Analysis of the Number of Tests, the Positivity Rate and Their Dependency Structure During COVID-19 Pandemic

1 Introduction

Dr. Li Wenliang, a 34-year-old ophthalmologist, warned his colleagues and set the alarm to the society about a new infection caused by a type of coronavirus in December 2019 in Wuhan, China (Jamshidi et al. 2020). Shortly after his warning, all over the world encountered this epidemic. WHO declared this fast speeding infection (COVID-19) in March 2020 as a pandemic. As of January 20, 2023, over 600 million cases, and around seven million deaths involving COVID-19 have been reported around the world. The epidemic COVID-19 is the most informative pandemic throughout history. These unprecedented recorded data give rise to some unprecedented concepts, relationships, analyses, discussions, and models (Jamshidi et al. 2021).

Modeling the dependence between the number of tests and the proportion of positivity (positivity rate) is one of these new issues. The proportion of positivity is a critical measure because it gives us an indication of how widespread infection is in the area of interest. The proportion of positivity helps public health officials answer questions such as:

1. What is the current level of SARS-CoV-2 (coronavirus) transmission in the community?

2. Are we doing enough testing for the people who are getting infected (Dowdy and D’souza 2020)?

According to the ratio nature, the high proportion of positivity is due to the high number of positive tests or the low number of total tests. Based on the first possibility, a higher positivity rate suggests higher transmission and that there are likely more people with coronavirus in the community who have not been tested yet. On the other hand, according to the second possibility, a high percentage of positivity means that more testing should probably be done. Accordingly, for policymakers, the high value for this parameter suggests either it is not a good time to relax restrictions aimed at reducing transmission, or it is a good time to add restrictions to slow the spread of disease (Dowdy and D’souza 2020). In this regard, an analytic report segregated by regions in the UK was presented by the Office for National Statistics (Coronavirus 2021).

This study is the first paper to highlight the importance of availability of the required tests and to model the effect of shortage of tests on the positivity rate. Our motivation is overcoming this ambiguity and giving a clear insight into the real status of the epidemic in the population. We want to emphasize the importance of fluctuations of positivity rate over the study time. We also aim to show the effect of available tests on these fluctuations. To highlight the meaningfulness of the criterion of interest, we use the data of 12 countries and compare them in terms of the other criteria. This study aims to investigate the time series of positivity rates individually and together with the time series of the number of tests. This investigation is conducted in two analytic methods: regional and temporal. The individual analysis is mainly undertaken based on the peaks of the spreading of the pandemic. For the regional aspect, among the 221 countries, we selected 12 countries: the USA, India, the UK, Italy, Iran, the UAE, Bolivia, Guatemala, Nigeria, Australia, South Korea, and South Africa. The reasons for selecting these 12 countries are

1. They are the top countries in the influential indices (Table 1).

2. Some of them are widely different from the others in some indices (Table 1).

3. Their positivity rates are greatly dispersed (Table 2).

4. The numbers and time of peaks are different about them (Table 3).

5. Their quality of healthcare systems are of different levels.

6. Their data, especially about the number of tests are relatively well recorded.

7. They are selected from all continents: the USA and Guatemala from North America; Bolivia from South America; India, Iran, the UAE, and South Korea from Asia; the UK and Italy from Europe; Nigeria and South Africa from Africa; and Australia from Australia.

Finally, to illustrate the dependency of the number of tests and the positivity rates, we apply copulas.

Sklar introduced the concept of copulas in 1959 (Sklar 1959). A copula – mainly parametric, partially semi-parametric, and rarely non-parametric – is a function that completely describes the dependency structure. It contains all the information to link the marginal distributions to their joint distribution. Accordingly, to obtain a valid multivariate distribution function, it suffices to combine several marginal distribution functions with any candidate for the copula function. Thus, for the purposes of statistical modeling, it is desirable to have a large collection of copulas at one’s disposal. Copula is widely applied in diverse fields, including environmental studies (Corbella and Stretch 2013; Zhang and Singh 2007) finance (Boubaker and Sghaier 2013; Wang et al. 2014), hydrology (Bekrizadeh et al. 2013), and medical studies (Bekrizadeh 2021; Bekrizadeh and Jamshidi 2017; Bekrizadeh et al. 2017; Chatrabgoun et al. 2020; Li and Fang 2012; Roman et al. 2012; Wienke 2011).

2 Data

The main data sources of the paper are the websites Worldometers (Worldometer website) and Our World in Data (Hasell et al. 2020). We summarize and illustrate all the relevant information about the 12 countries in three (twelve-row) tables and three (twelve-partitioned) figures created on Matlab R2015b.

Table 1 includes the key general indicators up to January 25, 2021. It is worth saying that the total indicators or even per-million indicators do not determine the quality of healthcare systems because there are observable underreported statistics about the countries Bolivia, Guatemala, Nigeria, Iran, and even India. On the other hand, we consider the indicator of the number of tests per one million (the 7th column of Table 1) as a criterion representing the level of facilities, therefore the quality of healthcare systems. Based on the information about this criterion, we define the lags (the distance between the test and diagnosis) for the different healthcare systems.

Table 2 represents the underlying properties of any country. As mentioned before, lag is the difference between the time of testing and the time of receiving the results of the tests, positive or negative, in days. The more facilities a healthcare system has, the more tests that system can do – therefore the lower positivity rate it has. Also, the more facilities a healthcare system has, the lower distance is between the tests and results. Based on the concept of lag, we pair the number of tests on the n th day with the number of results on the ( n + l a g ) th day to obtain the dependency structure by using the copulas. The last column is calculated based on the start and end dates of the period of recording data (the fourth and fifth columns) and the lag (the sixth column), and it displays the number of pairs that we use to obtain the dependency structure for each country.

Generally, during an epidemic wave, the number of new infected individuals increases rapidly to an epidemic peak and then falls more gradually until the epidemic wave is over, and the number of new cases be stabilized. Roughly speaking, the epidemic peaks are the – neighborhood of – time points that corresponds the local maximum of the number of newly infected cases.

3 Methods

3.1 Change Point Detection

We define the epidemic peak as the time neighborhood – or the time point – that X t : the number of new confirmed cases on the t th day, exceeds the mean plus three times standard deviation of the last three weeks for at least a week, that is,

t p e a k = { t   |   X t − i > ( m e a n   { X t − i − 21 , X t − i − 20 , … , X t − i − 1 } + 3 S D   { X t − i − 21 , X t − i − 20 , … , X t − i − 1 } ) , f o r   i = 1 , 2 , 3 , … , 7 } .

This definition is derived from the definition of outlier in regression analysis. These epidemic peaks are local maximums. In addition, it is noticeable that the distance between two successive epidemic peaks must be at least one month. According to this definition, the peaks of Table 3 are obtained for the countries of interest. It is remarkable that except for the peaks of Bolivia – which are almost the same –, the later waves are more acute than previous ones. We must add this point that the more acute peak means the more number of new confirmed cases, therefore the more intense spreading. Finally, it is possible that because of the lack of information at the beginning, this definition misses the epidemic peaks in the initial days.

Mathematically and logically, the number of positive tests (confirmed cases) is affected by the number of tests and positivity rate. The number of cases equals the number of tests multiplied by the positivity rate. Therefore, the increment of the number of cases (as a multiplication) equals the sum of these two:

1. The number of tests multiplied by the increment of positivity rate, and

2. The positivity rate multiplied by the increment of the number of tests.

Consequently, the intense changes in the count of cases are due to at least a remarkable change in one of these multiplications. About the countries with a regular increase in the number of tests like the USA, the increment of the proportion of positivity plays the principal role in the peaks.

Table 3 shows that the proportion of positivity is significantly better than the frequency of tests to indicate the peaks of the pandemic. The positivity rate is more associated with the number of cases than the number of tests (90% vs 45%). After moving average, these proportions reach 100% and 50%, respectively.

Countries of the southern and northern hemispheres faced a peak around July and November, respectively, possibly due to falling temperatures.

Figures 1, 2, and 3 consist of 12 subfigures, each of them belonging to one country. The arrangement of the subfigures in all three figures is identical. The horizontal axes in Figures 1 and 3 represent time in days from the start to the end of the period of study for the studied countries (the fourth and fifth columns of Table 2). The vertical axis of Figures 1 and 2 display the number of new tests – conducted on that day – and the proportion of positive tests – reported that day –, respectively. Figure 3 is the plot of the joint distribution of the number of tests on a day and the proportion of positivity on l a g days later.

Figure 1:

The time series of the number of new tests (daily) in the 12 countries. The peaks of the number of tests coincide with the epidemic peaks of COVID-19 in different countries. USA (r1 c1), India (r1 c2), UK (r2 c1), Italy (r2 c2), Iran (r3 c1), UAE (r3 c2), Bolivia (r4 c1), Guatemala (r4 c2), Nigeria (r5 c1), Australia (r5 c2), South Korea (r6 c1), and South Africa (r6 c2). r: row & c: column.

Figure 2:

The time series of the positivity rate (daily) in the 12 countries. The peaks of the positivity rate coincide with the epidemic peaks of COVID-19 in different countries. USA (r1 c1), India (r1 c2), UK (r2 c1), Italy (r2 c2), Iran (r3 c1), UAE (r3 c2), Bolivia (r4 c1), Guatemala (r4 c2), Nigeria (r5 c1), Australia (r5 c2), South Korea (r6 c1), and South Africa (r6 c2). r: row & c: column.

Figure 3:

Scatterplots of the relationship between the number of tests and the positivity rate. Generally, as the number of new tests increases, the positivity rate falls. USA (r1 c1), India (r1 c2), UK (r2 c1), Italy (r2 c2), Iran (r3 c1), UAE (r3 c2), Bolivia (r4 c1), Guatemala (r4 c2), Nigeria (r5 c1), Australia (r5 c2), South Korea (r6 c1), and South Africa (r6 c2).

Figure 1 shows that the peaks of the number of tests coincide with the epidemic peaks of COVID-19 in different countries. For example, in the USA, there are two peaks of the number of tests simultaneous with the second and the third epidemic peaks – mentioned in Table 3. Also, it is obvious that Bolivia has experienced two peaks for the number of tests around 150th and 300th days – from March 19, 2020 – which coincide with the epidemic peaks in Table 3.

The USA, the UK, and the UAE experienced some regularly rising time series. Except for some overruns in epidemic peaks, the patterns of Italy and South Africa are increasing too. The number of tests in Guatemala is increasing, accompanied by an increasing fluctuation. Owing to the restriction by the limited capacity of tests, Iran and Nigeria followed a stepwise trend. Apart from their peaks – one for each of them –, the plots of Australia and South Korea are stationary. In the case of Bolivia, the time series is proportional to the peaks. India is the only country whose time series is initially increasing, then stable, and after that decreasing. Generally, the counties have an increasing trend.

Figure 4 gives us a classification about the countries from the viewpoint of the number of tests: 1. The USA, 2. India, 3. The UK, 4. Italy, Australia, and the UAE, 5. South Korea, South Africa, and Iran, and 6. Nigeria, Guatemala, and Bolivia.

Figure 4:

The time series of the number of new tests (daily) in the 12 countries. The USA, India, and the UK had the greatest number of tests.

Figure 2 illustrates the time series of the positivity rate of the tests (the ratio of the number of positive tests on a day to the number of taken tests on l a g days ago). It is interesting that the subfigures of Figure 2 are more in accordance with the epidemic peaks than their analogous in Figure 1. For example, it is clear that the USA has encountered three peaks. It is worthwhile that the graph of Iran has three peaks while the first of them is missed in Table 3 because of the lack of information at the beginning. A similar situation (being missed by investigation of either the number of tests or the number of confirmed cases while discovered by the analysis of the positivity rate) happens to the epidemic peak in India in late March, the first and the second peaks of the UK, and the epidemic peaks in middle May and the November for the UAE.

Figure 5 illustrates a classification of the countries based on the positivity rate: 1. Nigeria, Guatemala, and Bolivia, 2. South Africa, and Iran, 3. The USA, India, the UK, and Italy, and 4. Australia, the UAE, and South Korea.

Figure 5:

The time series of the positivity rate (daily) in the 12 countries. Nigeria, Guatemala, and Bolivia had the greatest positivity rates among the 12 countries.

The horizontal and vertical axes of Figure 3 display the number of new tests and the proportion of positivity of them, respectively. Generally, as the number of new tests increases, the positivity rate falls. Since the epidemic peaks are opposing this general rule, it is not very clear to see the opposite direction of the changes. Guatemala, due to lack of epidemic peak, is a good example of this diversely proportional relationship.

If the reason for an increase be the rising number of tests, we expect not to return the previous channel in short term. In addition, the positivity rate does not undertake a remarkable change. On the other hand, it is normal to assume that entering a peak is accompanied by increasing the number of negative tests as well. Consequently, the lack of the growth of negative test results (rising the positivity rate while continuing the previous trend for the frequency of tests) is only reasonable if at least one of the factors of tests accuracy, testing policy, or the viewpoint of the population were changed around that time. Otherwise, there are a remarkable number of un-reported cases belonging the peak. It is noticeable that this company of risings causes the observed acceleration in growth regarding epidemic peaks.

4 Results

The presumptions to apply copula theory for a couple of variables are the existence of continuous marginal distributions accompanied with their correlation. Table 5 investigates whether the pair of the frequency of the tests and positivity rate meets the presumptions. The marginal distributions were obtained in EasyFit. It is observable that the generalized Pareto and Weibull distributions had good performance to fit the positivity rates. Also, the correlation in countries with the highest number of tests is negative and it is commonly between −0.2 and −0.3. In countries lacking enough tests, the correlation coefficient is significantly greater – possibly due to the low quality of data and under-reporting. It is noticeable that calculation over the data of Bolivia, Iran, and South Africa, lead even to positive correlations.

Based on Table 5, we are allowed to look for the suitable copula functions to connect the marginal distributions to find the desired joint distributions for nine of the countries. Notice that the countries without meaningful correlation (Italy, South Korea, and the UAE) were of the countries with the least proportion of positivity of the tests.

Table 6 represents the results of comparing the best candidates from the FGM, Clayton, Frank, and Gumbel families.

According to Table 6, Clayton copulas are suitable candidates for the countries with lower per million tests. In addition, Frank copulas can describe a wide variety of countries. Finally, the Gumbel family seems not to be a good option to couple the variables of the frequency of tests and the positivity rate.

We now discuss the simulation of data from the obtained copula models and perform comparisons between correlations in the simulated data and in the observed data based on 1000 simulations. We follow the simulation method proposed by Johnson (1987) and later Nelson (2006).

Figure 6 illustrates the scatter plots of the transformed observed data versus simulated samples of the CDFs of the frequency of tests and positivity proportion variables taken from the fitted copula models in Table 6. It can be seen that the simulated data and the original data have similar dependence patterns. To settle this concern, Table 6 shows the rank correlations between the frequency of tests and positivity proportion variables calculated from the original data and the simulated data of size 1000 taken from the fitted copula models. By comparing these correlations, we can conclude that the results show strong consistency of the estimated and real correlations.

Figure 6:

Scatter plots of the transformed observed values (·) versus simulated samples (∗) variables from subfamilies of the copula model. USA (r1 c1), India (r1 c2), UK (r1 c3), Iran (r2 c1), Bolivia (r2 c2), Guatemala (r2 c3), Nigeria (r3 c1), Australia (r3 c2), and South Africa (r3 c3). r: row & c: column.

Finally, we want to investigate the structure of dependency between the number of tests and positivity rate totally. By collecting the data of the 12 countries, 3877 pairs are obtained whose Kendall’s correlation is −0.1434 (P-value: 2.8464 * 10ˆ−19). In addition, we split the data into two parts: peaks and otherwise. This split restricted us to applying marginal distributions – then copulas – because it causes the gap in the number of tests. Table 7 represents the Kendall’s correlations for the countries of interest. It is worth saying that the correlation coefficient for the variables (the number of tests and positivity rate) is negative in both peaks and otherwise.

5 Discussion

Generally, at the beginning of an epidemic, the number of tests is low and the proportion of positivity is high. As time passes, the number of tests rises. Also, as the number of new tests increases, the positivity rate falls. The correlation in countries with high number of tests, higher quality of data, is negative and it is commonly between −0.2 and −0.3. By considering all the data as a set, the Kendall’s coefficients are −0.1434, −0.2132, and −0.1381 for total, peaks, and total after removing peaks, respectively. The positivity rate of the tests is significantly better than the frequency of tests to indicate the peaks of the pandemic. The positivity rate is more associated with the number of cases than the number of tests (90% vs 45%).

The proportion of positivity is more proportional than the number of tests to the number of infected cases. Approaching zero positivity rate is a good criterion to scale the success of a healthcare system in fighting against an epidemic. The number and accuracy of tests can play a vital role in the quality level of epidemic data. The policymakers can consider the factors affecting the positivity rate such as the testing policy, restricted facilities, peaks, fluctuations, and so on, and make decisions to prevent misleading because of them. Considering these factors altogether and inputting them in the models like (Khajanchi et al. 2022, 2021; Mondal and Khajanchi 2022) is the correct way to assess function of any intervention of interest.

The first limitation is the low quality of data for some countries because of the restricted facilities, the low number of tests, and non-organized data collection program. Also, some interpolation and moving average methods were applied to find some missing data regarding the countries of interest and calculating the correlation for the countries with poor data. Out of the 12 countries, Iran, South Africa, Nigeria, Bolivia, and Guatemala have got a relatively restricted number of tests. The data of Italy, the UAE, and South Korea showed no significant correlation. The highest quality and most significant correlations belong to the USA, India, the UK, and Australia.

The present approach using copulas is promising since it allows to take into account a wide range of correlation, frequently observed in medical studies. In fact, the classical multivariate models cannot reproduce all type of correlations. Moreover, the standard models are limited, especially because the choice of the marginal distributions is restricted. The crucial step in the modeling process is the choice of the copula function, which best fits the data. Further work is needed to choose the best copulas able to reproduce the dependence structure of bivariate medical variables. In clinical trials or medical studies, sample size is often an important consideration and is relatively small. The copula-based methodology overcomes this limitation as well, because the algorithm can be used to replicate data for any number of patients. The suggested copula-based methodology presented in this paper is simple and easy to implement.