Prediction of Cumulative Death Cases in Nigeria Due to COVID-19 Using Mathematical Models

Abstract

In this paper, we present various growth models such as Von Bertalanffy, Baranyi-Roberts, Morgan-Mercer-Flodin (MMF), modified Richards, modified Gompertz, modified Logistics and Huang in fitting and evaluating the COVID-19 epidemic pattern as of 15 July 2020 in the form of the total number of SARS-CoV-2 deaths in Nigeria. The MMF model was found to be the best model having the highest adjusted R2 value and lowest RMSE value. The values for the Accuracy and Bias Factors were near unity (1.0). The parameters derived from the MMF model include maximum growth rate (log) of 0.02 (95% CI from 0.02 to 0.03), curve constant () that affects the infection point of 1.61 (95% CI from 1.42 to 1.79) and maximal total number of death cases (Ymax) of 1,717 (95% CI from 1,428 to 2,123). The model estimated that the total number of death cases for Nigeria on the coming 15th of August and 15th of September 2020 were 940 (95% CI of 847 to 1,043) and 1,101 (95% CI of 968 to 1,252), respectively. The predictive ability of the model employed in this study is a powerful tool for epidemiologist to monitor and assess the severity of COVID-19 in Nigeria in months to come. However, like any other model, these values need to be taken with caution because of the COVID-19 uncertainty situation locally and globally.

KEYWORDS: COVID-19, SARS-Cov-2, MMF Model, Nigeria, Kinetics

Full text

BESSM, 2020, Vol 4, No 1, 20-24
- 20 -
Prediction of Cumulative Death Cases in Nigeria Due to COVID-19
Using Mathematical Models
Salihu Yahuza1, Ibrahim Alhaji Sabo2, Bilal Ibrahim Dan-Iya3, Mohd Yunus Abd. Shukor4*
1Department of Microbiology and Biotechnology, Federal University Dutse, PMB 7156,
Dutse, Jigawa, Nigeria.
2Department of Microbiology, Faculty of Pure and Applied Sciences, Federal University Wukari, P.M.B. 1020 Wukari,
Taraba State Nigeria.
3Department of Pharmacy Technician, College of Health Sciences, Kano, and Technology,
Kano, Nigeria.
4Department of Biochemistry, Faculty of Biotechnology and Biomolecular Sciences, Universiti Putra Malaysia,
43400 UPM Serdang, Selangor, Malaysia.
*Corresponding author:
Prof. Dr. Mohd Yunus Abd. Shukor
Faculty of Biotechnology and Biomolecular Sciences,
Universiti Putra Malaysia,
UPM Serdang 43400
Selangor,
Malaysia.
Email: yunus.upm@gmail.com
INTRODUCTION
The growth curve of viruses and other microorganisms on
substrates such as nutrients or other organisms, including
humans, typically followed a sigmoidal pattern, beginning with
the lag section just after t = 0, followed by the logarithmic section
and then enters the stationary phase and eventually moves into
the phase of decline. To describe the organisms’ growth curve,
various sigmoidal functions such as Von Bertalanffy, Baranyi-
Roberts, modified Richards, modified Gompertz and modified
Logistics [1] including Morgan-Mercer-Flodin (MMF) [2] could
be utilized. The important parameters of the growth curve inc lude
the maximum specific growth rate (μm), the lag period and
asymptotic values. Analyses of COVID-19 pandemic including
theoretical, quantitative and simulation for the total number of
death cases and deaths can be performed using different
mathematical models. Models such as the modified Gompertz,
Von Bertalanffy and logistics have been employed to model
COVID-19 pandemic [3] with good predictive ability. This study
is aimed at evaluating various available mathematical models
such as Logistic [1,4], Gompertz [1,5], Richards [1,6], Morgan-
Mercer-Flodin [2], Baranyi-Roberts [7], Von Bertalanffy [8,9],
Buchanan three-phase [10] and more recently Huang model [11]
in fitting and analyzing the epidemic trend of COVID-19 in the
HISTORY
Received: 18th July 2020
Received in revised form: 24th July 2020
Accepted: 28th July 2020
ABSTRACT
In this paper, we present various growth models such as Von Bertalanffy, Baranyi-Roberts,
Morgan-Mercer-Flodin (MMF), modified Richards, modified Gompertz, modified Logistics and
Huang in fitting and evaluating the COVID-19 epidemic pattern as of 15th of July 2020 in the
form of the total number of SARS-CoV-2 deaths in Nigeria. The MMF model was found to be
the best model having the highest adjusted R2 value and lowest RMSE value. The values for the
Accuracy and Bias Factors were near unity (1.0). The parameters derived from the MMF model
include maximum growth rate (log) of 0.02 (95% CI from 0.02 to 0.03), curve constant (
δ
) that
affects the infection point of 1.61 (95% CI from 1.42 to 1.79) and maximal total number of death
cases (Ymax) of 1,717 (95% CI from 1,428 to 2,123). The model estimated that the total number
of death cases for Nigeria on the coming 15th of August and 15th of September 2020 were 940
(95% CI of 847 to 1,043) and 1,101 (95% CI of 968 to 1,252), respectively. The predictive ability
of the model employed in this study is a powerful tool for epidemiologist to monitor and assess
the severity of COVID-
19 in Nigeria in months to come. However, like any other model, these
values need to be taken with caution because of the COVID-19 uncertainty situation locally and
globally.
KEYWORDS
COVID-19
SARS-CoV-2
MMF Model
Nigeria
Kinetics
BULLETIN OF ENVIRONMENTAL SCIENCE &
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BESSM, 2020, Vol 4, No 1, 20-24
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form of total death case of SARS-CoV-2 in Nigeria as of 15th of
July 2020.
MATERIALS AND METHODS
Data acquisition
Data for the cumulative or total number of death cases from
Nigeria as of 15th of July 2020 was acquired from Worldometer
[12]. Data were first converted to logarithmic values and the time
after first death was utilized for time zero.
Statistical analysis
The statistically significant difference between the models was
determined using various methods including the modified
determination coefficient (R2), accuracy factor (AF), bias factor
(BF), root-mean - square error (RMSE) and corrected AICc
(Akaike Knowledge Criterion) as before [13].
Calculation of RMSE was according to Eq. (1), where model
predicted values are Pdi and models experimental values are Obi
and n is the number of experimental data while p is the number
of models’ parameter.
()
pn
ObPd
RMSE
n
i
ii
−
−
=
∑
=1
2
(Eqn. 1)
The adjusted R2 is used to calculate the quality of nonlinear
models according to the formula where RMS is the Residual
Mean Square and Sy2 is the total variance of the y-variable,
calculated as follows;
( )
2
2
1
Y
s
RMS
RAdjusted −=
(Eqn. 2)
( ) ( )
( )
( )
1
11
1
2
2
−−
−−
−= pn
nR
RAdjusted
(Eqn. 3)
The Akaike information criterion (AIC) [14] was calculated as
follows;
()( )( )
2
212
12ln
2−−
++
+++






+
=pn
pp
p
n
RSS
npAICc
(Eqn. 4)
Where p is the number of parameters of the model and n is the
number of data points. The model with the smallest AICc value
preferable a difference of >5 arbitrary values is likely more
correct irrespective of the actual values [15].
Accuracy Factor (AF) and Bias Factor (BF) as suggested by Ross
[16] were calculated as follows;
( )








∑
=
=
n
i
ii
n
ObPd
1
/
log
10
factorBias
(Eqn. 5)
( )










=
∑
=
n
in
i
Ob
i
Pd
1
/
log
10
factor Accuracy
(Eqn. 6)
Fitting of the data
Fitting of the bacterial growth curve using various growth models
(Table 1) was performed using GraphPad Prism (v 8.0 trial
version).
Table 1. Models used in this study.
Model
p
Equation
Modified Logistic
3














+−+
=
2)(
4
exp1t
A
m
A
y
λ
µ
Modified
Gompertz
3


















+−−= 1)(expexp t
A
e
m
Ay
λ
µ
Modified
Richards
4





















−







++++= t
v
v
A
m
vvAy )(
1
1)1(exp)1exp(1
λ
µ
Morgan-Mercer-
Flodin (MMF)
4 = −
(

− )
1 + ()
Baranyi-Roberts
4
( )
















−
−





−−
−
−
+
−
+
+− Ay
e
hx
m
e
h
e
x
m
e
m
x
m
e
max
1
00
ln
1
1ln
µµ
µ
µ
Von Bertalanffy
3
3
3
1
3
/
exp
3
11




























−













−−=
K
x
m
K
A
Ky
µ
Huang
4
( )





−





−+−+= xB
m
e
A
e
Y
e
A
eyAy
µ
max
ln
max
( )
( )
αλ
λα
α
e
e
xxB
x
+
+
+=
−−
1
1
ln
1
Buchanan
Three-phase
linear model
3
Note:
A= maximum no of death cases lower asymptote;
ymax= maximum no of death cases upper asymptote;
µ
m= maximum specific growth rate of death;
v= affects near which asymptote maximum no of death cases occurs.
λ=lag time
e = exponent (2.718281828)
t = time after first death case is reported
α,β,δ and k = curve fitting parameters
h0 = a dimensionless parameter quantifying the initial physiological state of the
reduction process. The lag time (h-1) or (d-1) can be calculated as h0=
µ
m
When data at time zero is 0 (Day after 1st death cas e log 1=0 for COVID-19) the MMF is reduced
to a 3-parameter model
Y = A, IF X < LAG
Y=A + K(X ̶ λ), IF λ ≤ X ≥ XMAX
Y = YMAX, IF X ≥ XMAX

BESSM, 2020, Vol 4, No 1, 20-24
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RESULTS AND DISCUSSION
All the curves evaluated demonstrated an acceptable visual
fit with exception of the Buchanan-3-phase model (Figs 1
to 6). The best performance was found to be MMF model
having the lowest RMSE, AICc values and the highest
adjusted value, R2. The AF and BF values were also found
to be excellent for the model having their values closest to
1.0. The poorest performance was the Buchanan-3-phase
model (Table 2). The coefficients for the MMF model are
shown in Table 3.
Fig. 1. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the Huang model.
Fig. 2. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the Baranyi-Roberts model.
Fig. 3. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the modified Gompertz model.
Fig. 4. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the Buchanan-3-phase model.
Fig. 5. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the modified Richard model.
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
HG
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
BR
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
MG
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
B3P
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
MR

BESSM, 2020, Vol 4, No 1, 20-24
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Fig. 6. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the MMF model.
Fig. 7. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the modified logistics model.
Fig. 8. Total no of SARS-CoV-2 death cases in Nigeria as of 15th of July
2020 as modelled using the von Bertalanffy model.
Table 2. Statistical tests for the various models utilized in modelling the
total no of SARS-CoV-2 death cases in Nigeria as at 15th of July, 2020.
Model
p
RMSE
R2
adR2
AF
BF
AICc
Huang
4
0.064
0.995
0.995
1.017
1.00
-258.96
Baranyi-Roberts
4
0.068
0.995
0.994
1.017
1.00
-253.65
modified Gompertz
3
0.085
0.991
0.991
1.045
1.00
-234.55
Buchanan-3-phase
3
0.143
0.992
0.992
1.045
0.99
-182.92
modified Richards
4
0.086
0.991
0.991
1.019
1.00
-230.07
MMF
3
0.049
0.997
0.997
1.012
1.00
-289.13
modified Logistics
3
0.130
0.979
0.978
1.033
0.99
-192.33
von Bertalanffy
3
0.067
0.995
0.994
1.015
1.00
-259.15
Note: p is no of parameter
Table 3. Coefficients as modelled using the MMF model.
Parameters
Value
95% Confidence interval
µ
m
0.02989
0.02861 to 0.03108
δ
1.654
1.57 to 1.74
ymax
1,717
1,428 to 2,123
Table 4. Predictions of COVID-19 pandemic for Nigeria based on the
MMF model.
Prediction
Mean
95% Confidence
Interval
Maximum number of total death
cases by the end of Covid-19
1,717
1,428 - 2,123
Maximum number of total death
cases by 15
th
of August 2020
940
847 - 1,043
Maximum number of total death
cases by 15
th
of September 2020
1,101
968 - 1,252
The parameters obtained from the MMF model include
maximum growth rate (log) of 0.02 (95% CI from 0.02 to 0.03),
curve constant (
δ
) that affects the infection point of 1.61 (95% CI
from 1.42 to 1.79) and maximal total number of death cases (yma x)
of 1,717 (95% CI from 1,428 to 2,123). The MMF model
predicted that cumulative COVID-19 death will ceased to
increase about 307 days from 15th of July 2020 based on the
lower bound of the 95% CI from the calculated maximum
number of total cases (ymax) while the mean and upper 95% CI
bound values failed to be predicted by the software for their
number of days. The MMF predicted that the total number of
death cases for Nigeria on the coming 15th of August and 15th of
September 2020 were 940 (95% CI of 847 to 1,043) and 1,101
(95% CI of 968 to 1,252), respectively. This prediction has to be
taken with caution since the model failed to predict the number
of days for the mean and upper 95% CI values and the number of
days for COVID-19 to end may be much larger.
The MMF model was developed initially to describe a wide
variety of nutrient-response relationships in higher organisms
and has never been used to model infection cases for population’s
virus-infected [2]. The model has been popular and sometimes is
the best model in modelling the growth of animals such as rabbit,
sheep, horse and even microorganisms [17, 18, 19, 20,21]. The
model has also been used to model the yield of oil palm [22],
bacterial production of ethanol [23] and even in finance [2]. It has
to be remembered that the validity of the model results depend
on a case by case basis and the predictability of the model
especially the upper asymptote depends on the effectiveness of
lockdown, no mutation of the virus that increases the infectivity
rate of the virus to name a few. Certainly, the models must be
revisited on a few months interval to remodel the data in
anticipation of waves of new infection so that a better prediction
can be obtained. The existing asymptote can then be used as a
baseline data to predict new asymptotes of maximum infection.
CONCLUSION
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
MMF
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
ML
0
1
2
3
025 50 75 100 125 150
Day after 1st case
Log No of Infected
EXP
VB

BESSM, 2020, Vol 4, No 1, 20-24
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In the conclusion, the MMF model was found to the best model
in based on statistical tests such as corrected AICc (Akaike
Information Criterion), bias factor (BF), adjusted coefficient of
determination (R2) and root-mean-square error (RMSE).
Parameters obtained from the fitting exercise were maximum
growth rate (µm), the curve constants (
δ
) and maximal total
number of death cases (Ymax). The parameters obtained from the
MMF model include maximum growth rate (log) of 0.02 (95%
CI from 0.02 to 0.03), curve constant (
δ
) that affects the inflection
point of 1.61 (95% CI from 1.42 to 1.79) and maximal total
number of death cases (ymax) of 1,717 (95% CI from 1,428 to
2,123). The model allows estimation of the total number of death
cases, and this prediction varies depending upon the number of
different factors. Upon all this, the model's predictive capacity
used in this study is a powerful tool used by epidemiologists to
track and evaluate the extent of Nigeria’s COVID-19 in the
forthcoming months.
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