The First and Second Derivative Analysis of the Daily Cumulative

Coronavirus Disease 2019 Cases in Representative European

Countries

Liyan Lou

Ningbo Hanvos Kent School, Ningbo, Zhejiang, 315048, China

Keywords: Covid-19, First Derivative, Second Derivative Analysis, England, Ireland.

Abstract: Coronavirus disease 2019(Covid-19) had spread all around the world in 2020. For most countries, their

epidemic was still not stable. This study would analyze the epidemic of England and Ireland from different

aspects to find the reason for which their infection numbers are still increasing. Moreover, this study would

analyze them from three distinct geological strata, which are England, Ireland, and all countries in the Europe.

The author uses the first and second derivative graph of cumulative confirmed cases from June 1st 2021 to

August 3rd 2021. The data were collected from official websites that report daily confirmed cases. Moreover,

the author analyzes three different strata (England, Ireland, and all countries in the Europe). There are 5929786

cumulative cases in England, 305527 cases in Ireland and 51805897 cases in all European countries until

August 3rd, the change of F’’(x) in England and all European countries could be demonstrated in three periods:

Stable Period, Acceleration Period and Deceleration Period. Furthermore, the Ireland’s epidemic only

contained two Periods, which were Stable Period and Acceleration Period. The Acceleration Period could

reveal the relaxation of restriction, which increased the number of confirmed cases, in both Ireland and

England. Connecting to the local policy, the further vaccination that took place in England help them reduce

the Acceleration rate of confirmed cases.

1 INTRODUCTION

The coronavirus disease 2019(Covid-19), caused by

severe acute respiratory syndrome coronavirus

2(Sars-Cov-2, which had a genetic relationship with

the SARS disease happened in 2003 in China), took

place in China already for one year, and it spread

worldwide by mostly transportation, especially

international flights. Although the first and earliest

case happened on December 8th 2019, which was not

exact since it was the recall of the first patient of

Covid-19, the pandemic was carefully controlled by

start some anti-epidemic actions, quarantine. In

contrast, some other countries had more severe

epidemic situations than Chinese. For instance,

England was the country that had the 6th highest

number of diagnosed cases and 7th highest number of

deaths, These facts get attention globally. Until

August 30th, there were 6698486 diagnosed cases in

England alone, accounting for nearly 12.2 percent of

all cases in Europe. The total confirmed cases in all

European countries except England was

45876111(87.8%) cases. Moreover, Europe had total

number of 51805897 cases, accounting for 25.4% of

the world and ranking the 2nd of all continents.

Europe was the second most dangerous continent

since most European countries had high infection

density. The diagnosed rate was approximately

7.35%, implying that there would be over 70

thousand confirmed cases, second only to North

America (7.95%), where the majority of the cases

were amassed.

England has started several anti-epidemic policies

in order to control the pandemic like started

lockdown in the UK began on 16th March 2020 and

encouraged working from home on 15 April 2020.

Some of these actions definitely had effects, but some

anti-lockdown protests occurred, disturbing the

controlling process (Poppy 2021).

In this essay, the author aims to explain the

transmission of Covid-19 in England by explaining

the second derivative model from June 1st to August

3rd. In order to achieve a precise result, the writer

288

Lou, L.

The First and Second Derivative Analysis of the Daily Cumulative Coronavirus Disease 2019 Cases in Representative European Countries.

DOI: 10.5220/0011343400003437

In Proceedings of the 1st International Conference on Public Management and Big Data Analysis (PMBDA 2021), pages 288-293

ISBN: 978-989-758-589-0

investigated the epidemic in different strata,

including England, Ireland, and Europe Countries.

The result of this study would give suggestions to

the England and Ireland in order to help them monitor

the epidemic better by providing the trend of

pandemic and predict the trend afterward. These

pieces of evidence could help decision-makers

control their countries' pandemics better. The

research also enriches the field of pandemic study in

European Countries.

2 METHOD

Database: The author gain the data of daily confirmed

cases from only one source: Data from June 1st to

August 3rd were derived from an official website that

contains daily reported and officially finalized cases

(Max 2021)(American Library Association 2021).

In order to gain the data for study, the author

derives the daily data: (1) for England only, (2) for

Ireland only and (3) for Europe Only. Details could

be seen in Figure 1:

Figure 1: The number of cumulative confirmed cases of

COVID-19 in England, Ireland, and all European Countries

from June 1st to August 3rd.

To find the second derivative model, the author

firstly derived the cumulative diagnosed cases table,

which was denoted as F(x)

F(x)=



)(

(1)

where x(a) represent the daily confirmed cases in day

a(a=1,2,3...t)

According to this, the author gained the first

derivative F’(x). Since F’(x) means the velocity that

the growth of cumulative cases, everyday’s F’(x)s

were different. The author gain F’(x) from

F’(x)=



)(



)(

(2)

by subtracting the daily cumulative cases, the rate of

change of confirmed cases could be obtained. In

addition, since F’(x) is gained, the author found

F’’(x):

F’’(x)=F’(xa+1)-F’(x) (3)

where F’’(x) means the rate of change of the growth

rate, or the acceleration of confirmed cases.

The bigger F’(x), the faster daily cases increased,

whereas if F’(x) = 0, it means that the diagnosed cases

did not change during that day. In the cumulative

model, the rate of growth of confirmed cases would

always be non-negative since it grew almost every

day.

The First and Second Derivative Analysis of the Daily Cumulative Coronavirus Disease 2019 Cases in Representative European Countries

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When F’’(x) is bigger than 0, it means that the

speed of increasing the number of diagnosed cases is

accelerating, whereas the negative F’’(x) value means

the rate of change in daily new cases is decelerating.

If the F’’(x) = 0, it means the F’(x) does not change

during that time, and it further means the number of

cumulative confirmed cases is evenly increasing,

decreasing or stop growing. F’’(x) always be

represented as a straight line on the F(x) graph.

3 ANALYSIS

3.1 First Derivative Analysis

The first derivative graph of cumulative cases of three

different strata is presented in figure 2. From June 1st

to June 30th, the infection rate in England did not

fluctuate very much, which illustrates very slow

growth of epidemic. However, in July, the rate of

change of reported cases increased to its peak in these

three months. The highest F’(x) value reached 54130

cases per day on July 17th. This may be a result from

the new policy posted by the prime minister on June

14th, which delayed England’s lockdown easing by

four weeks. Moreover, on July 5th, the prime minister

set out a plan to ease Covid-19 restrictions in England

(Sarah 2021. Because of these policies, the infection

rate grew rapidly. After the peak, the infection rate

declined on July 27th, and after a few days increasing,

it went down again at the end of July

Unlike England, the F’(x) lines of Ireland showed

another picture, For the Ireland graph, since Ireland

is isolated from most countries and its lockdown

policy, the number of infections did not grow very

much during this time. It only experienced its peak on

June 28th because Ireland reach its highest rate of

vaccination throughout the whole epidemic. The rate

of vaccination reached 110673 new vaccinations per

day. Because of this, the number of contact among

people increased, which caused a short period of

increase in confirmed cases.

The first derivative graph for the whole of Europe

has an extremely different epidemic pattern from the

other two. It fluctuates numerous times during these

two months. After June 26th, it started to grow

rapidly, and it reached its peak on July 19th and

another people on July 22nd which is a little later than

England. Contrasting to all European countries, the

England and Ireland showed a much flatter pattern,

telling the researchers and decision makers that other

countries in the Europe show strengthen their

countries' anti-epidemic policies.

Figure 2: Epidemic curve of the first derivative of cumulative COVID-19 cases from June 1st to August 3rd in England,

Ireland and Europe.

3.2 Second Derivative Analysis

Figure 3 illustrates the second derivative graph for

different strata.

From the second derivative graph of England,

presented in figure 3, the author concludes that during

these two months, England epidemic experienced

three different period, which is categorized by the rise

or fall of F’’(x).

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First Derivative analysis

England ireland Europe

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Figure 3:

Second Derivative Curve of cumulative COVID-19 cases from June 1st to August 3rd in England.

The first period was called Stable Period (From

June 1st 2021 to June 22nd 2021). During this time,

the variation of F’’(x) did not exceed 2000

cases/day2. The situation was seemingly good to

England, until the lockdown was delayed by the

Prime Minister on June 14th. After that day, the

average change of infection rate became extreme and

intense.

The second period was called Acceleration Period

(From June 23rd 2021 to July 15th 2021). F’’(x)

became much less stable than Stable Period. After 4

days, F’’(x) began to fluctuate, reached its peak on

June 27th. The graph shows the relatively increasing

tendency from June 23rd to July 15th. This result tells

researchers and those decision makers in the UK that

delaying the lockdown time definitely negatively

influenced the England pandemic.

The Last period was called Deceleration Period

(From July 16th 2021 to August 3rd 2021), which

shows a decreasing trend of F’’(x). Some anti-

epidemic actions occurred during the period. Further

vaccination started in this period. Until July 16th,

there was already 35745635 people who had taken

the second vaccination (UK Health Security Agency

2021)(Johns Hopkins University 2021). This event

definitely decrease the growth of infection rate.

Furthermore, from these data, the author knew three

important information: first, the future tendency of

F’’(x) would be more stable and stay between ±2000

if England continues to maintain the same situation,

or more people be vaccinated. Second, if more people

be quarantined, or England continues lockdown,

another decline wave would occur after an incubation

period, which would last for 14 days. Third, the

actions that took place in England after Acceleration

Period led to the decline of F’’(x). Though less

evidence could support this fact, a longer time of

observation is needed to confirm this information.

The tendency of F’’(x) of Ireland was showed in

figure 3. Even though Ireland was the nearest country

to the England, the graph of F’’(x) of Ireland is very

distinct from England’s. It only had two periods:

Stable Period and Acceleration Period. There was no

Deceleration Period during these two months.

Additionally, the lengths of these periods were far

different from England’s. In Ireland, the Stable Period

only lasted for 18 days (From June 1st 2021 to June

17th 2021). After that, later days were considered as

Acceleration Period (From June 18th 2021 to August

3rd 2021). The main reason for which Ireland’s

epidemic during these two months did not have

Deceleration Period was because of the relaxation of

restriction around the end of July and the beginning of

August. According to Tully, the co-chairman of the

Behavior Change Group at Northern Ireland’s Public

Health Agency, he said: “We’ve also seen an increase

in the number of contacts between people here as well,

slightly higher than the rest of the UK.” (Chris 2021).

This fact had negatively affect Ireland’s epidemic.

Moreover, since there were more serious control and

anti-epidemic actions in Ireland than there were in

England, so less fluctuation as shown in the graph.

1160

378

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4441

2573

3212

-3718

8090

3460

1795

6569

-7191

6188

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5173

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England

The First and Second Derivative Analysis of the Daily Cumulative Coronavirus Disease 2019 Cases in Representative European Countries

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Figure 4. Second Derivative Curve of cumulative COVID-19 cases from June 1st to August 3rd in Ireland.

The trend of F’’(x) of all European countries was

showed in Figure 4. Unlike Ireland, the tendency of

all European countries is more similar to England’s.

Because England had the most infections at the time,

any change in England's infection would affect the

tendency of all European countries more than others.

It has three comparable periods to England: the Stable

Period (June 1st to July 3rd), the Acceleration Period

(July 4th to July 21st), and the Deceleration Period

(July 22nd to August 3rd).Its Stable Period was 11

days longer, and it had 5 days shorter Acceleration

Period and 6 days shorter Deceleration Period.

Figure 5. Second Derivative Curve of cumulative COVID-19 cases from June 1st to August 3rd in Europe.

327

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-100

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Ireland

9819

-19215

10491

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9467

10176

17277

36868

17698

-28694

29044

5536

-18536

46542

12687

-52754

39012

-51847

52105

15425

-40873

-60000

-40000

-20000

20000

40000

60000

Europe

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4 CONCLUSION

This first and second derivative analysis illustrates

that the main reason for the increase in the number of

infections was the relaxation of restriction, and that

caused the rise of contact among people. The fact that

all European countries’ F’’(x) tendency had the

unequal length of different periods from England

infer that other countries in Europe had more stable

epidemics than England. Furthermore, the result of

this study could help those decision makers make

more reasonable policies to control the pandemic.

This study, of course, has several limitations.

First, the main study region is England and Ireland,

which means that the author did not pay too much

attention to other countries. As the Covid-19 had

spread to other countries, more studies are needed to

help scientists and medical staff to monitor the

pandemic. Second, fluctuations Ireland’s epidemic

was too small that is hard to compare with other

countries, so it may be more effective to use other

countries' information, such as Russia, to compare

with England. Finally, the study period is relatively

shorter than the whole epidemic, and since the

derivative graph is very sensitive to the change, two

monthly analyses may not elaborate the big picture

comprehensively. Therefore, longer time research is

needed in order to confirm the tendency for each

country.

ACKNOWLEDGEMENTS

This study was conducted by the program “Dynamics

of Disease, taking covid-19 as an example”. The

author sincerely thanks to all people who strove on

study for the Covid-19 and who struggled in

overcoming the epidemic all around the world.

REFERENCES

American Library Association. (2021). Covid-19 statistics

of Ireland

https://www.worldometers.info/coronavirus/country/ir

eland/#graph-cases-daily. Accessed on August 29th

2021.

Chris Baraniuk. (2021). Covid-19: Why is Northern

Ireland’s death rate so high?

https://www.bmj.com/content/374/bmj.n2129.

Accessed on August 29th 2021.

Johns Hopkins University. (2021). Novel coronavirus

pneumonia epidemic analysis platform

https://www.arcgis.com/apps/dashboards/bda7594740f

d40299423467b48e9ecf6 Accessed on August 29th

2021.

Max Roser. (2021).

https://ourworldindata.org/coronavirus#growth-of-

cases-how-long-did-it-take-for-the-number-of-

confirmed-cases-to-double Accessed on October

30th 2021.

Poppy Bilderbeck. (2021). Police Break Up One-Man Anti-

Lockdown Protest

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commons-coronavirus-timeline/#july2021. Accessed

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https://coronavirus.data.gov.uk/details/vaccinations.

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