An estimation of the average, minimum and maximum ultimate death toll is given along with a predicted date. The expected ultimate death toll is estimated based on both forms of Hubbert equation, i.e. the Parabola and the Linearization (where appropriate), while the predicted date is determined based on a data forecast using the composite growth rate function (geometric sequence).

1. Data driven forecast of the Covid-19 death toll
Mohamed Bouanane
Management Consulting Director
Toulouse – France
Initial version of May 1, 2020
Updated version of May 3, 2020 (addition of European countries)
Updated version of May 6, 2020 (addition of USA)
COVID-19 is an emerging pandemic infection that has spread worldwide since January 2020
starting from Wuhan, Hubei province in China. Many epidemiologists and mathematicians are
trying to find the most accurate model in order to predict the magnitude and the end of the
pandemic as well as to advise governments define the better date for opening up the economies
after having established a lock-down.
We have started, on the 22nd of March, estimating the evolution of the deaths toll for some
European countries to satisfy a self-curiosity. As the cumulative death toll showed an exponential
curve, therefore we have used a geometric sequence function to estimate the next day total death
cases. The cumulative death (CD) toll is time-dependent function which requires a common ratio –
r-value – evolving each day for calculating the next term. Based on the observation of the
countries data, the best function for estimating the next death case would be a composite growth
rate function as follows:
CDt = CD0 ∗ (r-value)t
where t is time in days
r-value = 1 + Cagrt
Cagrt−1 =
(
CDt−1
CD0
)
1
(t−1)
− 1
The Compound average growth rate Cagrt at day t is then estimated using the previous Cagrt-1 and
the historic data (growth / decline) reported over time for each country (usually three to five
previous terms – days). The forecast was published as a PDF document for the first time on 26th of
March and updated several times1
.
Methodology
The next step was to predict the ultimate death toll and find out when it would happen. However,
estimating the ending date of the epidemic would be highly risky and usefulness. Moreover,
estimating the “ending date” is not straight-forward and presumes that the growth rate matches
the decline rate whereas the epidemic pattern should distinguish the two rates.
It would be hazardous to make any prediction of the Covid-19 epidemic infection based on the
number of infected cases, because different countries follow different policies to counter the
epidemic, thus the value of infected cases is not counted similarly everywhere, since it highly
depends on how patients are screened. Moreover, many people are infected and unaware
because asymptomatic and have not been tested. The most realistic and reliable data are that
count the death cases if it is reported in time and in a transparent manner.
M. Bouanane 1/12

2. Data driven forecast of the Covid-19 death toll
Covid-19 as any epidemic infection has a life-cycle pattern as a bell-shaped curve (Figure 1) for the
daily death (DD) curve over time, while the cumulative death toll has an S-shaped curve. The life-
cycle is composed of different phases: incubation, spreading, acceleration, inflection, deceleration
and flattening, and then ending.
Such life-cycle is very similar to that of extracting a finite mineral resource: a gradual rise from
zero resource production which then grows rapidly, reaching a peak representing the maximum
production, and then falls to approach zero production at a slow speed. The duration of this life-
cycle is highly dependent of each country’s strategy to counter the epidemic and the bell-shaped
curve is often not symmetrical simply because the growth rate does not match the decline rate.
The Hubbert bell-shaped curve has been used in modeling depletion of crude oil and predicting its
peak and its ultimately recoverable resource. Indeed, using such curve – a probability density
function of a Logistic distribution (a common S-shaped curve) – in modeling the Covid-19 death
toll will help to determine the peak and the ultimate total death cases.
We define the following parameters as per the Hubbert’s equation:
 CD(t) is cumulative death cases at day t;
 UCD is the ultimate cumulative death cases;
 DD(t) = d CD / dt is the daily death cases at day t;
 k is a Logistic growth rate.
Then, the Hubbert’s polynomial equation (EQ#1) can be expressed in a differential form:
dCD
dt
= DD(t ) = k ∗ CD(t ) ∗ (1 −
CD (t )
UCD ) ( EQ#1)
At the start of the epidemic, CD/UCD is too small then Equation (1) reduces to DD = k*CD showing
an exponential growth at a rate k. At the end of the epidemic, CD almost equals UCD, then
Equation (1) shows an exponential decline.
Dividing Equation (EQ#1) by CD we get the second form called the Hubbert Linearization2
Equation
(EQ#2):
DD(t )
CD(t )
= k ∗ (1 −
CD(t)
UCD ) ( EQ#2)
Equation (EQ#2) is linear in the (CD; DD/CD) plane. Consequently, a linear regression on the data
points gives the axis intercepts: The k parameter is the intercept of the Y-axis (=DD/CD) for CD=0,
and the UCD value is the intercept of the X-axis (=CD) for DD=0. We can as well derive the
Hubbert’s curve parameters from the value of the line slope calculated by -k/UCD (Figure 1).
The Hubbert’s equation can be extended to the second derivatives (EQ#3) by calculating the
derivative of equation (EQ#1) and where the left term, called the decline rate, represents the
death toll relative daily increase3
. Therefore Equation (EQ#3) is a linear function of the cumulative
death toll. In this case the Hubbert’s Linearization line intercepts the X-axis at half the value of the
Ultimate cumulative death toll (UCD).
M. Bouanane 2/12

3. Data driven forecast of the Covid-19 death toll
dDD
dt
∗
1
DD
= k ∗ (1 − 2
CD
UCD ) ( EQ#3)
Another way to plot the HL equation is to combine the equations (EQ#2) et (EQ#3) since they only
differ by a factor two on the slopes, their intercept with the Y-axis being the same and equal to k.
Therefore, the data points of the two representations could be mixed together in a unique
representation – Hybrid Hubbert’s Linearization – by multiplying the cumulative death by a factor
two in(EQ#3).
The next form of the Hubbert’s equation is simply the below polynomial function (EQ#4) in the
(CD; DD) plane where the points would follow the Hubbert Parabola passing through the origin
(0;0) and the point (UCD;0).
DD = k∗CD −
k
UCD
CD2
( EQ#4)
R. Canogar4
has used the Hubbert Parabola to model the oil depletion. We use the Hubbert
Parabola in this paper to model the Cumulative death (CD) toll of the Covid-19 pandemic
infection. In a first plot, we place the points (CD; DD) and determine the polynomial regression –
the Parabola – that passes through the origin of the plane (Figure 3). The intercept of this
parabola with the X-axis gives the estimated Ultimate cumulative death toll (UCD).
In a second plot, we study the evolution of the expected UCD over time. Thus, we define the
function UCD(t) as the estimated UCD for day t via the Hubbert’s equation (4) by placing all the
data points (CD; UCD) in the plot of .
At the beginning of the Covid-19 epidemic, it is obvious that the cumulative death toll CD(t) is too
low compared to the ultimate death toll UCD(t), thus the data points (CD(t); UCD(t)) are above
the line UCD=2*CD (except some strange data). When the epidemic reaches its peak, the CD(t)
reaches half of UCD(t) and then the data points (CD(t); UCD(t)) go below the line [UCD=2*CD].
After the peak and as the epidemic advances, the data points (CD(t); UCD(t)) approaches the line
[UCD=CD] and then CD(t) approaches the expected UCD(t) to reach the equality at the end of the
epidemic. According to the Logistic model, if the point (CD(t); UCD(t)) lies below the line
[UCD=2*CD] then the time is after the peak day, i.e. CD(t) > UCD(t)/2.
Results
We estimate the expected ultimate death toll using the model and the methodology explained
above and give a plausible date when it would be reached, keeping in mind that such estimate
may change the next day since the model is highly dependent of the changes in the complex real
life. Thus, each predictive data should be read with precaution.
An estimation of the average, minimum and maximum ultimate death tolli
is reported along with a
predicted date. The expected ultimate death toll is estimated based on both forms of Hubbert
equation, i.e. the Parabola and the Linearization (where appropriate), while the predicted date is
determined based on a data forecast using the composite growth rate function (geometric
sequence).
As a starting point, we report in Table1 below the predictive ultimate death toll for Europe, North
America and World wide, despite it is highly difficult to make reliable forecast for a whole region
i
Data source: Daily updated data from WHO, Wikipedia & BNO.
M. Bouanane 3/12

4. Data driven forecast of the Covid-19 death toll
composed of many different countries following very different, even sometimes contradictory,
policies. In the near future we will focus on North American and some European countries.
According to the plot in Figure 4 all the regions (World, Europe and North America) have reached
their peak respectively, having their recent data points (CD(t); UCD(t)) between the two dashed
lines. However, the decrease of the daily death cases is not yet stabilized and particularly in
Europe (as of 29 April).
Figure 5 shows that Italy, Spain and France have passed their peak date and their data points
(CD(t); UCD(t)) are approaching the dashed line UCD=CD which means that they are close to the
ending date. Exception is for the United Kingdom where data points (CD(t); UCD(t)) are not yet
stabilized due to a high number of deaths reported on 29 of April. The estimation for the expected
ultimate death toll is given in Figure 6 as per 10 of May. It confirms the trend pattern observed in
the previous figure, and shows that France would very slightly overtake Spain in terms of the
cumulative death toll by 6 of May. Table2 presents the predictive ultimate death toll for the four
countries.
The plots in Figure 7 and Figure 8 show almost the same pattern and trend for Belgium, Germany,
the Netherlands and Switzerland as the previous countries. However as of 30 April these countries
have less stabilized data points (CD(t); UCD(t)) between the dashed lines except for Switzerland
for which the data are too close to the dashed line UCD=CD and then to the “ending date”. Table3
presents the predictive ultimate death toll for the four countries.
While USA has passed its peak date, Figure 9 shows that the peak is shifting between the Orange
(actual data as of 6 May) and Red plots by more than 15k death cases (X-axis). The instability of
data points (CD(t); UCD(t)) is confirmed in the plot shown in Figure 10 for both actual and
predictive data. It is clear that USA is still far away from the dashed line U=CD, the indicator of the
“ending date”. Table4 presents the predictive ultimate death toll for the USA, reaching more than
103k death cases by 23 of May.
M. Bouanane 4/12

5. Data driven forecast of the Covid-19 death toll
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
51
53
55
57
59
61
63
0
2 000
4 000
6 000
8 000
10 000
12 000
0
25 000
50 000
75 000
100 000
125 000
150 000
175 000
200 000
225 000
WW Daily deaths Moyenne glissante (WW Daily deaths)
WW Cumulative deaths
Days (22nd Feb - 27th Apr)
Dailydeathcases
Cumulativedeathcases
Figure 1: WW Daily death (bell-shaped) & Cumulative death toll (S-shaped)
175000 180000 185000 190000 195000 200000 205000 210000 215000
1,00 %
2,00 %
3,00 %
4,00 %
f(x) = − 5,216E-07 x + 1,319E-01
R² = 7,284E-01
Hubbert Linearization for the WW Death Toll (21st - 27th April)
WW Cumulative death cases CD
WWDailydeathcasesDD/CD
Figure 2: Hubbert Linearization on the WW Daily death cases /
Cumulative death cases (k=13,2% & UCD=252 826)
M. Bouanane 5/12

6. Data driven forecast of the Covid-19 death toll
0
25 000
50 000
75 000
100 000
125 000
150 000
175 000
200 000
225 000
0
2000
4000
6000
8000
10000
12000
f(x) = − 4,31E-07 x² + 1,13E-01 x
R² = 9,63E-01
Hubbert Parabola on the WW Death Toll (22nd Feb - 27th Apr)
WW Daily deaths Polynome (WW Daily deaths)
WW Cumulative death cases
WWDailydeathcases
Figure 3: Hubbert Parabola on the WW Death Toll (k=11,3% & UCD=262 108)
0
20 000
40 000
60 000
80 000
100 000
120 000
140 000
160 000
180 000
200 000
220 000
0
50 000
100 000
150 000
200 000
250 000
300 000
350 000
400 000
Covid-19 Expected Ultimate Death Toll vs Cumulative Deaths
U=2*CD U=CD WW Expected UCD
EU Expected UCD NA Expected UCD
Figure 4: Estimated Ultimate Death Toll (as of 29 April 2020)
M. Bouanane 6/12

7. Data driven forecast of the Covid-19 death toll
Region World Wide Europe North America
Current CD
Date
226 470
29-Apr
135 659
29-Apr
66 425
29-Apr
Max. Ultimate
Date
346 374
11-May
228 922
> 13-May
91 819
> 11-May
Min. Ultimate
Date
296 806
06-May
158 990
04-May
81 198
> 11-May
Mean Ultimate
Date
321 590
09-May
193 956
> 13-May
86 508
> 11-May
Table 1. Estimation of Expected Ultimate Death Toll for Europe, North America and World wide
9000 11000 13000 15000 17000 19000 21000 23000 25000 27000
6 000
12 000
18 000
24 000
30 000
36 000
42 000
48 000
U=2*CD U=CD IT UCD
ES UCD FR UCD UK UCD
Figure 5: European Countries 1 – Expected Ultimate Death Toll vs CD (as of 30
April)
M. Bouanane 7/12

8. Data driven forecast of the Covid-19 death toll
18000 20000 22000 24000 26000 28000 30000
12 000
18 000
24 000
30 000
36 000
42 000
48 000
54 000
60 000
U=2*CD U=CD IT UCD
ES UCD FR UCD UK UCD
Figure 6: European Countries 1 – Estimation of Expected Ultimate Death Toll vs CD
(as per 10 May)
Country Italy Spain France United
Kingdom
Current CD
Date
27967
30-Apr
24543
30-Apr
24342
30-Apr
26771
30-Apr
Min. Ultimate
Date
29697
5-May
25555
4-May
25401
4-May
32669
10-May
Max. Ultimate
Date
34146
----
28768
----
27603
----
42122
----
Average Ultimate
Date
31687
----
27163
----
26710
8-May
39816
----
Table2. Estimation of Expected Ultimate Death Toll
M. Bouanane 8/12

9. Data driven forecast of the Covid-19 death toll
0 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000
0
2 000
4 000
6 000
8 000
10 000
12 000
U=2*UCD Séries anonymes 3 BE UCD
GE UCD NL UCD CH UCD
Figure 7: European Countries 2 – Expected Ultimate Death Toll vs CD (as of 30
April)
0 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000
0
2 000
4 000
6 000
8 000
10 000
U=2*UCD Séries anonymes 3
BE UCD GE UCD
Figure 8: European Countries 2 – Estimation of Expected Ultimate Death Toll vs CD
(as per 10 May)
M. Bouanane 9/12

10. Data driven forecast of the Covid-19 death toll
Country Belgium Germany Netherlands Switzerland
Current CD
Date
7594
30-Apr
6288
30-Apr
4795
30-Apr
1422
30-Apr
Min. Ultimate
Date
8192
4-May
7193
7-May
5320
6-May
1553
5-May
Max. Ultimate
Date
9581
----
8525
----
6842
----
1996
----
Average Ultimate
Date
8747
10-May
7833
----
6029
----
1739
----
Table3. Estimation of Expected Ultimate Death Toll
0
10 000
20 000
30 000
40 000
50 000
60 000
70 000
80 000
90 000
100 000
110 000
0
1 000
2 000
3 000
4 000
5 000
f(x) = − 7,26E-07 x² + 8,34E-02 x
R² = 8,52E-01
f(x) = − 1,44E-06 x² + 1,22E-01 x
R² = 8,79E-01
USA DC Polynome (USA DC)
USA Exp. DC Polynome (USA Exp. DC)
Figure 9: Hubbert Parabola on USA Death Toll (Daily deaths vs Cumulative
deaths)
M. Bouanane 10/12

11. Data driven forecast of the Covid-19 death toll
20 000
30 000
40 000
50 000
60 000
70 000
80 000
90 000
100 000
110 000
35 000
50 000
65 000
80 000
95 000
110 000
125 000 Covid-19 Expected Ultimate Death Toll vs Cumulative Deaths
U=2*CD U=CD
USA UCD USA Exp. UCD
Figure 10: USA – Estimation of Expected Ultimate Death Toll vs CD (Data as of 6
May – Est. as per 22 May)
Current CD
Date
Min. Ultimate
Date
Max. Ultimate
Date
Average Ultimate
Date
73431
06-May
84389
12-13-May
127202
----
103805
23-24-May
Table4. USA – Estimation of Ultimate Death Toll
M. Bouanane 11/12

12. Data driven forecast of the Covid-19 death toll
1 M. Bouanane, “Covid-19 – Forecast for Western European Countries & USA”, 26th Mar 2020.
2 M. King Hubbert, Techniques of Prediction as Applied to the Production of Oil and Gas, in: Saul
I. Gass (ed.): Oil and Gas Supply Modeling, National Bureau of Standards Special Publication 631,
Washington – National Bureau of Standards, 1982, pp. 16-141.
3 Khebab, "A Different Way to Perform the Hubbert Linearization", 18th Aug 2006.
4 Canogar Roberto, "The Hubbert Parabola". GraphOilogy, 06th Sept 2006.
M. Bouanane 12/12