key: cord-0716801-mtx9g1qu
authors: Hitchings, Matt D T; Dean, Natalie E; Garcia-Carreras, Bernardo; Hladish, Thomas J; Huang, Angkana T; Yang, Bingyi; Cummings, Derek A T
title: The Usefulness Of SARS-CoV-2 Test-Positive Proportion As A Surveillance Tool
date: 2021-02-12
journal: Am J Epidemiol
DOI: 10.1093/aje/kwab023
sha: fbfab588e3683acf4f69c1293aa15f56f7d79453
doc_id: 716801
cord_uid: mtx9g1qu

Comparison of coronavirus disease (COVID-19) case numbers over time and between locations is complicated by limits to virologic testing to confirm severe acute respiratory syndrome coronavirus 2 infection. The proportion of tested individuals who have tested positive (test-positive proportion, TPP) can potentially be used to inform trends in incidence. We propose a model for testing in a population experiencing an epidemic of COVID-19, and derive an expression for TPP in terms of well-defined parameters related to testing and presence of other pathogens causing COVID-19 like symptoms. In the absence of dramatic shifts of testing practices in time or between locations, the TPP is positively correlated with the incidence of infection. We show that the proportion of tested individuals who present COVID-19 like symptoms encodes similar information to the TPP but has different relationships with the testing parameters, and can thus provide additional information regarding dynamic changes in TPP and incidence. Finally, we compare data on confirmed cases and TPP from US states up to October 2020. We conjecture why states may have higher or lower TPP than average. Collection of symptom status and age/risk category of tested individuals can increase the utility of TPP in assessing the state of the pandemic in different locations and times.

indication that the reported cases represent the "tip of the iceberg" and that testing capacity should be increased to get a better understanding of transmission. Where capacity is limited, tests are given preferentially to those more likely to be positive, such as hospitalized patients, meaning that mildly symptomatic infections are likely to be undetected. On the other hand, a low TPP is viewed as an indication of a potentially effective surveillance and containment strategy (1) , and implies that increased testing would not reveal a substantial number of undetected infections. The WHO and US CDC made TPP part of their guidelines for easing lockdown restrictions, indicating that it can be used to assess readiness for releasing restrictions and recommending that communities should be below various thresholds (5-20%) for 14 days before they consider relaxing social distancing measures (2, 3) .

While these interpretations are broadly plausible, many aspects of testing strategies, including the rate of testing of symptomatic and asymptomatic individuals, the number of tests available, and the performance of the test, could change TPP independently of the true incidence. We aim to explore the relationship between TPP and testing parameters, and suggest additional metrics and data to aid interpretation of the TPP. We model the application of tests using compartments for available test kits and completed tests applied to SARS-CoV-2 positive/SARS-CoV-2 negative and symptomatic/asymptomatic individuals. We assume that an individual's rate of testing differs by symptom status. Recovered individuals are assumed not to seek testing. The test has sensitivity . Exposed and infectious individuals are isolated upon testing (efficacy of isolation is assumed to be perfect, represented by moving those individuals to the compartment).

A schematic for the natural history and testing model is shown in Figure 1 (see Web 

the entire population each day). The demand from any specific group is denoted using subscripts, e.g. for exposed individuals with CLI. We assume that anyone with CLI seeks testing at the symptomatic rate . We assume that test sensitivity for pre-infectious individuals is very low(4) so that the class contributes little to detected cases compared to the class, and the above inequality becomes an approximate equality.

Given the infectious prevalence at a point in time and assuming that the majority of detected cases are from the infectious compartment, the number of positive tests per capita (henceforth "confirmed incidence") is estimated by

We write TPP as a function of the confirmed incidence rate,

If the relevant parameters were known we could use Equation (2) to "estimate" the confirmed incidence from the TPP; we refer to this as the "TPP-estimated" incidence.

If the number of individuals with CLI (due to SARS-CoV-2 and/or other causes) among those seeking testing is known, the test-symptomatic proportion (TSP) provides a similar relationship between confirmed incidence and TSP, assuming :

In addition to the pre-symptomatic exposed period, some individuals remain asymptomatic throughout the course of infection. To model asymptomatic infections, we assume that a fraction of infected individuals become infectious with symptoms, with the remaining not developing CLI due to their SARS-CoV-2 infection. We assume that infectiousness is the same regardless of symptoms. The relationship in (3) becomes

This structure allows testing demand to vary for certain target groups, defined either by frequent exposure to infection (e.g. essential workers) and increased β, or increased probability of symptoms (e.g. the elderly) and increased p S . We assume that these risk groups can be tested at a different rate (i.e. d S and d A ) than the general population. The overall TPP will depend on the proportion of tested individuals in these target groups.

We use our previous formulas to assess TPP and TSP stratified by risk group, and the potential for bias in the overall TPP caused by testing of these groups.

Assessment of TPP-incidence and TSP-incidence relationships using simulations

To assess the accuracy of Equations (3) and (4), we simulate the SEIR model described above using an adaptive tau-leaping method (R package adaptivetau). From the simulations we calculate the weekly confirmed incidence, TPP, TSP, and

supply/demand ratio (stratified by risk group where relevant), varying testing parameters d A , d S , and p I . We then explore the impact of temporal variation in testing in two ways: by inducing linear changes in testing parameters, and by drawing available tests T* from a lognormal distribution to represent random fluctuations in capacity.

Finally, to understand the effect of targeting testing at high-exposure and highvulnerability populations, we vary the size of the risk groups, the relative hazard of infection and probability of symptoms, and relative rates of testing between the highand low-risk groups. In simulations, we used parameters from existing literature where possible(5-12) (Web Table 1 ), with initial R 0 drawn from a uniform distribution between 3 and 5. We modeled a lockdown with a reduction in R 0 to 0.8-1 at 21-35 days after the start of the simulation.

In addition to our simulations, we use data from the COVID Tracking Project (13) to examine the relationship between TPP and confirmed cases across states, and within states across time, with population data for states from the US Census Bureau (14) . We use the derived equations to plot the expected relationship between TPP and confirmed cases if all states had the same testing parameters, by fitting Equation (2) to the observed confirmed cases using ordinary least squares (OLS) regression. Similarly, we fit Equation (2) to observed confirmed cases for a time series within a single state to identify periods of time in which the trend in TPP is not expected given the trend in confirmed cases. In addition to national data, we use data from the Oregon Health Authority (15) , which reports the proportion of coronavirus disease cases with various symptoms to illustrate trends in symptoms over time.

In Figure 2 , we vary each model parameter in turn, with the other parameters fixed at default values (Web Table 1 ), to explore its univariate relationship with TPP. As the rate of confirmed infections increases (A), the TPP rises because there are more symptomatic, SARS-CoV-2 positive individuals in the population demanding tests. In addition, testing strategy can affect the TPP and TSP. Counter-intuitively, if individuals with CLI are tested at a higher rate the TPP will decrease (B) if the rate of confirmed infections is held constant. In this case, higher rate of testing individuals with CLI means that individuals with non-SARS-CoV-2 CLI will also be tested at a higher rate, leading to lower TPP. If more asymptomatic individuals are tested (e.g. through expanding eligibility for testing), the TPP will decrease, albeit modestly (C). If there is a shortfall of testing supply relative to demand (D), TPP will increase because the confirmed cases represent a smaller proportion of infectious individuals, and more overall demand from infectious individuals leads to higher TPP. Finally, factors independent of policy decisions can affect the TPP. The test sensitivity has a negligible effect on the TPP (E) when the confirmed incidence is held constant. On the other hand, if prevalence of non-SARS-CoV-2 CLI is higher (e.g. during an influenza outbreak), the TPP will decrease as more SARS-CoV-2 negative individuals will seek testing (F). Equation (2) provides intuition for how TPP and detected cases will change under different scenarios. During the exponential growth phase of an epidemic, TPP will rise as infectious prevalence rises rapidly (Figure 2A) . Similarly, if the rate of new infections is declining and testing strategies remain constant, TPP will decrease over time. This observation provides a simple way to understand whether falling case numbers is due to a true decline in incidence or a shortage of test kits. If the infectious prevalence declines, the rate of confirmed cases and the TPP will both decline (Figure 2A ),

whereas if the supply of test kits is falling relative to demand but the infectious prevalence remains constant, the rate of confirmed cases will decrease (Equation (2)) but the TPP will remain constant (Equation (1); with constant infectious prevalence, TPP is independent of test kit supply). Similarly, concurrently rising TPP and confirmed incidence is an indication that infectious prevalence is truly increasing. If a rise in confirmed cases were due to increases in testing capacity alone, we would not expect TPP to increase.

Relationship between TPP and TSP 

We assessed the accuracy of the TPP and TSP formulas using data generated from an SEIR model. We fixed the testing parameters for all simulations and sampled prelockdown R 0 uniformly from 3 to 5 (higher than observed values(6) to include parameter space in which supply of testing is limited), time of lockdown from 21 to 35 days, and post-lockdown R 0 from 0.8 to 1 in the absence of testing (5, 6) . Figure 3 shows that if the supply of test kits is sufficient to cover the demand from symptomatic and asymptomatic individuals, the relationship between TPP and confirmed incidence is as in Equation (2) (gray points), but that if there is limited supply of test kits the proportion of cases detected decreases while the TPP remains the same. Therefore, the TPP observed in the simulations is greater than predicted (black points). We observe a similar pattern for TSP. We found that the relationship between TPP and incidence was unaffected by the efficacy of quarantine (Web Appendix 2 and Web Figure 1) .

TPP was strongly correlated with the confirmed incidence rate (average Pearson correlation = 0.94 across all simulations in Figure 3 ). Correlation remained high in the presence of linear changes in testing parameters over time within a location, but decreased when there was large variability in daily available test kits. As the variance of testing availability increased, the correlation between TPP and confirmed incidence time series decreased (Web Appendix 2 and Web Figure 2 ), as decreases in confirmed cases due to testing shortage occurred without concurrent decreases in TPP.

High-exposure individuals are more likely to be infected, so infectious prevalence in this group will be higher than the general population. High-susceptibility individuals are more likely to have symptoms and thus more likely to be tested, so TPP will be higher in this group. However, the relationship between confirmed incidence rate and TPP remains the same unless the groups are tested at different rates.

We found that TPP at 35 days was positively correlated with relative testing rate in highexposure groups among symptomatic individuals (Pearson correlation 0.71) and

negatively correlated with relative testing rate among asymptomatic individuals (correlation -0.37). In contrast we found that TPP at 35 days was negatively correlated with relative testing rate in high-exposure groups among symptomatic individuals (Pearson correlation -0.27) and asymptomatic individuals (correlation -0.50). High rates of testing of symptomatic individuals in this group is highly effective at reducing transmission, as the clinical fraction was 0.75. Therefore, higher testing is associated with lower infectious prevalence among the high susceptibility group, and lower TPP.

If we had data on testing rates and cases stratified by risk group (e.g. job category, age), plotting confirmed incidence against TPP for each risk group can indicate whether there are testing differences between the groups. In simulations, the TPP-incidence relationship differed by risk group when the high-risk group was tested at a higher rate (Web Appendix 2 and Web Figures 3 and 4) . See Web Appendix 2 for more details.

Comparison of TPP and confirmed incidence across US states Figure 4 shows the relationship between confirmed incidence per 10,000 and TPP by state, at four different times relative to the start of the epidemic in each state. The parameter values that minimize the sum of squares are plotted as a regression line. If all states had the same testing parameters and sufficient supply of test kits, we would expect them to lie on the line as in Figure 3 (gray dots).

States that fall below the line have a lower TPP than expected given how many cases they have observed. Figure 2 shows that there could be several reasons for this:

increased prevalence of non-SARS-CoV-2 CLI; increased testing of asymptomatic individuals; or increased testing of individuals with CLI and correspondingly higher proportion of infections detected. States that fall above the line have a higher TPP than expected given how many cases they have observed. The reasons for this are the inverse of those above. In addition, as in Figure 3 , there could be a shortfall of supply of test kits relative to demand, leading to a lower rate of case detection than the average state. We observe a linear relationship between confirmed incidence and TPP, except in the first panel where there is some evidence of saturation of TPP at high incidence rates in over the summer was accompanied by a rise and fall in TPP, and thus TPP-estimated incidence rate, as we would expect if infectious prevalence were increasing then decreasing.

We present a simple transmission model incorporating testing of SARS-CoV-2 to derive an expression for TPP as a function of well-defined parameters. We use this expression to understand how TPP changes with the confirmed incidence as well as other parameters related to testing and the presence of other pathogens in the population. In particular, our work can be used to build hypotheses for why a location or point in time has a higher/lower TPP than expected.

When comparing TPP between locations, it is important to compare the rate of incident confirmed cases at the same time. Within the US, earlier in the epidemic New York and New Jersey were pointed to as examples of states that had very high TPPs compared to the country average, but this analysis shows that they were in line with the average after adjusting for the observed incidence. We showed that high variability in testing availability reduces the correlation between TPP and incidence, underscoring the need to evaluate smoothed trends in TPP.

Policies related to testing of high-risk groups in the population can drive changes in TPP. We showed that increased testing in groups with higher prevalence of infection can increase the TPP, although high testing rates can efficiently control infection and thus reduce the TPP. We note that high TPP in small subsets of the population (e.g. background incidence or testing rate is low, or the population is small (e.g. at the county level). Publicly available data on testing in different high-risk populations is critical in understanding changes in TPP. To our knowledge, the CDC website is the only source of such information stratified by age in the US (16) , and this information is only available for the whole country.

Other authors have attempted to infer the population prevalence in the US using case data and TPP (17, 18) . Our approach was not to make predictions or recommend absolute thresholds for TPP, but to explore the effect of varying testing parameters on TPP and its relationship with incidence; many parameters affect TPP only through their effect on incidence. We expect the relationship to be non-linear, but this non-linearity occurs at higher infectious prevalence than is observed in the data we have used, meaning that the model is not identifiable as we have five parameters to fit a single linear gradient. We note that TPP and confirmed incidence alone do not provide enough information to infer the true prevalence or incidence, and we cannot assume that simple relationships proposed between these variables will hold in different settings (e.g.

comparing March to June 2020 in the US) (19) .

The disease model presented here is a simplification of the true natural history. While we briefly presented an extended model, we could have considered further extensions. dynamics, but we would expect that within risk groups there would be strong correlation between TPP and incidence over time.

We included random variation in testing availability to account for short-term, unpredictable changes in testing availability. Another source of variability in data is in the reporting of tests performed, for example due to changes in guidelines or reporting delays, leading to variation in TPP that doesn't reflect true rates of testing. In addition, differences in which tests are included in the numerator and denominator affect the value of the TPP (e.g. the first test per person). We did not explicitly model repeated testing of individuals and thus did not examine the differences between available TPP metrics.

The assumption that allocation of test kits is proportional to demand implies that selection bias in the sample of individuals tested is independent of the number of test kits available. It may be that in the cases of extreme restriction in testing availability, more priority is given to sicker individuals seeking testing. Therefore, deviations from the expected TPP when testing availability is limited would be more extreme than observed in our simulations.

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