Numerical Implementation of a Susceptible - Infected - Recovered (SIR)
Mathematical Model of Covid-19 Disease in Nigeria
OGUNLADE TEMITOPE OLU1,*, OGUNMILORO OLUWATAYO MICHAEL1,
FADUGBA SUNDAY EMMANUEL1, OGINNI OMONIYI ISRAEL1,
OLUWAYEMI MATTHEW OLANREWAJU2,3,a, OKORO JOSHUA OTONRITSE3,4,
OLATUNJI SUNDAY OLUFEMI5
1Department of Mathematics, Ekiti State University,
Ado-Ekiti, 360001, Ekiti State,
NIGERIA
2Department of Mathematics and Statistics, Margaret Lawrence University,
Galilee, Delta State,
NIGERIA
3Landmark University SDG 4 (Quality Education Research Group), Landmark University,
Omu-Aran, Kwara State,
NIGERIA
4Department of Physical Sciences, Landmark University,
Omu-Aran, Kwara State,
NIGERIA
5Department of Mathematical Sciences, Federal University of Technology,
Akure,
NIGERIA
aORCiD: https://orcid.org/0000-0003-3170-6818
*Corresponding Author
Abstract: - In this study, we examine the dynamics of the Susceptible Infected Recovered (SIR) model in the
context of the COVID-19 outbreak in Nigeria during the year 2020. The model is validated by fitting it to data
on the prevalence and active cases of COVID-19, sourced from a government agency responsible for disease
control. Utilizing the parameters associated with the disease prevalence, we calculate the basic reproduction
number , revealing its approximate value as 10.84. This suggests an average infection rate of around 10
human individuals, indicating the endemic nature of the disease in Nigeria. The impact of variation of recovery
rate via treatment is examined, demonstrating its effectiveness in reducing disease prevalence when is
below or above unity. To numerically implement the model, we employ the Sumudu Decomposition Method
(SDM) and compare its results with the widely used Runge–Kutta fourth-order (RK4) method, implemented
through the Maple software. Our findings indicate a mutual efficiency and convergence between the two
methods, providing a comprehensive understanding of the COVID-19 dynamics in Nigeria.
Key-Words: - COVID – 19, Basic reproduction number, Runge Kutta Method, Sumundu Decomposition
Method, Susceptible-Infected-Recovered (SIR) model, Non-Pharmaceutical Strategies (NPIs).
Received: March 28, 2023. Revised: November 23, 2023. Accepted: December 27, 2023. Published: February 27, 2024.
1 Introduction
Deterministic models use the concept of
mathematical techniques to develop an accurate
depiction of a system. Epidemiologically, models
enable the description of the evolution and
transmission of infection, future-term behavior, and
possible control strategies to eradicate disease
spread. Several works on mathematical modeling of
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Ogunlade Temitope Olu,
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diseases have been developed together with
determinant factors like incidence, spread, and
persistence, [1]. The SIR model was first studied by
[2] and [3], where the impact of demographic
factors like births, deaths, and migration are studied.
Later, SIR models with vital dynamics, [4] and other
forms of extensions with vaccination, treatment,
relapse, susceptibility, etc., are studied by different
authors [3], [5], [6], [7], [8], [9].
A novel disease named coronavirus (COVID -
19) disease evolved in Wuhan, China, December
2019. This disease became the most devastating
health challenge experienced in the world after the
1918/1919 pandemic of influenza. The World
Health Organization (WHO) announced the disease
as a pandemic on March 11, 2020, and by the end of
the year 2020, over 90 million cases have been
recorded and more than two million lives lost, as a
result of the COVID – 19 menace. Nigeria is one of
the most affected countries in Africa with COVID-
19 cases. By the end of year 2020, 87607 and 1361
cases of COVID – 19 infection and casualties were
recorded, [10], [11], [12], [13], while efforts by the
WHO are ongoing to circulate vaccines and possible
drugs across the world to treat and minimize the
high rate of the infection spread.
Several deterministic and stochastic models
have been derived to explain and predict the
transmission of COVID – 19 in Nigeria. A study,
[14], formulated a model with Non-Pharmaceutical
Strategies (NPIs) fitted to the prevalence date as of
March 30, 2020. Their results show that COVID –
19 can be effectively mitigated using a moderate
level of compliance with NPIs to avoid a second
wave of the pandemic. In [15], derivation of a
model was done to forecast COVID – 19 dynamics
using the prevalence data as of March 16, 2020.
Their results reveal that if at least 55 percent of
humans can adhere to social distancing and face
mask usage, the disease will be eradicated. Also if
the case findings for humans with symptoms are
increased to 0.8 per day associated with social
distancing will lead to a reduction of COVID-19
disease incidence. The studies in [16] and [17],
considered the effect of optimal management in
minimizing COVID – 19 infection in Nigeria. Other
works on the formulation of COVID – 19 using
qualitative and quantitative techniques include the
works of [18], [19], [20], [21]. The SIR model is the
basic framework for describing disease spread in
population dynamics. The recent coronavirus
(COVID-19) disease across the world has majorly
been described using the SIR model, [22], as well as
other diseases in [23], [24] and [25]. The idea of
SDM was first conceived in [26]. Also, the studies
[2] and [27], employed hybrid methods of SDM and
Laplace to compute the system of ordinary
differential equations, other works on the
application of SDM can be seen in the works of
[27], [28] and [29], while works on the modification
of SDM, using the other semi-analytical approaches
can be seen in [30], [31], [32] and [33].
Inspired by the cited works on the mathematical
modeling approach to COVID – 19 disease spread
in Nigeria together with different applications of
numerical methods to obtain approximate solutions
of models, in this work we consider fitting a SIR
model to the COVID – 19 prevalent and active cases
in Nigeria in relation to year 2020 utilizing the non-
linear least square method by the use of MAPLE
computational software, such that the estimated and
fitted values were used to analyze and obtain the
value of [25]. Also, the numerical solution of
the model using the SDM in comparison with the
RK4 method is obtained. It is to the best
understanding of the authors that this has not been
done by the aforementioned authors. The
subsequent parts of the article are sectionalized.
Section 2 involves the model formulation and
analysis and data fitting analysis. Section 3 involves
the numerical implementation of the model
equations by the use of SDM and RK4 methods,
while Section 4 discusses the results and conclusion.
See also a study in [34] as a case study of Lagos
State, Nigeria.
2 Model Formulation
The model is divided into the Susceptible;
Infected ; and Recovered , where the
whole human population yields
. ∏ denotes the crude birth
rate, β represents the transmission rate per COVID –
19 infective, ϕ is the recovery rate for COVID – 19
infection, κ is the mortality related to COVID – 19
infection and µ is the natural death rate. Using these
descriptions, the model is expressed as:
=
(1)
Analytically, Eq. (1) is positively invariant and
well posed in the region:
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(2)
The average time of COVID - 19 infection is
and since the infectious individual
transmits COVID - 19 disease at the rate β, then
is computed to be
. measures the
number of secondary COVID-19 infectious humans
per COVID-19 index case in a naive population of
vulnerable human population. Solving for
equilibrium solutions in Eq.(1) when independent of
time, yields
(3a)
(3b)
Where and represent the COVID – 19 free
and endemic equilibrium points respectively. If
, then the disease vanishes, but if ,
then the infection becomes persistent in the human
and environment host population.
2.1 Model Fitting
To validate the model system Eq. (1), Data on
cumulative and active cases of COVID-19 in
Nigeria, reported by NCDC [12] for the year 2020 is
applied. The parameters estimated are the
transmission rate and progression rate. The
parameters were obtained from the literature. For
instance, is the crude birth rate of Nigeria , which
is estimated to be 37.269 per 1000 people, and the
death rate in Nigeria is taken to be 11.577, so that
= 3.2192 year-1 is the restricted human population in
a COVID-19 free community and
= 0.0863 year-1.
Also, the total COVID-19 induced death rate in
Nigeria for year 2020 is 1,361, where
day [12]. To apply the non-linear least
square method for model validation using the
available data, at time t, vector z of Eq. (1) and
vector of the unknown parameter, Eq. (1) follows
the form:
(4)
Also, the residual form of Eq.(4) is expressed as:
Residual (5)
and the error is given by:
Error
, (6)
where is denoted by the actual data and
is the solution to Eq. (4) for . In
addition, the minimization function is given by:
min error based on Eq. (4) (7)
is applied to compute the optimal parameters.
2.1.1 A Non Linear Algorithm For Estimation
of the Parameter
The non-linear algorithm governing the parameter
estimation applied to obtaining the fitted values is
given below.
1. Take up state and parameter values.
2. Compute Eq.(1) by the use of RK4 method
together with step 1.
3. Check the error.
4. Minimize to derive a new set of parameter
values of Eq.(1) to agree with actual data.
5. Investigate convergence. If it doesn’t converge,
return to 2.
6. Continue iteration, till the convergence for the
acquired parameters are achieved.
The model fit simulation using the data in [25],
via the non-linear least square algorithm is
displayed in Figure 1 and Figure 2.
Fig. 1: Model fit of cumulative cases of COVID –
19 in Nigeria relative to the year 2020
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Fig. 2: Model fit of active cases of COVID – 19 in
Nigeria in relation to the year 2020.
3 Numerical Implementation
3.1 The SDM and RK4 Method
To understand the essentials of SDM, we consider
the denotation of in-homogenous non-linear
ordinary differential equation with its initial data as:
. (8)
Where represents the derivative of first order,
represent a differential operator, denotes the
non-linear term, while is the source term, [24].
On the application of SDM to Eq. (8) yields:
(9)
Also, following the same procedure, Eq. (9) yields:
.
(10)
The inverse of Eq. (10), yields:
.
(11)
So that Eq. (11) is denoted as an infinite series given
by:
. (12)
Also, the non-linear term in Eq.(11) can be
expressed as:
(13)
Where are Adomian polynomials of …
such that:
,….
(14)
Putting Eqs. (13) and (14) into Eq. (12), we have:
. (15)
Where is the expression arising from the
source term and the initial data.
Making use of the Adomian Decomposition Method
(ADM) to Eq.(15), we obtain:
, (16)
, (17)
, (18)
. (19)
From Eqs. (16-19), we obtain the values of , ,
. We implement the SDM on Eq. (1) by using the
values; 0863, ϕ
and . For the
purpose of illustration, let
, .
The SDM of Eq. (1) yields:
(20)
.
Further, the inverse SDM of Eq. (20) yields
(21)
Assuming the solution in Eq. (21) as infinite series
of unknown function then:
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(22)
,
, (23)
So that:
0.87607
0.13610
(24)
0.67507
Further computation of Eq. (23) together with Eq.
(24) yields the following series solution as:
+… (25)
+… (26)
+… (27)
Moreover, the RK4 scheme is applied to Eq.
(1). The RK4 scheme is given by:
(k1 + 2 k2 + 2 k3 + k4). (28)
Where
= hf (xn, yn),
= hf(xr +
h, yr +
),
= hf(xr +
h, yr +
), (28)
= hf(xr + h, yr + ).
and
= +
(+ 2+ 2k3 + k4) h,
= +
(l1 + 2l2 + 2l3 + l4) h, (29)
= +
(m1 + 2m2 + 2m3 + m4) h.
and
= ∏ - β - µ,
L1 = β – (µ + ϕ + k + r) , (30)
M1 = ϕ - µRc.
and
= ∏ - β( +
)
( +
) - µ( +
),
L2 = β( +
) ( +
) (31)
– (µ + ϕ + k + r) (+
),
M2 = ϕ(+
) - µ( +
).
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and
= ∏ - β( +
) ( +
)
- µ( +
),
L3 = β( +
) ( +
) (32)
– (µ + ϕ + k + r) (+
),
M3 = ϕ(+
) - µ( +
).
and
K4 = ∏ - β( +
) ( +
)
If n = 0
= +
(+ 2+ 2+ )h,
= +
(L1 + 2L2 + 2L3 + L4)h,
= +
(m1 + 2m2 + 2m3 + m4)h, (34)
= +
(+ 2+ 2+ )h,
= +
(L1 + 2L2 + 2L3 + L4)h,
= +
(m1 + 2m2 + 2m3 + m4)h.
4 Discussion of Results and Conclusion
4.1 Discussion of Results
In the course of simulation, the parameter and
variable values given in Section 2 are adopted to
obtain the numerical results in Table 1, by
comparing SDM and RK4 method, while, the errors
between the two methods are given in Table 2.
L4 = β( +
) ( +
) (33)
– (µ + ϕ + k + r) (+
),
M4 = ϕ(+
) - µ( +
).
Table 1. Comparison between SDM and RK4 for Approximate Solutions of Model Eq. (1).
Time(Months)
(SDM)
(RK4)
(SDM)
(RK4)
(SDM)
(RK4)
1
2.4636
2.4640
1.2124
1.2315
0.7015
0.7020
2
2.6863
2.6875
6.8525
6.8555
1.0683
1.0693
3
29.9452
29.9463
20.8694
20.8703
2.0905
2.1001
4
92.9325
92.9341
47.0761
47.0779
4.0831
4.0850
5
205.2676
205.2685
89.2856
89.2871
7.3611
7.3630
6
308.5699
308.5710
151.3109
151.3121
12.2395
12.2405
7
632.4588
632.4599
236.9650
236.9668
19.0333
19.0372
8
974.5537
974.5570
350.0609
350.0615
28.0575
28.0620
9
1420.4740
1420.4770
494.4116
494.4120
39.6271
39.6352
10
1983.8391
1983.8402
673.8301
673.8312
54.0571
54.0582
11
2678.2684
2678.2715
892.1294
892.1301
71.6625
71.6637
12
3517.3813
3517.3831
1153.1225
1153.1225
97.7583
97.7590
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Table 2. Errors between RK4 and SDM approximate solutions for model Eq. (1).
Time(Months)
(RK4 – SDM)
(RK4 – SDM)
(RK4 – SDM)
1
0.0004
0.0011
0.0005
2
0.0012
0.0030
0.0010
3
0.0011
0.0009
0.0016
4
0.0006
0.0018
0.0019
5
0.0009
0.0011
0.0019
6
0.0011
0.0012
0.0010
7
0.0011
0.0018
0.0039
8
0.0033
0.0006
0.0045
9
0.0030
0.0004
0.0081
10
0.0011
0.0011
0.0011
11
0.0031
0.0007
0.0012
12
0.0018
0.0075
0.0007
The graphical representations in Figure 1 and
Figure 2 depict the model fit for cumulative and
active COVID-19 cases in Nigeria throughout the
year 2020. Notably, a consistent upward trend is
evident over time, attributed to a significant lack of
adherence to COVID-19 protocols. This underscores
the imperative for stringent enforcement of non-
pharmaceutical interventions to curb the rapid
spread of the disease. Additionally, the examination
of Table 1 and Table 2 reveals a harmonious
agreement between the two numerical methods,
displaying minimal errors. Furthermore, it is
noteworthy that the Sumudu Decomposition Method
(SDM) exhibits better performance in both
efficiency and convergence when compared to the
Runge–Kutta fourth-order (RK4) method, while
Figure 3 and Figure 4 describes the effect of
recovery rate on in 12 months the host
community. It is observed that the curve converges
to the disease – free and endemic equilibrium when
When the recovery rate
through treatment is increased, that is, ,
reduces but not below unity. This highlights the
challenge of completely eradicating the virus
through treatment alone. The relevance of umerical
modeling techniques and recovery rate through
treatment for the containment of the spread of
COVID-19 has been shown in this work. Though
effective treatment is needed to ameliorate the
impact of the virus, preventive measures is essential
to reduce the spread due to the endemic nature of
the disease.
Fig. 3: Behavior of recovery rate on
when <1 and
Fig. 4: Behavior of recovery rate on
when <1 and
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5 Conclusion
The compartmental model has been used in this
study to analyze COVID-19 cumulative and active
cases in Nigeria throughout the year 2020. The
simulations, based on fitted and estimated
parameters from existing literature, showed a basic
reproduction number of approximately 10.84.
This finding reveals the endemic nature of COVID-
19 in Nigeria, with an average infection rate of at
least 10 individuals. Furthermore, the investigation
into the impact of the recovery rate on showed
that an increase in the recovery rate through
treatment can reduce, although it remains above
unity. This suggests that treatment alone may not be
sufficient to effectively combat the disease. The
numerical implementation of the model equations
using the Sumudu Decomposition Method (SDM)
and the Runge–Kutta fourth-order (RK4) method
demonstrated their efficiency, with SDM exhibiting
better convergence. Consequently, health policy-
makers in Nigeria are advised to intensify the
implementation of Non-Pharmaceutical
Interventions (NPIs) recommended by the World
Health Organization (WHO). This strategic scaling
up of NPIs is crucial to mitigate the spread of
COVID-19 and reduce below unity, ultimately
aiming to eliminate the disease. Mathematically, this
study suggests potential extensions into spatial,
fractional order, stochastic, and optimal control
problems. These avenues of research could further
enhance our understanding of the dynamics of
COVID-19 and contribute to the development of
more effective strategies for disease control and
prevention.
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DOI: 10.37394/23208.2024.21.7
Ogunlade Temitope Olu,
Ogunmiloro Oluwatayo Michael et al.
E-ISSN: 2224-2902
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WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2024.21.7
Ogunlade Temitope Olu,
Ogunmiloro Oluwatayo Michael et al.
E-ISSN: 2224-2902
74
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