Statistically-based methodology for correcting delay-induced errors on the evaluation of COVID-19 pandemic

05/25/2020 ∙ by Sebastián Contreras, et al. ∙ 0

COVID-19 pandemic has reshaped our world in a timescale much shorter than what we can understand. Particularities of SARS-CoV-2, as its persistence in surfaces and the lack of a cure for COVID-19, have pushed authorities to apply restrictive policies to control its spreading. As data drove most of the decisions made in this global contingency, its quality is a critical variable for decision-making actors, and therefore should be carefully curated. In this work, we analyze the sources of error in the typically reported epidemiologic variables and the different tests used for diagnosis, and their impact on our understanding of COVID-19 spreading dynamics. We address the existence of different delays in the report of new cases, induced by the incubation time of the virus and testing-diagnosis time gaps, and other error sources related to the sensitivity/specificity of the tests used to diagnose COVID-19. By a statistically-based algorithm, we perform a temporal reclassification of cases to avoid delay-induced errors, building up new epidemiologic curves centered in the day where the contagion effectively occurred. We also statistically enhance the robustness behind the discharge/recovery clinical criteria in the lack of a direct test, which is typically the case of non-first world countries, where the limited testing capabilities are fully dedicated to the evaluation of new cases. Finally, we applied our methodology to assess the evolution of the pandemic in Chile through the Basic Reproduction Number R_0, identifying different moments in which data was misleading governmental actions. Doing so, we aim to raise public awareness of the need for proper data reporting and processing protocols.



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1 Introduction

Since the outbreak of novel SARS-CoV-2 in late 2019, the spread of COVID-19 has changed nearly every aspect of our daily life, challenging modern society to find a way to function under conditions never seen before. Governmental plans on public health have played a crucial role in its control in the absence of an effective treatment to cure COVID-19 or a vaccine to prevent it. Typically reported variables in the COVID-19 pandemic, available in public repositories (as [1, 2], among others), are the total cases , active cases , discharged/recovered , and deaths . These variables serve as input for the development and evaluation of governmental plans and to fit the vast variety of SIR-like mathematical models recently proposed (see,e.g., [3, 4, 5, 6] and references therein for a brief review of them). Among the several factors conditioning the quality/reliability of the variables mentioned above are those related to the sensitivity and specificity of diagnostic tests, delays between sampling and diagnosing, and the delay between presenting symptoms and getting tested. The latter varies from country to country, depending heavily on the local government’s testing strategy and resources.

Different parameters can be used to evaluate the evolution of the SARS-CoV-2 outbreak. Among them we may find the documentation rate [7], secondary infection rate, serological response to infection[8], number of vacancies at ICU [9], and the Basic Reproduction Number [10, 11], which is one of the most widely used. This parameter () represents the number of persons a single infected individual might infect before either recovering or dying [12]

. Traditional forms to estimate the

are rather complex, heavily depending on the fitting of SIR models to local data [13, 14, 15, 16]. In a previous work [17], we proposed a methodology to obtain real-time estimations of directly from raw data, which was satisfactorily applied to evaluate the panorama of the COVID-19 spread in different countries and to forecast its evolution [18]. Nevertheless, its heavy dependence on the reported data required the study of common error sources affecting it, and the development of methodologies to control, correct, and quantify their impact [17].

In this work, we analyze the sources of error in the typically reported epidemiologic variables and their impact on our understanding of COVID-19 spreading dynamics. We address the existence of different delays in the report of new cases, induced by the incubation time of the virus and testing-diagnosis time gaps, and provide a straightforward methodology to avoid the propagation of delay-induced errors to model-derived parameters. Using our statistically-based algorithm, we perform a temporal reclassification of individuals to the day where they were -statistically- most likely to have acquired the virus, building a new smooth curve with corrected variables. We present an analogous methodology to estimate the number of discharged/recovered individuals, based on the reported evolution of the viral infection, the performance of the different tests for its diagnosis, and the case fatality, which can be easily adapted for a particular country. We used our methodology to assess the evolution of the pandemic in Chile, identifying different moments in which data was misleading governmental actions.

2 On the performance of tests for diagnosing COVID-19

Different methods for diagnosing COVID-19 have been developed and reported in the literature, with real-time RT-PCR being the standard applied in the globe [19]. Nevertheless, techniques as the IgG and IgM rapid tests, the Chest Computed Tomography (Chest CT), and CRISPR-Cas systems are also being used. In this section we provide a brief analysis of them, highlighting the different characteristics of both the techniques and their basic approach to the viral infection.

2.1 Real-time RT-PCR

Real-time reverse transcription polymerase chain reaction (RT-PCR) is a mechanism for amplification and detection of RNA in real time (see [20]

for an exhaustive description of the technique). Initially, RNA obtained from samples is retrotranscribed to DNA using a reverse transcriptase enzyme. By applying temperature cycles, the conditions are created for new copies of the DNA to be synthesized from the initial one. The lower the initial DNA concentration, the lower the probability of a synthesis reaction in a given cycle. It is presumed that the minimum time from contagion until testing positive in the RT-PCR test is

( CI, ) before the onset of symptoms [21], which typically appear 5.2 days ( CI, ) after contagion [22]. Unsurprisingly, [21, 23] showed that to of the total infections occur in the pre-symptomatic period. It has been inferred that the viral load reaches a peak value before 0.7 days from the onset of symptoms ( CI, ), from which falls monotonically together with the infectivity [21]. Finally, the virus has been detected for a median of 20 days after the onset of symptoms [24], but infectivity may decrease significantly eight days after that moment.

A high false-negative rate [25, 26] and a sensitivity of 71 to  [27, 28] have been reported for the real-time RT-PCR technique, and several vulnerabilities of it have been identified and quantified [19]. Considering sample obtention, handling, testing, and reporting, the total time necessary to obtain the RT-PCR results may range between 2 to 3 days [29]. However, the time it takes to perform the RT-PCR experiment takes about 2 to 3 hours [30].

2.2 IgG and IgM rapid tests

Part of the immune response to the SARS-CoV-2 infection is the production of specific antibodies against it, including IgG and IgM [31]. Serological tests detect the presence of those antibodies and, unlike the other detection methods, take only to produce results [30]. These tests have a sensitivity of and a specificity of [30]. This technology was developed for the SARS-CoV epidemic, which was caused by a virus belonging to the same family of coronaviruses as SARS-CoV-2, providing satisfactory results after 2–3 days from the onset of symptoms (for IgG), and after eight days (for IgM) [32].

2.3 Chest Computed Tomography

The principle behind the Chest Computed Tomography (Chest CT) is the analysis of cross-sectional lung images to identify viral pneumonia characteristics, like ground-glass opacity, consolidation, reticulation/thickened interlobular septa or nodules [33]. Chest CT has shown a sensitivity between and  [28, 27]. However, due to the similarities between CT images accounting for COVID-19 and CT images from other viral types of pneumonia, false-positive are likely to occur. Compared to RT-PCR, Chest CT tends to be more reliable, practical, and quick to diagnose COVID-19 [33]. Nevertheless, requiring the presence of the potentially infected patient in a health center lacks the flexibility that rapid tests provide, and can backfire on movement restriction measures. COVID-19 pneumonia manifests with abnormalities on computed tomography images of the chest, even in asymptomatic patients [34].

2.4 CRISPR-Cas systems

In CRISPR-Cas systems, a guide RNA (gRNA) is designed to recognize a specific RNA sequence, like any particular gene of SARS-CoV-2 coronavirus. Endonuclease enzymes of the Cas family and the specific gRNA will search for the sequence match. This match will deliver a signal that confirms the presence of SARS-CoV-2 RNA in the sample [35]. DETECTR and SHERLOCK are two examples of CRISPR-Cas technologies for the detection of SARS-CoV-2, being able to obtain results in less than 1 hour at a significantly lower cost compared to the RT-PCR technique. DETECTR showed a 95% positive predictive agreement and 100% negative predictive agreement [36], while SHERLOCK has not been validated using real patient samples and is not suitable for clinical use at this time [37].

3 Methodology

Our work aims to expose and quantify, both theoretically and in a case study, the impact of different sources of error in commonly reported data of the COVID-19 spread, as the newly reported cases , the total cases , the infected , and recovered fractions of the population.

First, we define random variables associated with the delay in both sampling and diagnosing new cases, and by modeling their probability distribution functions, we derive a method to re-classify accordingly the newly reported cases. As the reclassification occurs backward, for re-evaluating the current scenario through our methodology, we cast predictions on the reported new cases using an ARIMA auto-regression model. Having the corrected variables, we evaluate differences on the values of

, following the methodology presented by [17]:


Auto-regression models for the forecast of were implemented using the statsmodels Python library [38]. All other calculations and visualizations were made in MATLAB R2018a.

4 Results

4.1 Temporal misclassification of new cases

There exists an incubation period for the development of SARS-CoV-2-related symptoms, which average has been reported to range between 5.2 days [39, 40] and 6.4 days [41]

. We can assume the incubation period follows an exponential distribution with



This incubation time is especially relevant in the case of a symptoms-based testing strategy or when the spread has reached the non-traceability stage. Moreover, even though the required time for performing the test is short [30], delays between testing and diagnosis have been reported [29]. We will sum up secondary delays, such as the symptom-testing and testing-diagnosing time gaps, into a random variable

, which, for the sake of simplicity, will be assumed to follow a uniform distribution between

and :


Consequently, we may postulate a reclassification for obtaining the real new contagions occurred in a day as the contribution of the cases reported with a delay of days:


where represents the fraction of patients that were notified at time but had acquired the virus at . Note that the different delays are referred to the random variable . The probability distribution function for is obtained by the convolution method, assuming and are independent and combining equations 2 and 3:


Assuming that data is reported on a daily basis, we can calculate the probability associated to having a delay of days:


For practical reasons, we can define a threshold for truncating the probability mass distribution (equation 6), which otherwise would assign a probability to every . Let be the first positive integer for which equation 7 holds,


we may rewrite equation 4 as:


The magnitude of the total delay between infection and diagnosis can be estimated through the expected value of equation 5 (or equivalently, equation 6). A schematic representation of the proposed methodology is presented in Figure 1. Assuming the lowest reported value for the average incubation time, , and a conservative timeframe for the delay between the appearance of symptoms, testing and diagnosing, (2–5 days), the expected delay is about .

Result: Corrected differential and cumulative total cases .
Officially reported new cases per day. ;
Parameters of the and distributions;
Probability threshold for the maximum possible delay;

, respectively the probability and cumulative distribution functions for

idx = <

; logical indexes of the values of vector

smaller than ;
= (idx); ;
Define , delay indexes of in idx. Renormalize ;
Forecast the next values for , using an auto-regression (ARIMA) model;
Define an empty vector to record the corrections;
for i = 1:length(dT)  do
end for
Delete the last (incomplete) values from ; ;
Round values to the nearest integer; ;
Algorithm 1 Statistically-based temporal reclassification algorithm for correcting delay-induced errors in the report of new COVID-19 infections
Figure 1: Schematic representation of the temporal reclassification methodology proposed herein.

4.2 Case discharge/recovery criteria

As discussed previously, errors in the amount of discharged/recovered patients are likely to be greater only when no quantitative criteria are applied. In such cases, some countries (like Chile) have adopted the following criteria (officially reported in [42]), possibly based on the recommendations published by the [43].

  • If there were no previously existing pathology, a patient would be discharged 14 days after testing positive for COVID-19.

  • If there were previously existing pathologies, the patient would be discharged 28 days after testing positive for COVID-19.

This criterion turns out to be quite simplistic, especially considering the existence of uncertainties regarding the diagnosis and contagion days. If we try to model the probability of recovering from COVID-19, some assumptions are necessary. Let be the random variable for the time of discharge/recovery:

Further assumptions are necessary to estimate the probability distribution function , as it depends on local diagnosis criteria, testing strategy, and the fraction of the population having preexisting pathologies. In particular, depending on the test applied for diagnosis and its sensitivity/specificity – which were carefully described in Section 2– the probability profile would change. The simplest form that can be assumed, and which, for clarity reasons, is adopted herein, is a triangular distribution:

Result: Estimated discharged/recovered cases per day .
corrected new cases per day. ;
Officially reported daily deaths due to COVID-19. ;
Parameters of the probability distribution function for the time of discharge/recovery .;
time frame for discharging an infected patient.;
, width of the time-frame of discharge;
Calculate the probability distribution function for ; ;
Define an empty vector to record the corrections;
for i = 1:n-  do
end for
Delete the last (incomplete) values from ; ;
Subtract the daily deaths due to COVID-19 to obtain the final estimator, round to the nearest integer. Confirm the result is positive or zero.;
Algorithm 2 Statistical estimation of , daily patients that have been discharged/recovered from COVID-19

5 Case study: COVID-19 spreading dynamics in Chile

The spread of COVID-19 in Chile is far from being controlled, as shown by the exponential growth that new cases have had in recent weeks [1]. In order to apply our methodology, we need to cast predictions on the trends of . Figure 2 presents the current and forecast trends, using an auto-regression ARIMA model.

Figure 2: Current and forecasted evolution of in Chile. The official data (blue curve) was obtained from [1], while the red curve was generated using an auto-regression ARIMA model.

First, we perform the temporal reclassification of new cases to obtain , presented in Figure 3. It can be seen that our methodology, besides exhibiting an horizontal semi-displacement, generates an smooth curve. The last part of the red curve is dashed because it partially contains contributions of the forecast of , and therefore might change in the upcoming days, when the required data for completing the reclassification of cases would be available.

Figure 3: Statistically-driven reclassification of new cases (red curve) compared with the raw data (blue curve), both differential a) and cumulative b). The last dashed part of the red curve accounts for the values which depends in the forecast shown in Figure 2, and therefore might change over time. Official data obtained from [1].

After obtaining statistically-based corrections of both and by following algorithms 1 and 2, and knowing the daily deaths due to COVID-19 , we proceed to calculate the variation on the active cases:


Using such values, we proceed to calculate , using equation 1 with raw data, mobile averages of raw data, and the methodology proposed herein. As shown in Figure 4, an abrupt growth in was evidenced around April 22nd, consistently with the relaxation of restrictive measures that were applied in Santiago and the apogee of the governmental plan for a “safe return to work”. Even though different trends seem to decrease again in the second week of May, it is not totally clear, as the forecast of strongly influences the statistical correction for that week, and it is well-known that forecast models fail to predict exponential growth [44, 45].

Figure 4: Effect of data processing on the evaluation of the spread of COVID-19 through . The noisy raw data (blue curve) can be smoothed through mobile averages (dark red curve), but the trends are the same. A significantly different scenario is shown by the statistically-corrected trend (green curve). Highlighted dates associated with iconic governmental actions in Chile: March 31st (second week of quarantine for the rich districts of Santiago, capital of Chile), April 7th (obligatory use of facemask in public transport), April 24th (governmental call for a “safe return to work”), April 30th (quarantine –poor districts of Santiago–), May 15th (total quarantine in Santiago).

The different iconic dates highlighted in Figure 4 were obtained from the chronology presented in [46] and references therein. We can assess both the success and the misleading effect that the different governmental actions have had on the spreading dynamics of COVID-19 in Chile by analyzing the corrected trends of . We can observe that the different actions had a strong relationship with the locally observed values from raw-data, yet appear to be too late according to the statistically-corrected trend. In particular, the apogee of the governmental plan for a safe return to work happened in the zone where the raw-data driven values were at a minimum, but the corrected trends showed a steep growing trend.

6 Conclusions

We have presented an exhaustive assessment of error sources in reported data of the COVID-19 pandemic and provided a methodology to minimize –and correct– their effect on both reported variables and model-derived parameters by applying a statistically-driven reclassification of newly reported cases, and corrections to discharge/recovery criteria. By using the corrected variables, SIR-like models could be fitted directly, as every value would represent the real dynamics. We present the methodology in a general framework, aiming to provide a useful tool for researchers and decision-making actors looking to adapt it for their particular interests.

In a case study on the spreading dynamics of COVID-19 in Chile, we observed the effects that different iconic actions taken by the government had on and discussed the reasons behind them under the eye of raw data and our methodology. The delay-induced error in raw data slowed-down the reaction time, so the actions taken were too late. Our statistically-driven method corrected such error, exposing the real dynamics at a given time. The proposed methodology also serves as a non-invasive smoothing process, as it does nothing but the temporal re-sorting the cases according to their most likely delay.

We expect our methodology to serve as a valuable input for researchers trying to add statistical value to their calculations and to raise public awareness on the need for a proper (and standardized) strategy for the report and curation of data in the COVID-19 pandemic.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Author Contributions

Conceptualization, SC, HAV; methodology, SC; validation JPB-L, DM-O, AO-N; investigation, HAV, JPB-L, NL-K; writing, original draft preparation, JPB-L, HAV, NL-K, DM-O, SC; writing, review and editing, SC, JPB-L, DM-O, AO-N; supervision, SC, HAV; project administration, AO-N; funding resources, AO-N.


The authors gratefully acknowledge support from the Chilean National Agency for Research and development through ANID PIA Grant AFB180004, and the Centre for Biotechnology and Bioengineering - CeBiB (PIA project FB0001, Conicyt, Chile). DM-O gratefully acknowledges Conicyt, Chile, for PhD fellowship 21181435.


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