A Novel Approach to Calculate the Case Fatality Rate of COVID-19 in European countries

As a result of the global epidemic of coronavirus disease 2019 (COVID-19), many European countries and regions have been strongly affected. The case fatality rate (CFR) is the most important index to evaluate the hazards associated with an epidemic situation. We aimed to present a method to calculate the instant CFR and to evaluate and compare the instant CFR of COVID-19 in the four European countries that have been most impacted: Italy, Spain, France and Germany. The daily COVID-19 case data from January 30, 2020, to July 9, 2020 in Italy, Spain, France and Germany were collected from WHO reports. Death time was calculated as the difference between the peak dates of the number of daily con�rmed cases and the number of deaths in each of the four countries. The estimated dates of diagnosis of the declared deaths were a death time prior to the dates of death. The instant fatality rate of COVID-19 was calculated as the ratio of the number of cumulative deaths to the number of cumulative con�rmed cases; these deaths and con�rmed cases occurred on the same estimated dates of diagnosis. As of July 9, 2020, the COVID-19 death time was 6, 4, 6 and 12 days in Italy, Spain, France and Germany, respectively. The instant CFR of COVID-19 was 14.4%~27.6%, 2.2%~14.7%, 8.2%~25.0% and 2.0%~10.5% in Italy, Spain, France and Germany, respectively. The average CFR of COVID-19 was highest in France (16.7%) and lowest in Germany (5.0%). Since late April 2020, the CFR has stabilized at approximately 15%, 20% and 5% in Italy, France and Germany, respectively. Since early June, 2020, the CFR in Spain has stabilized at approximately 11%. We have established a more accurate way to calculate the CFR that may provide a basis for the prevention and control of infectious diseases.


Introduction
Presently the outbreak of coronavirus disease 19 (COVID-19) 1 in China has been well controlled.
Unfortunately, the global epidemic has spread to 212 countries and territories.The number of con rmed COVID-19 cases is sharply increasing every day worldwide and the real-time reports can be tracked on websites provided by international forums such as Johns Hopkins University 2 .At the end of January 2020, the WHO has declared that the COVID-19 became a Public Health Emergency of International Concern (PHEIC) [3][4][5] .As of July 9, 2020, over 12.2 million COVID-19 cases and over 550,000 deaths were con rmed worldwide.Many European countries also had early epidemics after China, but the current situation of some countries is becoming even worse than it was in China.From February 2020, Italy became the rst country to face COVID-19 6,7 .Other European countries, such as Spain, the United Kingdom, France, Germany, Russia and the Netherland, were also caught in the epidemic.
It is important to assess the hazards of an infectious disease since it is a vital concern in epidemiology 7 .The case fatality rate (CFR) is one of most important indexes to re ect the hazard of the infectious disease.It is of great signi cance to estimate and predict the fatality rate of COVID-19 in different countries during the epidemic.It can help us analyze the pros and cons of different countries' responses to the epidemic.It can also provide an important reference to decide on e cient strategies to face the huge public health challenge posed by the epidemic.
Since the COVID-19 rst appeared in December 2019, many scientists and researchers have studied this disease from different scienti c perspectives at different levels, some of which have been related to the CFR.However, we recently found that many studies calculated the CFR directly as the ratio of the number of cumulative deaths to the number of cumulative con rmed cases, without any correction 8 .Giangreco 9 and Stafford 10 reported the CFR of COVID-19 in Italy and Germany, respectively, where the CFR was obtained by dividing the number of cumulative deaths by the number of cumulative con rmed cases.
However, this crude calculation of the CFR could be misleading.When the number of total deaths is divided by the number of total con rmed cases, the calculation ignores that the date of the deaths and the date of initial diagnosis were not on the same day.There was a period of time for the patients underwent the whole COVID-19 disease course, including developing symptoms, being diagnosed and reported, and the nal clinical outcome (death or recovery).There was a time lag between the diagnosis and outcome.The date of death should be a few days later than the date of diagnosis.As the denominator, the total number of con rmed cases contains some cases that would die a few days later, and these deaths should have added to the total number of deaths as the dividend.During a growing epidemic, the nal clinical outcome of most of the reported con rmed cases is unknown.Simply dividing the cumulative reported number of deaths by the cumulative number of reported con rmed cases will therefore underestimate the CFR, especially in the early stage of an epidemic 7,11 .
To avoid the abovementioned biases, we established a method to estimate the instant CFR of COVID-19 12 at different stages.In the present study, the method was expected to resolve the problems of COVID-19 patients with unknown outcomes and the time lag between the diagnosis and outcome.The analysis and comparison of the COVID-19 CFRs in different European countries during the epidemic has a signi cant role in providing a basis for the prevention and control of COVID-19 in Europe.

Data sources and collection
The daily data of newly con rmed cases/deaths/cured cases and the daily data of cumulative con rmed cases/deaths/cured cases of COVID-19 from Jan 30, 2020 to July 9, 2020 in Italy, Spain, France and Germany, were respectively collected from websites of the WHO (https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports),Johns Hopkins University (https://coronavirus.jhu.edu/map.html) and worldometer (https://www.worldometers.info/coronavirus)websites.Two researchers collected the data independently every day and then checked and corrected the data.We collected data from the o cial website, which was considered exempt from approval.

Research design
The con rmed COVID-19 patients would eventually have one of two outcomes, death or recovery.After diagnosis, the patients would rst experience a death time or a recovery time and then reach the end of disease, death or recovery.In the present study, death time was de ned as the period of time from the diagnosis date to the death date.Similarly, recovery time was de ned as the period of time from the diagnosis date to the recovery date.The death time was calculated as the difference between the peak dates of the number of deaths and the number of con rmed cases in the four different countries.The estimated dates of diagnosis of the declared deaths cases were one death time earlier than the dates of death.The instant fatality rate for COVID-19 was calculated as the ratio of the number of cumulative deaths to the number of cumulative con rmed cases, and these deaths and con rmed cases occurred at the same estimated dates of diagnosis.

Data analysis
The daily variation histograms of con rmed cases and deaths in the four different countries were plotted.
Their trend curves of these histograms were tted, and the dates corresponding to the highest point of the trend curves were found.The difference in the peak dates between the number of deaths and the number of con rmed cases on the trend curves was considered as the mean lag time from diagnosis to death, namely, death time.On the basis of the above lag times (death time), the initial diagnosis dates of the declared COVID-19 death from daily reports were calculated.
The estimated diagnosis date of declared deaths were n days prior to the death date.That is, the estimated diagnosis date (EDD) of the deaths was the date n days before the reported death date.The estimated diagnosis date (EDD) of deaths was obtained from the death date reported on the daily noti cation data (DND) minus the death time (DT).(Fig. 1).

EDD of deaths = date on DND -DT
The daily CFR of the four countries was calculated from the number of cumulative deaths (NCD) divided by the number of cumulative con rmed cases (CCC).

CFR = NCD / CCC × 100%
Since recovery and death are a pair of competing events, the recovery rate and CFR are opposing gures.The daily recovery rates (RR) were calculated from the following equation: RR = 1-CFR The instant incidence rates were calculated on the basis of the number of cumulative con rmed cases in the population, and the instant mortality rates were calculated on the basis of the number of cumulative deaths in the population in each of the four different countries, respectively.

Statistical analysis
GraphPad Prism 8 and Excel 2016 were used to record, calculate and analyze the data, and to draw gures of different patterns.The unit of time for the data collection of COVID-19 epidemic was a day.Fit

Incidence rates and mortality
The incidence rates and mortality rate of COVID-19 in Italy, Spain, France and Germany are shown in Fig. 3. From late January to early March, the incidence rates among these countries grew slowly, and then they accelerated from early March.Italy had the highest incidence rate among the four countries before late March, and Spain overtook Italy in late March to have the highest incidence rate of 523/1,000,000.
The incidence curves in France and Germany were similar, with incidence rates of 228/1,000,000 and 225/1,000,000, respectively.The mortality rate increased slowly from late January to early March, and then began to accelerate.The mortality rate curves in Italy and Spain are similar to the growth track of the incidence curve.In late March, Spain became the country with the highest mortality rate of 64/1,000,000.France's mortality has been much higher than that of Germany, which had the slowest growth rate and the lowest mortality rate of 10.6/1,000,000.The mortality rates of Italy and France were 57/1,000,000 and 44/1,000,000, respectively.

The lag time
There was a lag time from diagnosis date to death date, namely, death time.Figure 4 shows daily variation histograms of con rmed cases, and deaths.The trend curves of the histograms were tted, and the dates that corresponded to the highest point of the trend curves were found.The dates of the highest point in the trend curves of daily con rmed cases, and the deaths were March 24, and March 30, 2020, respectively, in Italy; 31 March and April 4, 2020, respectively, in Spain; April 3 and April 9, 2020, respectively, in France; and April 3 and April 15, 2020, respectively, in Germany.The lag times from diagnosis to death (death time) were 6 days, 4 days, 6 days, and 12 days in Italy, Spain, France, and Germany, respectively.

Estimation of diagnosis dates of the deaths
The con rmed COVID-19 patients had one of two outcomes: death or recovery.From the above estimation, we assumed that the average death time of COVID-19 in Italy was 6 days, that is, the initial diagnosis date of the declared deceased patients should be an average of 6 days prior to the death date.
In the same way, the diagnosis dates of declared deaths were estimated to be 4 days, 6 days and 12 days earlier prior to the death date in Spain, France and Germany, respectively.Figure 5 shows the curves of the numbers of cumulative deaths and the number of cumulative con rmed cases of COVID-19 on the same estimated diagnosis dates.The deaths were not diagnosed on the same date as con rmed cases.Instead, the deaths were initially diagnosed as con rmed cases a period of time (death time) ago.
Therefore, the number of daily cumulative con rmed cases should be the cases diagnosed on the same date when the deaths were diagnosed as con rmed cases.

Instant CFRs
Figure 6 shows that instant CFRs and recovery rates of COVID-19 in the four countries.In Italy, the instant CFR ranged from 14.4% to 27.6%, and the average CFR was 16.0%.In late April, 2020, the CFR stabilized at approximately 15.0%.In Spain, the highest instant fatality rate was 14.7%, the lowest fatality rate was 2.2%, and the average CFR was 11.5%.In early June, 2020, the CFR stabilized at approximately 11.0%.In France, the highest CFR was 25.0%, the lowest CFR was 8.2%, and the average fatality rate was 16.7%.In late April, 2020, the CFR stabilized at approximately 20.0%.In Germany, the highest instant fatality rate was 10.5%, the lowest CFR was 2.0%, and the average CFR was 5.0%.As of late April, 2020, the CFR stabilized at approximately 5.0%.

Instant recovery rates
The trends in the recovery rate and fatality rate are opposite because they are a pair of competitive events.In Italy, the highest instant recovery rate was 85.6%, the lowest recovery rate was 72.4%, and the average recovery rate was 84.0%.In late April, 2020, the recovery rate has stabilized at approximately 85.0%.In Spain, the highest instant recovery rate was 97.8%, the lowest cure rate was 85.3%, and the average recovery rate was 88.5%.In early June, 2020, the recovery rate stabilized at approximately 89.0%.
In France, the highest instant cure rate was 91.8%, the lowest cure rate was 75.0%, and the average recovery rate was 83.3%.In late April, 2020, the recovery rate stabilized at approximately 80.0%.In Germany, the highest instant recovery rate was 98.0%, the lowest recovery rate was 89.5%, and the average recovery rate was 95.0%.In late April, 2020, the recovery rate stabilized at approximately 95.0% (Fig. 6).

Discussion
In view of the misunderstanding of the CFR calculation in many published articles, we have, for the rst time, established a method to calculate the instant CFR of COVID-19.This method resolved the problems of COVID-19 patients with uncertain outcomes and the time lag between the diagnosis and outcome.By using this new calculation, we determined and compared the CFRs of COVID-19 in Italy, Spain, France and Germany, which are the European countries where the epidemic was most severe.The results showed that the average CFR of COVID-19 was highest in France and lowest in Germany.
At present, the global epidemic of COVID-19 is still ongoing.Assessing the hazards of infectious diseases is a vital concern in epidemiology 13 .Generally, an accurate CFR of an infectious disease can be obtained after the epidemic is over on the basis of the number of total deaths and the number of total con rmed cases.However, the number of con rmed cases, deaths and cured cases are constantly changing 14 during the epidemic.
Recently, some scientists reported the "CFR" of COVID-19 in their publications.However, we found that this crude CFR 9,10 was mostly obtained by dividing the number of cumulative deaths by the number of cumulative con rmed cases, which would underestimate the real CFR of COVID-19.For example, Zhang extracted a total of 72,314 COVID-19 cases reported in China's Infectious Disease Information System 15 .
There were a total of 1,023 deaths out of 44,672 con rmed cases for an overall CFR of 2.3%.While this may seem reasonable on the surface, the fact that there were still 38,909 hospitalized cases among the con rmed cases was ignored.The outcomes of these hospitalized patients remained uncertain.Some patients died a few days later.Therefore, this number of future deaths should be added to the dividend when calculating the CFR.In addition, Sun et al. reported that the CFR was 2.1% (361/17,205) 16 in China, which also misled the public as it was estimated by using the same calculation as Zhang et al.These results may represent a large deviation from reality.Interestingly, Yang et al. used a simple linear regression model to estimate the CFR of COVID-19 in mainland China 17 ; the reported CFR was 2.10% (95% CI: 2.05-2.14%).However, at present, the outbreak has been basically controlled in China.Therefore, the CFR can be obtained by dividing the total deaths (4,644) by the total con rmed cases (84,451); the current estimated CFR in China is approximately 5.5%, which is obviously higher than expected in all above papers.Therefore, estimating the CFR is challenging 18 .
We assumed the CFR during the epidemic process to be a collection of many successive instant CFRs.We set the instant unit of the CFR as one day and calculated the daily CFR.The daily CFR was equal to the number of daily cumulative deaths being divided by the number of daily cumulative con rmed cases.The difference from the above calculation is that the two numbers in the equation (deaths and con rmed cases) occurred on the same estimated diagnosis date.The deaths (dividends) were not diagnosed as con rmed cases (divisors) on the calculation date.Instead, they were diagnosed as con rmed cases a period of time (death time) ago.Therefore, in the CFR equation, when the dividend was the number of daily cumulative deaths, the divisor should be the number of daily cumulative con rmed cases diagnosed on the same date when the deaths (in the dividend) were diagnosed as con rmed cases.We used this method to estimate the CFR (5.7%) of COVID-19 in China 11 ; the estimated CFR is close to its actual value (5.5%), supporting that this method is reliable.COVID-19 is a topic of general concern to all humankind.Due to the differences in attention, medical facilities, public consciousness of self-protective measures, and economics, there are some differences in the epidemic situation in different countries.The number of con rmed cases/deaths/recovery cases is changing every day, so it is di cult to use a simple number or average number to show the mortality rate and CFR.Hence, we used the instant CFR approach to estimate the impact of the epidemic and changes in the hazards associated with COVID-19 in different European countries.
According to the different levels of the CFR in different places, we are able to determine the status of the COVID-19 situation in different countries.The CFR could be an important reference and indicator for realtime measurement of epidemic inside a country and even for comparison among different countries.Germany has shown the lowest CFR of COVID-19, demonstrating that Germany has provided the best medical facilities for COVID-19 among these four European countries.France has the highest average CFR (16.7%) and it has surpassed that in Italy (16.0%) after June, indicating that the French needs to take more stringent measures to control COVID-19.Meanwhile, France has maintained the highest instant CFR after stabilization (20.0%).Since late April, 2020, the instant CFR has stabilized at approximately 15% in Italy, and 20% in France, respectively, which means that the French and Italian people and government should pay more attention to the epidemic right now.
During the outbreak of severe acute respiratory syndrome (SARS) in 2003, a total of 8,422 SARS cases were reported in 32 countries and regions.The global death toll from SARS was 919, with a CFR of nearly 11% for SARS worldwide.However, the current total number of COVID-19 cases and deaths for now is far more than the respective reported numbers for SARS.The SARS epidemic was most severe in China (Beijing, Hong Kong) and some countries in Southeast Asia, while SARS epidemic in Europe was relatively mild, and SARS had no widespread impact in European countries.The number of SARS cases in Italy, Spain, France and Germany was 4, 1, 7, 9, respectively, out of which there was only 1 death (from France).As a comparison, as for June 18, 2020, the instant (average) CFR of COVID-19 was 14.4%~27.6%(16.0%), 2.2%~14.7%(11.0%), 8.2%~25.0%(16.7%) and 2.0%~10.5 (5.0%) in Italy, Spain, France and Germany, respectively.Obviously, the current status of the COVID-19 epidemic in Europe is much more serious than it was for SARS in 2003.We have also determined that the incidence rates and mortality rates of COVID-19 for the four European countries are signi cantly higher than those for SARS in 2003.It seems that even though COVID-19 was rst seen in Wuhan, China, the situation of China is currently well YXC and LC were responsible for the conceptual design and led the team.QQ performed the modeling, responsible for data collection and management.QQ, LC and YXC drafted the manuscript, revised the manuscript.All authors read the manuscript and contributed to editing.

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