using the vaccination model, the 2-by-2 table is modified in the presence of vaccination along with the expression for PE S (Figure 2E).We assumed that vaccination is being linearly scaled up to reach the vaccine coverage attained in Qatar during the duration of the simulation.We also assumed that protection of natural immunity and of vaccine immunity act independently of each other, as suggested recently for the effect of hybrid immunity. 53Accordingly, protection of hybrid immunity of prior infection (PE S ) and vaccination (VE S ) combines as a multiplicative protection effect 53 -hybrid immunity of prior infection and vaccination is superior to that of either prior infection or vaccination separately. 53,54,60ince different forms of bias may act synergistically when present together, the impact of the above biases was also investigated by applying all of them together at the same time.

Sensitivity analyses
Four sensitivity analyses were conducted.In the first sensitivity analysis, presented analyses were repeated using the real-world, detailed reference mathematical model that was used to describe the epidemic and forecast its progression in Qatar, to inform policy decision-making (the Qatar model; Figure S1). 12,36,38This model stratified the population into compartments according to age group, infection status (uninfected, infected, reinfected), infection type (asymptomatic/mild, severe, and critical), COVID-19 disease type (severe or critical disease), and vaccination status (vaccinated, unvaccinated).38][39][40][41]61 Model fitting was used to estimate key epidemiologic indicators including the ascertainment rates among others.Detailed description of the model, its input data, and fitting are available in the references. 12,36,38he second sensitivity analysis investigated the representativeness of PE S as derived using the test-negative study design of the true PE S , over the full spectrum of possible PE S values.The third sensitivity analysis investigated whether the PE S estimate can vary by using incidence instead of instantaneous prevalence in deriving the estimate.The fourth sensitivity analysis investigated the impact on PE S of full misclassification bias of those latently infected.That is, none of those latently infected are being diagnosed; only those in acute infection are being diagnosed.

Real-world application: effectiveness of prior infection in preventing reinfection in Qatar
To validate the test-negative design, PE S was estimated in Qatar using national-level routine PCR testing data.Databases include all SARS-CoV-2-related data, with no missing information since pandemic onset, such as PCR tests and vaccinations.Only persons being PCR tested for clinical suspicion of infection due to symptoms between March 8 and April 21, 2021, were eligible for inclusion in this analysis.This study duration was chosen because there are existing estimates for PE S during this time but using a conventional, cohort study design. 4This allows validation of the estimate generated using the test-negative design.
Prior infection was defined as a PCR-confirmed infection ≥90 days before a new PCR-positive test. 2,6Individuals infected during the 90 days preceding the PCR test were thus excluded.Based on existing evidence [62][63][64] and viral genome sequencing, 3,21 a SARS-CoV-2 Alpha variant case was defined as an S-gene "target failure" case using the TaqPath COVID-19 Combo Kit (Thermo Fisher Scientific, Waltham, MA 65 ) applying the criterion of a PCR cycle threshold (Ct) value ≤ 30 for both the N and ORF1ab genes but a negative outcome for the S gene. 3,4,64With essentially only Beta and Alpha cases identified between March 8 and April 21, 2021, [21][22][23][24][25][26] a Beta case was proxied as the complement of the Alpha criterion, that is, any case with a Ct value ≤ 30 for the N, ORF1ab, and S genes.
Only the first PCR-positive test during the study was included for each case, and only the first PCR-negative test during the study was included for each control, per established protocol for the test-negative design. 21,22,25,33No Beta-positive cases were included as Alpha-negative controls, or vice versa.The negative controls in both the Alpha and Beta analyses were chosen from the same population of those who tested negative during the study.Alpha and Beta cases were exact-matched 1-to-1 to controls (PCR-negative persons) by sex, 10-year age group, nationality, and calendar week of PCR test.][59] This applied test-negative design, including these specific inclusion and exclusion criteria, was developed over a series of studies 17,21,22,25,52,66 to minimize effects of potential bias, such as retesting after a positive test to check for clearance of infection, or to control the effect of repeat testers. 25Extensive sensitivity and additional analyses were conducted in these prior studies to investigate effects of different kinds of potential bias in this design, including investigating different adjustments in the analysis, different approaches for matching, 67 different approaches for factoring prior infection in the analysis, and other different study inclusion and exclusion criteria. 17,21,22,25,52,66The applied test-negative design is an outcome of these analyses to optimize the design by minimizing different sources of bias in real-world data.The design was also validated using studies that used control groups to test for null effects, 22,25,52,68,69 and also validated using cohort study designs applied to the same population and that yielded findings similar to those of the test-negative design. 21,22,66,42,54 Sociodemographic characteristics of study samples were described using frequency distributions and measures of central tendency.The odds ratio, comparing odds of prior infection among cases versus controls, and its associated 95% confidence interval (CI) were derived using conditional logistic regression, that is, factoring matching in the study design.This analytical approach is done to minimize potential bias due to variation in epidemic phase 15,70 and other confounders. 12,42,57-59,71,72PE S and its associated 95% CI were calculated by applying the following equation:

PE S = 1−odds ratio of prior infection among cases versus controls
Statistical analyses were performed using STATA/SE, version 17.0. 73The study was approved by the Hamad Medical Corporation and Weill Cornell Medicine-Qatar institutional review boards with waiver of informed consent.The study was reported following the Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) guidelines.The STROBE checklist is found in Table S1.

Protection of prior infection using the test-negative design and impact of bias
Figure 3 shows simulated evolution of the SARS-CoV-2 epidemic in its 2 waves (Figure 3A), the proportion of the population ever infected (Figure 3B), and vaccine coverage (Figure 3C). Figure 4A shows the estimated PE S using the test-negative design (labeled as PE test-negative S ), by application of the expression in Figure 2A, compared with the true PE S (labeled as PE true S ), here assumed at 80% (Table 1).Apart from the very early phase of the epidemic, when the number of reinfections was minimal, the difference between PE test-negative S and PE true S was no more than several percentage points.The difference became negligible as the epidemic progressed.
Assuming that only 25% of prior infections are documented (Table 1), Figure 5A shows the impact of misclassification of prior infection, by application of the expression in Figure 2B.This form of bias resulted in underestimation of PE true S .When the proportion of the population ever infected was below 50% (Figure 3B), PE test-negative S was only few percentage points lower than that of PE true S .However, the underestimation increased to as much as 30 percentage points when the proportion of the population ever infected was approximately 75%.Therefore, PE test-negative S would provide only a lower bound for PE true S in situations where nearly everyone is infected, such as for inf luenza.
Misclassification of latent infection (Figure 5B), misclassification of current active infection (Figure 5C), and scale-up of vaccination (Figure 5D) all resulted in negligible bias in estimated PE test-negative S .Application of the above forms of bias at the same time suggested that there is no synergy when biases are combined (Figure S2).Applying the waning-of-immunity model, Figure 4B shows PE test-negative S versus PE true S , month by month after prior infection, assuming that there is a gradual linear waning in protection of prior infection against reinfection.This comparison was done after the second wave at day 600 after the virus introduction (Figure 3A).PE test-negative S provided a robust approximation of PE true S and its waning month by month.
Above analyses were repeated in the first sensitivity analysis that used the real-world Qatar model.The analysis confirmed the same findings as those of the main analysis using the par-simonious models (Figure S3).Impact of bias due to scale-up of vaccination was not investigated using the Qatar model, as this model's fitting already factors in the scale-up of vaccination in Qatar. 36he second sensitivity analysis showed that PE  S4).The third sensitivity analysis showed that the PE test-negative S estimate using incidence is similar to that using instantaneous prevalence (Figure S5).The fourth sensitivity analysis showed that

Application: effectiveness of prior infection in preventing reinfection in Qatar
Figure 6 presents a f lowchart describing the population selection process for estimating PE S in Qatar using the test-negative design.The median age of study subjects was 32-34 years, at least half were male, and they came from diverse countries (Table 2).Study samples were broadly representative of Qatar's demographic distributions. 42,74mong the 4645 Alpha cases (PCR-positive persons), 7 had a record of prior infection, compared with 232 among their matched controls (PCR-negative persons).PE S against Alpha was estimated at 97.0% (95% CI, 93.6-98.6).Among the 13 753 Beta cases, 124 had a record of prior infection, compared with 815 among their matched controls.PE S against Beta was estimated at 85.5% (95% CI, 82.4-88.1).
There were 239 discordant pairs and 4406 concordant pairs in the Alpha analysis and 925 discordant pairs and 12 828 concordant pairs in the Beta analysis.The analyses were conducted on large samples of paired cases and controls and should not be affected by bias due to small samples or sparse data. 75uring the study duration (March 8 to April 21, 2021), we conducted 2 earlier matched cohort studies to estimate PE S for Alpha and for Beta. 4 For Alpha, cohort-study estimates were 97.6% (95% CI, 95.7-98.7)and 96.4% (95% CI, 92.1-98.3). 4 For Beta, cohortstudy estimates were 92.3% (95% CI, 90.3-93.8)and 86.4% (95% CI, 82.5-89.5). 4

Power analysis
The above application for Alpha and Beta protections demonstrates an actual empirical application, but the number of cases may not be sufficient in other applications to provide a precise and meaningful estimate for PE S .Therefore, we conducted a power analysis to provide an estimate of the sample size necessary to apply this method using Power and Sample Size, version 3.1.2, 76following Dupont principles. 77ssuming the proportion of controls with prior infection at 25% and a high correlation between cases and controls of 0.5, 78 an estimated sample size of 71 individuals for each of cases and controls is needed to detect an odds ratio of 0.2, that is, assuming PE S of 80%, at 2-sided type I error probability of 5% and power of 80%.
Assuming an attrition of 80% due to exclusion for study ineligibility and an additional attrition of 5% from loss to matching, as informed by the above applications for Alpha and Beta protections, the required sample size would be 374 for each cases and controls.If PE S was 50% instead (an odds ratio of 0.5), the required sample size would be 1474 for each of cases and controls.

Discussion
This study's results show that the test-negative design can be used to generate rigorous estimates for protection afforded by prior infection against reinfection, even though most prior infections are undocumented.Estimates were robust despite several forms of potential bias, and even under rather extreme assumptions for these biases.The test-negative design was also applied to Qatar's routine PCR testing data, and results were validated by comparing test-negative estimates with those generated using conventional cohort study designs. 4pplication of the test-negative design should be feasible in different countries as long as there are databases for infection testing that are of reasonable quality and that can be linked to documented prior infection status (and preferably to vaccination status).Such databases are available and have been used extensively in vaccine effectiveness studies using the testnegative design, such as for SARS-CoV-2 infection, [17][18][19][20][21][22]33 and recently to estimate PE S for the Omicron variant. 79 his is a key strength for test-negative studies in that such studies are typically implemented on full eligible routine datasets where the large sample sizes optimize the statistical precision of the estimates.
Of the considered biases, only misclassification of prior infection status could have a large effect on PE S estimation, but mainly where more than 50% of the population already had a prior infection.This situation is not likely to have been reached for SARS-CoV-2 infection before the introduction the Omicron variant in most countries. 56Even in such situations, the direction (and magnitude) of bias is known; it underestimates PE S .Therefore, the test-negative design can still provide a lower bound for the true PE S , which may be sufficient to inform public health decision making, such as in relation to differential application of restrictions according to prior infection status, timing of vaccination following documented infection, and protocols for isolation and quarantine.Thus, this bias may not restrict the utility of this method.
The test-negative study design has strengths that conventional designs may lack.Cohort study designs can be affected by bias resulting from different infection testing frequencies in the different arms of the study.This bias does not affect the test-negative design, as it uses only those who are tested.An example can be seen in comparing the results of the test-negative design with the results of our earlier cohort design. 4In the cohort design, adjustment for testing frequency reduced PE S from 97.6% (95% CI, 95.7-98.7) to 95.8% (95% CI, 92.5-97.7)for Alpha, 4 very similar to the test-negative estimate of 97.0% (95% CI, 93.6-98.6).Similarly for Beta, adjustment for testing frequency reduced PE S from 92.3% (95% CI, 90.3-93.8) to 86.5% (95% CI, 83.0-89.2), 4 very similar to the test-negative estimate of 85.5% (95% CI, 82.4-88.1).Accordingly, the test-negative design may provide a more representative estimate than the cohort design.The test-negative design may also be preferable to the cohort design for other reasons.Cohort designs rely on cohorts that may not be strictly comparable, and it may not be possible to control for all differences in risk of exposure to the infection by matching and analysis adjustments.For example, in our earlier cohort study, 4 we compared those who had a record of a prior PCR-confirmed infection with those who had an antibodynegative test, but these groups may differ in ways that cannot be controlled.Meanwhile, the test-negative design is perhaps less susceptible to such differences, as cases and controls are selected to meet certain clinical criteria that presumably imply the same health care-seeking behavior.That said, use of administrative databases may still be prone to bias due to unmeasured differences in health care-seeking behavior.Last, while the testnegative design can be biased by misclassification of prior infection, the cohort design is perhaps more affected by this bias.The odds ratio metric in the test-negative design is less affected by this bias than the relative risk, incidence rate ratio, or hazard ratio metrics in the cohort design.
With regard to limitations, we used a heuristic approach to motivate the test-negative design through mathematical modeling, but this approach may not exactly match an actual empirical test-negative-design application.The ultimate validity and utility of this design rests on actual validation studies, including comparison with results of other conventional designs.We provided 2 such validation studies in the present study for each of the Alpha and Beta variants.Considering the demonstrated utility of this design in providing timely results in emergent situations during the COVID-19 pandemic, 53,[79][80][81] this study should be seen as a call for further investigation and methodological development to enhance this design and its applications.
Specific forms of bias were investigated, but other sources of bias are possible, and these may also depend on the database being analyzed. 25There is already a volume of literature investigating other forms of bias for the test-negative design in the context of vaccine effectiveness estimation, 15,16,[27][28][29][30][31][32] some of which may also apply in the context of PE S estimation, such as for issues relating to testing and applicability of this design for different Abbreviations: PCR, polymerase chain reaction; SMD, standardized mean difference.a Cases and controls were matched 1-to-1 by sex, 10-year age group, nationality, and calendar week of PCR test.b SMD is the difference in the mean of a covariate between groups divided by the pooled standard deviation.An SMD < 0.1 indicates adequate matching.c Values are expressed as median (interquartile range).d SMD is the mean difference between groups divided by the pooled standard deviation.e Nationalities were chosen to represent the most populous groups in Qatar.f These comprise 61 other nationalities in Qatar in the Alpha variant analysis and 78 other nationalities in the Beta variant analysis.g Given our interest in quantifying differentials in the odds of exposure to prior infection between cases and controls, this variable was not included as a matching factor.
testing modalities. 25More studies are needed to investigate different methodological aspects of this design and other sources of bias, such as the uncertainty/power to estimate effect and validity of the assumption of proportional random sampling of the different epidemiologic classes/compartments.While this study demonstrated use of the test-negative design to estimate PE S , other factors need to be considered in actual application.For instance, the algorithm for matching 67,82 needs to be developed with knowledge of the local epidemiology to ensure that matching can effectively control differences in the risk of exposure to the infection.Of note, with Qatar's young population, the estimates presented here for PE S may not be generalizable to other countries where elderly citizens constitute a larger proportion of the total population.
The models used to investigate applicability of the testnegative design were not structured by age, nor by infection type and severity.However, the sensitivity analysis that used the real-world Qatar model, with its detailed stratifications, confirmed the same findings as those of the study's parsimonious models.Moreover, the 3 other sensitivity analyses confirmed the applicability of the test-negative design regardless of the value of PE true S , irrespective of whether incidence is used instead of instantaneous prevalence in the estimation, and whether or not there was full misclassification bias of those latently infected.
In conclusion, the test-negative design offers a feasible and robust method to estimate protection of prior infection in preventing reinfection.This method should be considered to provide rapid, rigorous estimates of protection offered by prior infection for different variants of SARS-CoV-2, including those that emerged recently.

Figure 1 .
Figure 1.Schematic diagrams of mathematical models used in this study.A) Classic susceptible-exposed-infectious-recovered (SEIR) model extended to allow for reinfections (baseline model).B) Baseline model extended to include vaccination (vaccination model).C) Baseline model extended to include waning in protection of prior infection against reinfection (waning-of-immunity model).

Figure 2 .
Figure 2. The 2-by-2 tables and equations used to estimate effectiveness of prior infection in preventing reinfection (PE S ) using the test-negative, case-control study design.A) PE S estimated in absence of bias.B) PE S estimated in presence of misclassification of prior infection.C) PE S estimated in presence of misclassification of latent infection.D) PE S estimated in presence of misclassification of current active infection.E) PE S estimated in presence of vaccination scale-up.
test-negative S ref lects the value of PE true S regardless of the actual value of PE true S and over the full spectrum of possible PE true S values (Figure

Figure 3 .
Figure 3. Simulated severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) epidemic through 2 epidemic waves.A) Daily number of new infections.B) Proportion of the population ever infected.C) Scale-up of vaccine coverage.

Figure 4 .
Figure 4.Estimated effectiveness of prior infection in preventing reinfection using the test-negative study design (PE test-negative S ) compared with the

Figure 5 .
Figure 5. Impact of bias in estimating effectiveness of prior infection in preventing reinfection using the test-negative study design (PE test-negative S ).A) Impact of misclassification of prior infection.B) Impact of misclassification of latent infection.C) Impact of misclassification of current active infection.D) Impact of scale-up of vaccination in the population.This figure was generated using the instantaneous prevalence at each time point for each population.

Figure 6 .
Figure 6.Flowchart describing the population selection process to estimate effectiveness of prior infection in preventing reinfection using the test-negative study design, using data from Qatar, March 8 to April 21, 2021.Individuals with a polymerase chain reaction (PCR)-confirmed infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) Alpha or Beta variant were exact matched on a 1:1 ratio by sex, 10-year age group, nationality, and PCR test calendar week to the first eligible PCR-negative individual.Prior infection records were retrieved for all matched individuals.

Table 1 .
Model parameters and assumptions.

Table 2 .
Demographic characteristics of subjects in the samples used to estimate effectiveness of prior infection in preventing reinfection using the test-negative study design, Qatar, 2021