A Critical Analysis of Corona Related Data: What the More Reliable Data Can Imply for Western-Europe - applied sciences
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applied
sciences
Article
A Critical Analysis of Corona Related Data: What the
More Reliable Data Can Imply for Western-Europe
Robert J. Meier
Pro-Deo Consultant, 52525 Heinsberg, North-Rhine Westphalia, Germany; r.meier@planet.nl
Received: 17 April 2020; Accepted: 7 May 2020; Published: 14 May 2020
Abstract: We present a less common type of discussion about COVID-19 data, beginning with the
observation that the number of people reported deceased following COVID-19 infection is currently
the most reliable dataset to be used. When the available real-life data are visualized for a number
of European countries, they reveal the commonly seen exponential increase, though with different
absolute rates, and over time different periods. More interesting information is obtained upon
inspection of the daily increments in deaths. These curves look very similar to those for China,
and seem to indicate that in European countries that have imposed more strict human–human contact
measures, in particular Italy and Spain, where we have seen a decrease in daily deaths since early
April, it is to be expected it will take 40–50 days from the end of March until this number has fallen to
negligible levels. Taking the initial increase in the number of deaths for Germany, and combining this
with typical values for the mortality reported in the literature and the published number of daily
contacts for the working population, we calculated an initial increase in infections of 20 per day by
a single infected person with an average human–human contact number of 22, decreasing to 5.5
after the first 10 days. The high number at the outset is likely related to outbreaks in a high local
concentration of people.
Keywords: coronavirus 19; CoV-19; critical data analysis; similarity
1. Introduction
Since the beginning of 2020, we have witnessed the pandemic caused by coronavirus-19 (CoV-19
or SARS-CoV-2), a virus structurally related to the SARS (severe acute respiratory syndrome) virus.
The coronavirus can cause acute respiratory diseases (COVID-19) and it has been reported that its
spreads 0.8%–3.0% more than normal influenza [1]. When the virus infects the respiratory tract,
it induces the release of pro-inflammatory cytokines. The binding of CoV-19 to the TLR (Toll-like
receptor) causes release of such pro-inflammatory components, and therefore one of the therapies
to suppress lung inflammation is to suppress the pro-inflammatory species [2]. Conti et al. recently
published several papers in which a large number of factors, including mechanisms, prohibitive
actions, and specific sensitivity (e.g., gender and pre-vaccination) are summarized [1–3]. Regarding the
origin of this type of viruses, there seems to be a common understanding that they originate from
zoonic transfer [2]. SARS-CoV-2 is the seventh coronavirus known to infect humans [4]. From a
comparative analysis of genomic data, Anderson et al. [4] reported an analysis claiming clear evidence
that SARS-CoV-2 is not a laboratory construct or a purposefully manipulated virus, and that wild
animals such are bats are the more likely origin. Coronaviruses similar to SARS-CoV-2 have been
identified in such wild animals.
In addition to papers on the mechanism of the action of viruses including CoV-2-SARS,
the recent CoV-2-SARS pandemic has obviously led to a large number of publications by public
health organizations, including the WHO, and magazines and newspapers reporting the number of
people who tested positive for the COVID-19 virus, the number of people deceased, and, for many (but
Appl. Sci. 2020, 10, 3398; doi:10.3390/app10103398 www.mdpi.com/journal/applsciAppl. Sci. 2020, 10, 3398 2 of 11
not all) countries, the number of people who have recovered. Both the press and official institutions in
different countries have issued daily statements on the situation regarding the spread of the coronavirus
CoV-19. Some of these statements erroneously suggested the situation in a country was stabilizing or
improving, and often had to be corrected several days later when infections were reported to have
increased. The same holds for the daily number of deaths. From a forward-looking perspective, it is
therefore more useful to look at the development over time. This is the objective of various scientific
publications, e.g., Ref. [5], in which the cumulative number of infected people, in different countries,
is used to describe the evolution of the spread of the infection from the outset. When the behavior of
the virus, from its initial spread until the point at which transmission has fallen to practically nil (such
as the case of China), is known fully, the data can be fitted to models, such as those referred to in [5] or
to the generalized Richards model [6], which is an extension of the original Richards model [7].
In the present work, we take a somewhat different approach. Because the situation in Europe
remains one characterized by further spreading, we will not fit models describing the full behavior, as
too many parameters need to be fitted that would lead to arguable predictions. Therefore, we look at
the time-evolution of real data reported thus far, using the number of deaths, which, as we will argue,
constitutes the more reliable data set.
2. Data Selection and Approach
One of the potential problems with analyzing the time evolution of the spread of the CoV-19 virus
is that the number of new infections is often taken as the basis for such statements, but it is questionable
whether this is the best way to monitor what is happening and what may eventuate. In the present
work, which spans the timeframe 10 March until 24 April 2020, we present a discussion starting from
what should be considered the more reliable data (to be discussed below), even though uncertainties
always remain. We focus on a number of West European countries, including those that are heavily
suffering from COVID-19. Although Italy had the misfortune to be first in line when CoV-19 reached
Europe, it could have served as an early warning for other nations. However, next to Italy, multiple
countries have had a relatively large number of deceased compared to their population size, including
Spain, France, the United Kingdom and the Netherlands. Based on the number of deceased per
inhabitant over the course of time (or mortality), the outbreak in Spain, France, and the Netherlands,
to name a few, was in essence as heavy as in Italy. On the contrary, Germany had, relatively speaking,
significantly fewer deaths from CoV-19. This is also true of Austria. We will not discuss the possible
reasons for these differences; this stage of the disease does not permit detailed comparative analyses
since the available data consist of the number of infected patients in different geographic areas with
different social, political, and economic structures [5].
Statements on whether the effects of the virus slow down can only be made on sufficient and
proper data. All of the data used in this paper were taken from the publications on the website of
the Berliner Morgenpost [8], with the data on that site originating from John Hopkins University [9],
the German Robert Koch Institute (RKI) [10], and the various health organizations in the different
regions of Germany (see the RKI reference [10]). The collected data can also be found on the WHO
site [11] or that of the ECDC (European Center for Disease Prevention and Control) [12]. The data
published comprise (i) number of confirmed infections, (ii) number of people reported healed, and (iii)
number of people deceased.
The number of confirmed cases, i.e., the number of people tested positive, is highly dependent on the
number of tests performed in each individual country, and this varies largely. Several news agencies
reported on the low number of tests in the Netherlands, e.g., the live-blog coronavirus of the Dutch
newspaper Het Parool at 13:45, 1 April 2020, reported: ‘Germany carries out 70,000 tests per day,
whereas in the Netherlands this number was as low as about 1000 tests per day.’ On 1 April, there were
less than 13,000 persons who had tested positive in the Netherlands, whereas more than a thousand
had died, implying that the mortality resulting from corona infections was about the same as in Italy
and Spain [13]. A mortality rate of 2%–3% has recently been reported [2], suggesting that for EuropeanAppl. Sci. 2020, 10, 3398 3 of 11
countries this was also a clear indication that testing was, thus far, inadequate. John P.A. Ioannidis,
a professor of epidemiology at Stanford University, has branded the data we have about the epidemic
“utterly unreliable”. “We don’t know if we are failing to capture infections by a factor of three or 300,”
he wrote recently [14]. A recent study on data from Iceland support this view [15]. Data from Iceland,
a relatively isolated country, are ideal for specific studies and revealed that, whereas there was much
more testing than elsewhere (5% of the population was tested, whereas for other countries this was
0.1% or often less), around 50% of those tested positive had no symptoms at all. Moreover, Iceland’s
strategy does not include a lock-down but extensive testing and contact tracing, enhanced by a very
aggressive policy of quarantine for individuals that are or might be infected.
There is another reason not to undertake discussions on the basis of the daily growth of infected
people, as it seems inconsistent with policies followed in several countries (e.g., Sweden, as well as the
first stage in the Netherlands, where the policy was changed later), namely, aiming at mass immunity.
It has been reported in multiple countries that eventually 60%–70% of the population will be infected,
but this must be achieved in a controlled way in order not to overload the medical sector. Nonetheless,
the aim would be mass immunity, which implies the majority of people are infected and requires,
in turn, an increase, rather than a decrease, in the number of infected people.
Finally, a genuine fall in the true number of infections (a number we do not know), inevitably
implies a decrease in the number of deaths. There will be a time-lag of several weeks between
these events, comprising the weeks during which a person is tested, subsequently taken to hospital,
and eventually dying. Relaxing safety measures too early, based on highly unreliable infection rates
and disregarding the possibility of a new local outbreak, is therefore a serious risk.
The second set of data available regards the number of people healed. As it is generally not compulsory
to report when a person recovers from the coronavirus, the spread between countries can be extremely
large: the collected data set we consulted reports almost no official recoveries for the Netherlands,
whereas in other countries this is up to 25%–50% of those infected. Consequently, these numbers are
the least useful to be used for analysis.
Infectious diseases are present year-round globally, with the flu one of the more deadly examples,
infecting numerous people on any given day. However, the stand-out characteristic of COVID-19 is the
associated high number of deaths that have occurred in a relatively short period of time. Thus, at the
present time, in the absence of extensive testing of the populations of all countries, the most reliable and
most relevant analysis needs to be made using the number of people that have died (measurement errors and
individual cases where people have died from, for example, a heart-attack, while also being infected
with coronavirus, are often not counted as a coronavirus-related death; however, such deviations
are likely to be considered as systematic deviations and therefore do not modify the overall trends).
With an increasing number infections, the number of deaths will also increase; thus, the mortality will,
in a first-order approximation, reflect true infection levels. Consequently, to avoid statements that
would more likely involve some kind of speculation, we believe the focus should be placed on the
number of deaths.
As mentioned earlier, models aiming at describing the entire cycle from beginning to end require
a series of parameters to be fitted [6,16]. When the outbreak of the infection is still in the exponential
growth state, this fitting of the overall curve may lead to different predictions for when the spread will
flatten, depending on parameter choice (as in many kinds of modelling of experimental data, purely
mathematical fitting and selecting the best fit as the best solution does not necessarily reflect reality).
Therefore, for the European data we discuss in the present work, we only apply exponential fitting
(in this work we use an exponential form for the mathematical function Np , where p is the exponent;
a quadratic function is defined when p = 2 and a cubic function when p = 3) for the initial period
when the infection rate is still high, as this is appropriate for the circumstances in Western Europe until
24 April (the last date for which we show collected data), although for some countries we see the onset
of a change as we will show below. It is important to note that the fitting of these curves is only to
reveal their steepness; in other words, the curves reveal exponential behavior, and the precise functionAppl. Sci. 2020, 10, 3398 4 of 11
has no concrete physical meaning at this moment. An exception is discussed in Section 4 (‘More on
Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 11
infection rate’).
The way we have analyzed the data is primarily by critical assessment and visual inspection
The way we have analyzed the data is primarily by critical assessment and visual inspection of
of the time series. Data visualization can be a very useful and powerful tool, as recognized from
the time series. Data visualization can be a very useful and powerful tool, as recognized from practical
practical experience
experience in the process
in the process industry.
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for example, is such aissoftware
such a software
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successfully usedused for very
for very manymany
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in thatin that
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[17]. The[17].
onlyThe only mathematics
mathematics applied inapplied in the
the current
current work are the exponential curves overlaying experimental data to show
work are the exponential curves overlaying experimental data to show exponential growth. Often exponential growth.
Often statistics
statistics is applied
is applied to determine
to determine the levelthe level of correlation,
of correlation, e.g., by the
e.g., by calculating calculating thecoefficient
correlation correlation
R2. This,R2however,
coefficient . This, however, is not always
is not always useful,useful, and depends
and depends very very
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as discussed
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elsewhere ononthethebasis
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real-lifepractical
practical examples [18].
examples [18].
3. Results from
3. Results Time
from TimeSeries
SeriesofofNumber
Number of
of People Deceased
People Deceased
LetLet
usus first
firstlook
lookatata aplot
plotofofthe
the data
data obtained
obtained on on China
China(Figure
(Figure1).1).The
Thegenerally
generally assumed
assumed
exponentialmodel
exponential modelfor for virus
virusoutbreaks
outbreaks (see, e.g.,e.g.,
(see, [6,16]) nicely
[6,16]) fits thefits
nicely actual
the number of deaths.of
actual number Asdeaths.
we
know from the press, the Chinese government decided to impose a lock-down
As we know from the press, the Chinese government decided to impose a lock-down for Wuhan and for Wuhan and some
other
some cities
other as of
cities as 23 January
of 23 2020.
January From
2020. FromFigure
Figure1, during thethe
1, during three weeks
three following
weeks the the
following lock-down,
lock-down,
the number of deaths increased because they were already infected, and, obviously,
the number of deaths increased because they were already infected, and, obviously, people were people were alsoalso
infected in other parts of China that were not in lock-down status. Thereafter, however,
infected in other parts of China that were not in lock-down status. Thereafter, however, the number of the number
of daily deaths began to drop, as shown by a clear bend in the curve. The number of deaths has
daily deaths began to drop, as shown by a clear bend in the curve. The number of deaths has remained
remained at around 3300 for a country with more than a billion inhabitants (in the interim, this
at around 3300 for a country with more than a billion inhabitants (in the interim, this number has been
number has been corrected to 4632 deaths, which remains a low number compared to the number of
corrected to 4632 deaths, which remains a low number compared to the number of inhabitants).
inhabitants).
4
number of people deceased
3
Thousands
2
1
0
0 20 40 60 80
d a y s (sta r ti n g J a n 2 7 )
Figure
Figure 1. 1.
TheThe numberofofdeaths
number deathsin inChina
China asas of
of 27
27 January
January2020
2020(red
(redsquares)
squares)and
anda cubic function
a cubic fit fit
function
(separate solid, green, curve) to the first part of the data set. The cubic function reads N 3/15, where N
(separate solid, green, curve) to the first part of the data set. The cubic function reads N3/15, where N is
theisnumber
the number of days
of days fromfrom
the the start
start (27(27 January
January 2020).
2020).
Turning
Turning to Western
to Western Europe,
Europe, data
data forfor several
several countries
countries areare shown
shown inin Figure2.2.ItItisisclearly
Figure clearlyseen
seenthat
thethat the number
number of deaths
of deaths steadily
steadily rises
rises for for each
each country,
country, untiluntil the last
the last dayday of this
of this study
study i.e.,i.e.,
24 24 April
April 2020.
2020. In all countries we see a clear exponential increase from the outset.
In all countries we see a clear exponential increase from the outset.
Selected data for a few countries are shown in Figure 3. The fits shown for France and the
Netherlands are cubic functions (N3 ), and similar behaviour is observed for the other countries.
These curves reveal, when compared to the experimental data (number of deaths), that the death rate
indeed shows exponential behaviour as commonly assumed in a virus outbreak situation. What may
appear in Figure 3 as an eventual flattening is often an optical illusion, a well-known phenomenon.
Furthermore, the exponential fit to the curve for France for the first 23 days then becomes steeper,Thousands
number of people
Italy
Spain
Franc e
10 Germany
Netherlands
Appl. Sci. 2020, 10, 3398 UK 5 of 11
0
0 10 20 30 40 50
whereas around day 35 we see a gradual decrease again. Similar observations can be made for the data
da y (sta rt M a rch 10)
on Germany, for example. Nonetheless, in all countries the growth is still exponential (see also specific
discussions for Germany further below, viz. Section 4).
Figure
Appl. Sci. 2020,2.10,
Number of deaths
x FOR PEER (ordinate) in several European countries (as indicated) during the period5 of 11
REVIEW
10 March–24 April 2020. All curves show exponential behaviour from the outset, with possible
30
indication of flattening at a later stage (see text for discussion).
number of people deceased
Selected data for a few countries are shown in Figure 3. The fits shown for France and the
Netherlands are cubic functions (N 20 ), and similar behaviour is observed for the other countries. These
3
Thousands
curves reveal, when compared to the experimental data (number of Italy
deaths), that the death rate indeed
shows exponential behaviour as commonly assumed in a virus outbreak Spain situation. What may appear
in Figure 3 as an eventual flattening is often an optical illusion, Franc e a well-known phenomenon.
10
Furthermore, the exponential fit to the curve for France for the Germany first 23 days then becomes steeper,
Netherlands
whereas around day 35 we see a gradual decrease again. Similar observations can be made for the
UK
data on Germany, for example. Nonetheless, in all countries the growth is still exponential (see also
specific discussions for Germany 0further below, viz. section 4).
0 10 20 30 40 50
The cubic function for France is N3/3 and for the Netherlands is N3/12, where N the number of
da y (sta rt M a rch 10)
days from the start (10 March 2020).
Figure 4 shows in more detail the data for the three most seriously affected German regions to
Figure
2. 2. Numberofofdeaths
Number deaths(ordinate)
(ordinate) in in several
several European
European countries (as indicated) during thethe
period
Figure
date, in addition to the Netherlands. Independent of thecountries
number(as ofindicated) during
people tested period
positive, the
10 10 March–24
March–24 April
April 2020.
2020. All
All curves
curves show
show exponential
exponential behaviour
behaviour from
from the outset,
the with
outset, possible
with possible
number of deaths in the Netherlands was initially almost 10 times higher per inhabitant compared to
indication of flattening at a later stage (see text for discussion).
indication
Germany, of flattening
whereas in theatlater
a later stagethis
phase (seedecreased
text for discussion).
to a factor of about 5.
Selected data for a few countries are shown in Figure 3. The fits shown for France and the
25
Netherlands are cubic functions (N3), and similar behaviour is observed for the other countries. These
curves reveal, when compared to the experimental data (number of deaths), that the death rate indeed
shows exponential behaviour20as commonly assumed in a virus outbreak situation. What may appear
number of people deceased
in Figure 3 as an eventual flattening is often an optical illusion, a well-known phenomenon.
Furthermore, the exponential 15fit to the curve for France for the first 23 days then becomes steeper,
Thousands
whereas around day 35 we see a gradual decrease again. Similar observations
Model f it Nether lands can be made for the
data on Germany, for example. Nonetheless, in all countries theFrgrowth anc e
is still exponential (see also
10 Model f it Fr anc e
specific discussions for Germany further below, viz. section 4).
Netherlands
The cubic function for France is N3/3 and for the Netherlands is N3/12, where N the number of
Ger many
5
days from the start (10 March 2020).
Figure 4 shows in more detail the data for the three most seriously affected German regions to
date, in addition to the Netherlands.
0 Independent of the number of people tested positive, the
0 10 20 30 40
number of deaths in the Netherlands was initially almost 10 50
times higher per inhabitant compared to
d a y (sta rt M a rch 10)
Germany, whereas in the later phase this decreased to a factor of about 5.
Figure
Figure 3. 3.Number
Numberofofdeaths
deaths (ordinate)
(ordinate) during
during the
the period
period10 10March–24
March–24April
April 2020
2020forfor
France, thethe
France,
Netherlands, and Germany, 25as well as exponential fits, in the present case cubic, to the data for France
Netherlands, and Germany, as well as exponential fits, in the present case cubic, to the data for France
and the Netherlands. The cubic fit to the German data is shown and discussed at a later stage in this
and the Netherlands. The cubic fit to the German data is shown and discussed at a later stage in this
paper (section 4). 20
paper (Section 4).
number of people deceased
The cubic function for France
15 is N3 /3 and for the Netherlands is N3 /12, where N the number of
Thousands
Model f it Nether lands
days from the start (10 March 2020).
Fr anc e
Figure 4 shows in more detail
10 the data for the three most seriously affected German regions to
Model f it Fr anc e
date, in addition to the Netherlands. Independent of the number Netherlands
of people tested positive, the number
of deaths in the Netherlands was5
per
initially almost 10 times higherGer manyinhabitant compared to Germany,
whereas in the later phase this decreased to a factor of about 5.
0
0 10 20 30 40 50
d a y (sta rt M a rch 10)
Figure 3. Number of deaths (ordinate) during the period 10 March–24 April 2020 for France, the
Netherlands, and Germany, as well as exponential fits, in the present case cubic, to the data for France
and the Netherlands. The cubic fit to the German data is shown and discussed at a later stage in this
paper (section 4).Appl. Sci. 2020, 10, 3398 6 of 11
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 11
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 11
6
6
deceased
deceased
4
Thousands
of people
4
Thousands
of people
number
2
2
number
0
0 10 20 30 40 50
0 day (starting March 10)
0 10 20 30 40 50
day (starting March 10)
Figure
Figure 4. 4. Numberofofdeaths
Number deaths(ordinate)
(ordinate) during
during the
the period
period1010March–24
March–24April
April2020
2020in in
three German
three German
‘Bundesländer’ (federal
Figure 4. Number states)—North-Rhine
of deaths Westphalia
(ordinate) during Westphalia (red
the period 10 squares),
March–24 Bavaria
April (green
2020 in diamonds),
three diamonds),
German
‘Bundesländer’ (federal states)—North-Rhine (red squares), Bavaria (green
and Baden-Wűrttemberg (blue triangles)—withWestphalia
Germany (brown open squares)
Bavariaand the Netherlands
and‘Bundesländer’ (federal (blue
Baden-Wűrttemberg states)—North-Rhine (red squares),
triangles)—with Germany (brown open squares) (green
and thediamonds),
Netherlands
(pink diamonds).
anddiamonds).
Baden-Wűrttemberg (blue triangles)—with Germany (brown open squares) and the Netherlands
(pink
(pink diamonds).
Compared to other infectious diseases, the global problem centers on mortality and thus the total
Compared to other infectious diseases, the global problem centers on mortality and thus the total
number of people other
that could die from this infection.
the globalTherefore, the bottom line is that the number of
numberCompared
of peopleto that couldinfectious
die fromdiseases,
this infection. problem centers
Therefore, on mortality
the bottom line isandthatthus
the the total
number of
deaths
number should stabilize
of people thatand and
could subsequently decrease
die from thisdecrease over
infection.over time.
Therefore, From curves like those shown before,
deaths should stabilize subsequently time. the
Frombottom
curveslinelike
is that theshown
those numberbefore,
of
namely,should
deaths Figures 2–4, this
stabilize cannot
and be observed
subsequently in an unambiguous
decrease over time. Fromwaycurves
unlesslike
thethose
process has reached
shown before,
namely, Figures 2–4, this cannot be observed in an unambiguous way unless the process has reached
the current
namely, state 2–4,
Figures of China, as shown
this cannot in Figure in
be observed 1. To
an reveal whetherway
unambiguous stabilization
unless the has begun,has
process wereached
display
thethe
current
dailystate of China,
increments as shown
in Figure in Figure
5, i.e., 1. To reveal
the cumulative whether
number stabilization
of stabilization
deaths has begun,
on a certain daywe we display
minus the
the current state of China, as shown in Figure 1. To reveal whether has begun, display
thecumulative
daily increments
number in
ofFigure
deaths 5,
oni.e.,
the the cumulative
previous day. number
These data of deaths
show thatonfor a certain
the day
Netherlandsminus
and the
the daily increments in Figure 5, i.e., the cumulative number of deaths on a certain day minus the
cumulative number
Germany, innumber of deaths
total theofnumber on the previous day. These data show that for the Netherlands and
cumulative deaths continues to rise, day.
on the previous although
Theseatdata
a slower
showrate than
that for inthethe early phases.
Netherlands and
Germany, in total the number continues to rise, although at a slower rate than in the early phases.
Germany, in total the number continues to rise, although at a slower rate than in the early phases.
400
deceased
400
deceased
300
300
of persons
200
of persons
200
100
number
100
number
0
0 10 20 30 40 50
0
0 d
10a y (sta
20rtin g M30
a rch 10)
40 50
d a y (sta rtin g M a rch 10)
Figure 5. Daily increments of deaths (ordinate) during the period 10 March–24 April 2020 in three
German5. Bundesländer—North-Rhine-Westphalia
Figure Daily increments ofofdeaths (ordinate) (redthe
during squares),
period Bavaria
10 (green
March–24 diamonds),
April 2020 and
in three
Figure 5. Daily
Baden-Wűrttemberg
increments
(blue
deaths
triangles)—with
(ordinate)
Germany
during
(brown
theopen
period 10 March–24
squares) and the
April 2020 in
Netherlands
German Bundesländer—North-Rhine-Westphalia (red squares), Bavaria (green diamonds), and
three German
(pink Bundesländer—North-Rhine-Westphalia
diamonds). The larger fluctuations seen, (red squares),
in particular, for the Bavaria
data on (green diamonds),
Baden-Wűrttemberg (blue triangles)—with Germany (brown open squares) andGermany and the
the Netherlands
andNetherlands
Baden-Wűrttemberg
were also (blue triangles)—with
comparatively larger in Germany
the early (brown
days but open
less squares)
visible in and
this the because
graph Netherlands
of
(pink diamonds). The larger fluctuations seen, in particular, for the data on Germany and the
(pink
the diamonds).
lower numbers The larger fluctuations
(ordinate). The seen,
fluctuations in particular,
might be real for the between
differences data on subsequent
Germany and days, the
Netherlands were also comparatively larger in the early days but less visible in this graph because of
Netherlands
but lower were
they are also
more likelycomparatively
the result larger in the early days but less visible in this graph because
the of
the numbers (ordinate). The of the time when
fluctuations mightdata are officially
be real differences reported.
betweenIndependent
subsequent of days,
thecause,
loweroverall
numbers (ordinate).
it does not the The fluctuations
influence thethe
overall might be time
real as
differences between subsequent days,
but they are more likely result of time trends over
when data discussed
are officially in this
reported. paper.
Independent of the
but they are more likely the result of the time when data are officially reported. Independent of the
cause, overall it does not influence the overall trends over time as discussed in this paper.
cause, overall it does not influence the overall trends over time as discussed in this paper.Appl. Sci. 2020, 10, 3398 7 of 11
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 11
The corresponding increments for Italy and Spain, shown in Figure 6, look somewhat different as
of aroundThe25corresponding
March (day 15 increments
in Figurefor
6). Italy and Spain, shown
The lock-down in Figure
in northern Italy,6,where
look somewhat different
people stayed home
as of around 25 March (day 15 in Figure 6). The lock-down in northern Italy, where people
largely from the end of February, seems to have resulted in a flattening of the curve revealed in the stayed
home largely from the end of February, seems to have resulted in a flattening of the curve revealed
number of deaths about four weeks later. Indeed, in Italy, the country that was affected first in Europe
in the number of deaths about four weeks later. Indeed, in Italy, the country that was affected first in
and that imposed restrictions first, the number of daily deaths had begun to stabilize by 25 March
Europe and that imposed restrictions first, the number of daily deaths had begun to stabilize by 25
(day 15 in Figure 6) and drop from early April (day 25 in Figure 6). The same is observed for Spain,
March (day 15 in Figure 6) and drop from early April (day 25 in Figure 6). The same is observed for
where severe restrictions were also imposed (we avoid the word lock-down in the case of Spain, as this
Spain, where severe restrictions were also imposed (we avoid the word lock-down in the case of
word is used
Spain, forword
as this different levels
is used of restrictions
for different levels in
of different EU
restrictions incountries).
different EU countries).
1400
1200
mumber people deceased
1000
800
600
400
200
0
0 10 20 30 40 50
day (start March 10)
Figure
Figure 6. Daily
6. Daily incrementsduring
increments duringthe
theperiod
period 10
10 March–24
March–24April
April2020
2020ofofdeaths
deathsin in
Spain (red
Spain squares)
(red squares)
and Italy (green diamonds).
and Italy (green diamonds).
When
When a comparisonisismade
a comparison madebetween
between thethe curves
curves showing
showingthe theincrements
increments inin
Europe
Europe with those
with those
from China, namely, Figure 7, an interesting similarity can be seen. This similarity
from China, namely, Figure 7, an interesting similarity can be seen. This similarity is not just is not just thethe
exponential
exponential growth
growth atat thebeginning
the beginningfollowed
followed byby aa gradual
gradualdecrease
decreaseafter
afterthe maximum
the maximum was attained,
was attained,
but also an interesting correspondence in time lines. For the countries where more strict contact
but also an interesting correspondence in time lines. For the countries where more strict contact
restrictions were imposed, after the daily deaths began to accelerate, the maximum is reached after
restrictions were imposed, after the daily deaths began to accelerate, the maximum is reached after
about 21 days, as observed from the plots for China, Italy, and Spain. The overall shape of the curves
about 21 days, as observed from the plots for China, Italy, and Spain. The overall shape of the curves
showing the increments is much the same for these three countries. These data seem to suggest that,
showing the increments is much the same for these three countries. These data seem to suggest that,
for European countries such as Italy and Spain, it can be expected to take another 40–50 days after 25
forMarch
European countries
to reach suchsimilar
a situation as Italy andin
to that Spain,
Chinaitwhere
can be expected
the infectionto take
rate hasanother 40–50
essentially gone days
down after
25 March to reachnegligible
to practically a situation similarOftocourse,
values. that in China where
this might the be
only infection rateassuming
achieved has essentially gone down
restrictions on
to practically negligible values. Of course, this might only be achieved assuming
hygiene and human–human distance are maintained in full for that period of time. Furthermore, restrictions on hygiene
andbecause
human–human
of the timedistance are maintained
lag between infection and in death,
full forthe
that periodofofnew
number time. Furthermore,
infections shouldbecause of the
have fallen
timeto lag between
essentially nilinfection
some 2–3and death,
weeks theassuming
earlier, number the of new
sameinfections should
testing rate have fallen
is maintained to essentially
for appropriate
nil monitoring.
some 2–3 weeks earlier, assuming the same testing rate is maintained for appropriate monitoring.Appl. Sci. 2020, 10, 3398 8 of 11
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 11
300
number of people deceased
200
100
0
0 20 40 60 80
days (starting Jan 27)
Figure 7. Daily
Figure increments
7. Daily of deaths
increments in China
of deaths duringduring
in China the main
theperiod
main of the corona
period of thecrisis.
corona crisis.
4. More
4. More onon InfectionRate
Infection Rate
TheThe discussionabove
discussion aboverelates
relates toto the
the data
data revealing,
revealing, by byfitting
fittingmathematical
mathematical functions,
functions, common
common
exponential behavior. To learn more from the data, we need to look
exponential behavior. To learn more from the data, we need to look more carefully to understand more carefully to understand the
cause of curves like those shown in Figures 1–4. The models,
the cause of curves like those shown in Figures 1–4. The models, as mentioned, reveal exponential as mentioned, reveal exponential
behavior; here, we found cubic behavior for all countries examined (N3). This, however, is still a
behavior; here, we found cubic behavior for all countries examined (N3 ). This, however, is still a purely
purely mathematical description. To understand what could be an important underlying mechanism
mathematical description. To understand what could be an important underlying mechanism we
we consider what makes the virus spread. The most important factor is human–human contact,
consider what makes the virus spread. The most important factor is human–human contact, which is
which is underpinned by the observation that when this is forbidden by authorities, the number of
underpinned by the observation that when this is forbidden by authorities, the number of infections
infections and, consequently, the number of deaths, falls and eventually reaches zero (the cases of
and, consequently,
China and South Korea the number
illustrate of this
deaths,
well).falls
Let and eventually
us start reachesa zero
by considering single(the
personcases of China
infecting, onand
South Korea
average, m illustrate
other personsthis well).
per day; Lettheus following
start by considering
day, each ofa these singlempersonpersonsinfecting, on average,
infects another m
m other persons per day; the following day, each of these m
persons, and so on. This can be expressed in mathematical form and, by fitting this expression to and
persons infects another m persons, the so
on.experimental
This can be death expressed
rate, wein mathematical
can obtain a typical form and,valueby forfitting
m. When this weexpression
implement to the
this experimental
and fit an
death rate, we can
experimental obtain
curve a typical
of the number value for m. in
of deaths When we implement
Germany, we obtainthis and fit an
a good to theexperimental
data (similar curve
of the number
results were of deaths for
obtained in Germany, we obtain
other countries) a good
as shown byfit
the tosolid
the data (similar
red curve in results
Figure 8. were
We obtained
obtained for
m = countries)
other 0.24, meaning that oneby
as shown death leads red
the solid to an additional
curve 0.24 8.
in Figure deaths m = 0.24, itmeaning
per day. Although
We obtained might appear
that one
strange
death leads that a deceased
to an additional person
0.24 leads
deaths toperanother deceased person,
day. Although it might this is to be
appear tracedthat
strange back to the
a deceased
number
person leadsof infections.
to anotherAs there is person,
deceased no fixedthis percentage
is to be for the number
traced back to the of deaths
number starting from the As
of infections.
number of people infected, we reach this conclusion. At a later point
there is no fixed percentage for the number of deaths starting from the number of people infected, in time, around day 25 in Figure we
8, we see that the behavior changes. This is the consequence of countries
reach this conclusion. At a later point in time, around day 25 in Figure 8, we see that the behavior taking restrictive measures,
for example,
changes. This primarily reducing contact.
is the consequence of countriesAfter some
taking time, one should
restrictive not assume
measures, m new infections
for example, primarily
per infected person per day, but a lower number p.
reducing contact. After some time, one should not assume m new infections per infected person per
Based on a mortality rate (derived from the number of infected people and the number of deaths)
day, but a lower number p.
of 4% (there are a range of sources quoting different mortality rates; for regions where an outbreak
Based on a mortality rate (derived from the number of infected people and the number of deaths)
began and with a high number of deaths, mortality is in the range of 5% and higher, see Ref. 19 [19]),
of an
4%increase
(there are a range of sources quoting different mortality rates; for regions where an outbreak
of 0.24 deaths per day corresponds to an increase of 25 × 0.24 = 5.5 of additional infected
began and with
people per day (wherea high number of deaths,
actual infection mortality
occurred is in
in the the range
weeks before). ofWith
5% and higher, daily
an average see Ref. 19 [19]),
average
an number
increaseofofcontacts
0.24 deaths per dayofcorresponds
per person around 22 for to an
theincrease 25 × 0.24 =[20],
workingofpopulation 5.5 these
of additional
numbersinfected
(5.5
people
versus 22) reveal that the virus is highly infectious during the period shown in Figure 8 daily
per day (where actual infection occurred in the weeks before). With an average (Noteaverage
that
number
some of of the
contacts
numbers per used
person of around
might 22 for the
differ between workingand
countries population
change over [20],time.
these numbersin(5.5
However, suchversus
a
22)case
reveal thatthe
where thecalculated
virus is highly
value infectious
is actually,during the period
for example, 0.12 or shown in Figure
0.36, rather than8 0.24,
(Noteallthat some of the
conclusions
numbers
and generalusedobservations
might differ presented
between countries
are still valid.and Although
change over time. However,
a different rate applies, in such a case where
transmission is
thestill shown to
calculated be exponential,
value is actually, illustrating
for example, that theorvirus
0.12 0.36,israther
highlythan infectious.
0.24, allThis also applies
conclusions andtogeneral
the
number of presented
observations daily contacts for the
are still working
valid. Although population;
a differentconclusions
rate applies, are unchanged,
transmissionirrespective
is still shownof to
be whether
exponential,the actual number
illustrating thatis 15
theorvirus
35, or the 25 we
is highly used here
infectious. This(based
also on the literature)).
applies to the number In places
of daily
contacts for the working population; conclusions are unchanged, irrespective of whether the actualAppl. Sci. 2020, 10, 3398 9 of 11
number is2020,
Appl. Sci. 15 or10,35, or PEER
x FOR the 25 we used here (based on the literature)). In places where real outbreaks
REVIEW 9 of 11
occurred—the market in Wuhan, the area of Heinsberg in Germany, Ischgl in Austria (and, for example,
thewhere real outbreaks
follow-up in Norway occurred—the
from peoplemarket in Wuhan,
that were theIschgl),
skiing in area of and
Heinsberg in Germany,
the region in NorthIschgl in
Italy—the
Austria (and, for example, the follow-up in Norway from people that were skiing in Ischgl), and the
density of people at the beginning of the spread of infection was much higher. Thus, it is possible that
region in North Italy—the density of people at the beginning of the spread of infection was much
the mortality is lower than what has been reported to date. However, in such a case, the number of
higher. Thus, it is possible that the mortality is lower than what has been reported to date. However,
infected people is higher than currently detected and, therefore, new infections are likely to continue
in such a case, the number of infected people is higher than currently detected and, therefore, new
over an extended period of time.
infections are likely to continue over an extended period of time.
6
number pf people deceased
4
Thousands
2
0
0 10 20 30 40 50
d a y (sta r t M a r c h 1 0 )
Figure
Figure 8. Number
8. Number of of deaths
deaths as as a function
a function of time
of time forfor Germany
Germany during
during thethe period
period 10 10 March–24
March–24 April
April 2020.
2020. The experimental data (solid triangles) are compared to a fitted curve (red curve) using the
The experimental data (solid triangles) are compared to a fitted curve (red curve) using the assumption
assumption that each death leads to the eventual death of another 0.24 persons per day (the
that each death leads to the eventual death N−1 of another 0.24 persons per day (the mathematical fitted
mathematical fitted function reads 8*(1.24) , where N is the number of days from the start of the
function reads 8*(1.24)N−1 , where N is the number of days from3the start of the series, 10 March 2020).
series, 10 March 2020). Initially the increase follows a cubic (N ) function; after day 30 the increase
Initially the increase 2follows a cubic (N3 ) function; after day 30 the increase reflects quadratic
reflects quadratic (N /0.38) behaviour.
(N2 /0.38) behaviour.
Examining the data more closely for the first 10 days in Figure 8 (not explicitly shown in detail),
Examining the data more closely for the first 10 days in Figure 8 (not explicitly shown in
the agreement between the experimental number of deaths and the model with an increase of 0.24
detail), the agreement between the experimental number of deaths and the model with an increase
per day is not strong. However, the fit is satisfactory when an increase of 0.8 per day is adopted,
of 0.24 per day
compared is not
to 0.24 strong.
per day after However, the fitWith
the first 10 days. is satisfactory when an
an average number increase
of human of 0.8 of
contacts per
22 day
per is
adopted,
day, thecompared
increase of to 0.8
0.24implies
per day anafter the first
increase 10 days.
in infected Withof
people an20average
per day.number
Thus, theof human contacts
experimental
of 22
data and this analysis explain why the virus spread so rapidly at mass meetings earlier this year withthe
per day, the increase of 0.8 implies an increase in infected people of 20 per day. Thus,
experimental
large numbersdataofand thisseveral
deaths analysis explain
weeks later.why the virus spread so rapidly at mass meetings earlier
this yearThese
withobservations,
large numbers of deaths
namely, severalofweeks
an increase later.
0.8 at the early stages and 0.24 starting after about 10
These
days, canobservations,
have different namely, an increase
interpretations. One mayof 0.8 at the
be that early
these stages
models andthat
reveal 0.24thestarting
measuresafter about
taken
by politicians
10 days, can havetodifferent
reduce human–human
interpretations.contactOne may are be
effective. Nonetheless,
that these and importantly,
models reveal the
that the measures
spread
taken of the outbreak
by politicians to reduceis exponential,
human–human and, contact
therefore,arethere is no Nonetheless,
effective. reason to believe that attitudesthe
and importantly,
should
spread be outbreak
of the relaxed. The other possibility,
is exponential, however,there
and, therefore, is that thereason
is no steeper
to curve
believeatthat
the attitudes
outset is should
the
consequence of a sudden outbreak in a dense population (Wuhan Market, Heinsberg
be relaxed. The other possibility, however, is that the steeper curve at the outset is the consequence of a carnival, etc.),
with many
sudden more
outbreak in human–human
a dense population contacts than average,
(Wuhan which is incarnival,
Market, Heinsberg agreement with
etc.), withthe many
changed more
behavior of the curve after 10–15 days. The latter could be related to the typical incubation time of
human–human contacts than average, which is in agreement with the changed behavior of the curve
the virus [21]. The more intense contact in such a situation, combined with the change in the curve
after 10–15 days. The latter could be related to the typical incubation time of the virus [21]. The more
behavior after about two weeks, suggest this is the more likely reason. The observation that the
intense contact in such a situation, combined with the change in the curve behavior after about two
increase becomes less steep after 25 days seems to suggest that restrictive measures are effective.
weeks, suggest this is the more likely reason. The observation that the increase becomes less steep after
25 days seems to suggest that restrictive measures are effective.
5. Conclusions
The most relevant COVID-19 data for humanity, namely, the death-toll, is also arguably the most
relevant data for current analyses of the COVID-19 pandemic in Western Europe. ExponentialAppl. Sci. 2020, 10, 3398 10 of 11
5. Conclusions
The most relevant COVID-19 data for humanity, namely, the death-toll, is also arguably the
most relevant data for current analyses of the COVID-19 pandemic in Western Europe. Exponential
behavior at the outset of the epidemic was observed for all countries considered. Although the spread
of infection started on different dates and, in particular, with different steepness, all countries showed
cubic (N3 ) behaviour. Interestingly, for countries/regions in which a strict no human–human contact
policy was imposed at an early stage, behavior over time resembles that in China, where infection has
essentially fallen to nil after 60–70 days, from a peak at around 25 days. For Spain and Italy, the data
indicate the peak has been passed, with daily deaths falling for the past 20 days, suggesting that
infections leading to deaths began to fall several weeks earlier. This reveals the effectiveness of the
measures taken by the governments in these countries. Other countries do not appear to be at that
point yet (as indicated by the data), but this state can be expected to be reached assuming restrictions
remain in full force. It remains to be seen whether the more severe restrictions in countries like China,
Italy, and Spain, resulted in a more timely outcome, as is suggested by the present analysis.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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