A Dynamical and Sensitivity Analysis of the Caputo Fractional-Order Ebola Virus Model: Implications for Control Measures

Abstract

The recurrence of outbreaks in cases of Ebola virus among African countries remains one of the greatest issues of concern. Practices such as hunting or consumption of contaminated bush meat, unsafe funeral practices, and environmental contamination have all been implicated as possible contributors. This paper investigates the transmission dynamics of the Ebola virus model in the setting of a Caputo fractional-order derivative that accounts for both direct and indirect transmissions of the virus. By employing the concept of fixed theorems, we derived the existence and uniqueness results of the model. Moreover, we analyzed the forward normalized sensitivity indices to identify the critical parameters for controlling the infection and found that reducing the contact rate between infected individuals and susceptible vectors is vital to limiting the virus's spread. Comparing the proposed fractional-order model with those of the previously developed integer-order model numerically, we found that the proposed model provides more reliable information on the model's dynamics. Thus, we conclude that the Caputo fractional-order operator is a precise tool for describing the proposed model behavior and can help understand the complexities of Ebola virus disease outbreaks.

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P-ISSN 2586-9000
E-ISSN 2586-9027
Homepage : https://tci-thaijo.org/index.php/SciTechAsia Science & Technology Asia
Vol.28 No.4 October - December 2023 Page: [26-37]
Original research article
A Dynamical and Sensitivity Analysis of the
Caputo Fractional-Order Ebola Virus
Model: Implications for Control Measures
Idris Ahmed1,*, Abdullahi Yusuf2,3, Jessada Tariboon4,
Mubarak Muhammad5, Fahd Jarad6, Badamasi Bashir Mikailu7
1Department of Mathematics, Sule Lamido University, Kafin Hausa 741103, Nigeria
2Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey
3Department of Mathematics, Federal University Dutse, Jigawa 720223, Nigeria
4Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology
North Bangkok, Bangkok 10800, Thailand
5Graduate School (Neurosience program), Khon Kaen University, Khon Kaen 40002, Thailand
6Department of Mathematics, Cankaya University, Ankara 06790, Turkey
7Department of Mathematics, Bayero University Kano, Kano 700006, Nigeria
Received 24 May 2023; Received in revised form 19 October 2023
Accepted 3 November 2023; Available online 27 December 2023
ABSTRACT
The recurrence of outbreaks in cases of Ebola virus among African countries remains
one of the greatest issues of concern. Practices such as hunting or consumption of contam-
inated bush meat, unsafe funeral practices, and environmental contamination have all been
implicated as possible contributors. This paper investigates the transmission dynamics of the
Ebola virus model in the setting of a Caputo fractional-order derivative that accounts for both
direct and indirect transmissions of the virus. By employing the concept of fixed theorems,
we derived the existence and uniqueness results of the model. Moreover, we analyzed the
forward normalized sensitivity indices to identify the critical parameters for controlling the
infection and found that reducing the contact rate between infected individuals and suscep-
tible vectors is vital to limiting the virus’s spread. Comparing the proposed fractional-order
model with those of the previously developed integer-order model numerically, we found
that the proposed model provides more reliable information on the model’s dynamics. Thus,
we conclude that the Caputo fractional-order operator is a precise tool for describing the
proposed model behavior and can help understand the complexities of Ebola virus disease
outbreaks.
Keywords: Ebola virus; Fixed point theorems; Mathematical model; Numerical simula-
tions; Sensitivity analysis
*Corresponding author: idris.ahmed@slu.edu.ng doi: xx.xxxxx/scitechasia.xxxx.xx

I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
1. Introduction
Ebola virus disease, one of the dead-
liest viral diseases, is markedly character-
ized by hemorrhagic fever [1]. The virus
was first discovered in 1976 in the Demo-
cratic Republic of the Congo, and secondly
within the same year in Sudan, some miles
away from the vicinity of the first case [2]
(Fig. 1). The name ‘Ebola’ originated from
the name of a river near the village of Yam-
bku, where the virus was first discovered.
From the date of discovery to March 25,
2020, the virus has accounted for the deaths
of an estimated 2,267 people, and an out-
break was reported in about 38 predomi-
nantly African countries [3].
Fig. 1. Heaths from the Ebola virus in Africa
from 1976 to 2014.
The Ebola virus disease remains an
important health problem that poses a threat
to public health. First is the high case-
fatality rate associated with the disease,
with an average case fatality rate of about
90 being reported in African species of the
virus [4] (Fig. 2). Moreover, the absence of
definitive treatment as a cure or a defined
vaccine makes only supportive treatment
the viable treatment option [5–8]. Lastly,
there is the high transmissibility of the virus
via direct contact with a patient or the pa-
tient’s bodily fluids of an infected individ-
ual, thereby prompting quarantine measures
with any form of contact with a confirmed
infected individual [9] (Fig. 3).
The recurrence of outbreaks in cases
Fig. 2. A fact file on how the Ebola virus at-
tacks. Picture: AFP.
Fig. 3. Ebola-transmission-medium.
of Ebola virus among African countries re-
mains one of the greatest issues of con-
cern. For instance, the most recent was
on September 20, 2022, when an outbreak
was reported in Uganda after a suspected
case was confirmed at the Uganda Virus
Research Institute (UVRI). The reasons be-
hind the recurrence and resurgence of the
Ebola outbreak are not far-fetched. Prac-
tices such as hunting or consumption of
contaminated bush meat, unsafe funeral
practices, and environmental contamination
have all been implicated as possible contrib-
utors [10–15].
A comprehensive understanding of
the nature of a pandemic is essential for ef-
fectively limiting its spread and reducing
infections. To gain insights into the trans-
mission dynamics and potentially work to-
wards eradicating the disease, numerous
researchers have developed mathematical
models for the spread of infectious diseases
[16–18]. These models provide valuable
tools for studying and analyzing the dy-
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I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
namics of transmission, predicting future
trends, evaluating the impact of interven-
tions, and developing strategies for con-
trol and prevention. By utilizing mathe-
matical models, researchers can gain valu-
able insights into the mechanisms of dis-
ease transmission and identify effective
measures to mitigate its impact on pub-
lic health. Most commonly, mathemat-
ical models for the spread of infectious
diseases are based on classical differen-
tial equations. However, fractional-order
differential equations have gained promi-
nence and outperformed standard mathe-
matical modeling approaches. The use of
fractional-order differential equations has
become significant due to their wide ap-
plications across various fields such as sci-
ence, engineering, finance, and epidemiol-
ogy [19–22]. Recent research has demon-
strated the effectiveness of fractional-order
modeling in capturing complex dynamics
and providing more accurate descriptions
of real-world phenomena [23–31] and ref-
erences cited therein. These advancements
highlight the potential of fractional-order
modeling to enhance our understanding of
epidemiological processes and improve the
accuracy of disease transmission predic-
tions.
The paper is organized into seven
sections. Section 2 presents the preliminary
concepts of Caputo fractional-order deriva-
tives. Sections 3 and 4 provide the develop-
ment of the Caputo fractional-order math-
ematical model and analysis of the basic
properties of the model, including the ex-
istence and uniqueness of solutions, pos-
itivity, and boundedness. The sensitivity
analysis in relation to the basic reproduction
number of the model is discussed in sections
5. In section 6, a dynamically consistent
numerical scheme is utilized, and numeri-
cal simulations are presented to support the
theory. Finally, in section 7, the concluding
remarks on how their findings relate to ex-
isting literature and potential extensions of
the model were provided.
2. Some Basic Background of Caputo
Fractional-Order Derivatives
The main aim of this section is to
recall some basic background and notions
of the Caputo fractional-order derivative,
which are key for the theoretical analysis.
Definition 2.1 ([20]).The fractional oper-
ator defined by
𝐼𝑟
0𝑓(𝑡)=1
Γ(𝑟)𝑡
0
𝑓(𝑡)(𝑡−𝑧)𝑟−1𝑑 𝑧, (2.1)
is called the Riemann-Liouville fractional
integral of order 𝑟(0< 𝑟 < 1)of the func-
tion 𝑓∈𝐿1[0, 𝑇 ]for 0< 𝑡 < 𝑇 . Moreover,
Γ(𝑟)=∞
0
𝑧𝑟−1𝑒−𝑧𝑑𝑧, 𝑟 ∈C/{0,−1,−2, . . .},
is the gamma function.
Definition 2.2 ([20]).Suppose the function
𝑓∈𝐶𝑛[0, 𝑇 ], 𝑛 ∈Nand 𝑡 > 0.The frac-
tional operator
𝐶𝐷𝑟
0𝑓(𝑡)=1
Γ(1−𝑟)𝑡
0
1
(𝑡−𝑧)𝑟
𝑑
𝑑𝑡 𝑓(𝑧)𝑑𝑧,
(2.2)
is referred to the Caputo fractional deriva-
tive of order 𝑟(0< 𝑟 < 1). Note that if
𝑟→1then 𝐶𝐷𝑟
0𝑓(𝑡)=𝑑
𝑑𝑡 𝑓(𝑡).
Lemma 2.3 ([20]).Suppose 𝑓∈
𝐶([0, 𝑇 ],R)and any 𝑢∈𝐶1[0, 𝑇 ].
Then, 𝑢(𝑡)is a solution of:
𝐶𝐷𝑟
0𝑢(𝑡)=𝑓(𝑡), 𝑡 ∈ [0, 𝑇 ],0< 𝑟 ≤1,
𝑢(0)=𝑢0,
if and only if 𝑢(𝑡)satisfies the integral equa-
tion:
𝑢(𝑡)=𝑢0−1
Γ(𝑟)𝑡
0
𝑓(𝑡)(𝑡−𝑧)𝑟−1𝑑 𝑧.
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I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
3. Description of the Ebola Virus
Model
In the context of Caputo fractional
derivatives, we investigate the dynamic
transmission of the Ebola virus model pro-
posed in [32]. The proposed model is based
on the following setting:
𝐶𝐷𝑟𝑆=𝜋− (𝛽1𝐼+𝛽2𝐷+𝜆𝑃)𝑆−𝜇𝑆,
𝐶𝐷𝑟𝐼=(𝛽1𝐼+𝛽2𝐷+𝜆𝑃)𝑆− (𝜇+𝛿+𝛾)𝐼,
𝐶𝐷𝑟𝑅=𝛾𝐼 −𝜇𝑅,
𝐶𝐷𝑟𝐷=(𝜇+𝛿)𝐼−𝑏𝐷,
𝐶𝐷𝑟𝑃=𝜎+𝜉𝐼 +𝛼𝐷 −𝜂𝑃.
(3.1)
with the initial conditions
𝑆(0)=𝑆0≥0, 𝐼 (0)=𝐼0≥0, 𝑅 (0)=𝑅0≥0,
𝐷(0)=𝐷0≥0,and 𝑉(0)=𝑉0≥0.
(3.2)
Tables 1-2 provides the meaning of each of
the state variables as well as the parameters.
Table 1. States variables.
Compartment Description
𝑆Susceptible human pop-
ulation
𝐼Infected human popula-
tion with Ebola
𝑅Recovered human popu-
lation
𝐷Infected and deceased
human population
𝑃Ebola virus pathogens
4. Theoretical Analysis of the Ebola
Virus Model
This section presents the existence
and uniqueness of solutions to model (3.1)
via the techniques of fixed point theorems.
Table 2. Meaning of the parameters.
Parameters Biological Meanings
𝜋Recruitment rate of
susceptible individuals
𝛽1Infectious individuals
effective contact rate
𝛽2Deceased individuals
effective contact
𝜆Ebola virus effective contact
rate
𝜇Natural death rate
𝛿Disease-induced death rate
𝛾Recovered rate
1/𝑏Deceased individuals mean
caring duration
𝜎Recruitment rate of Ebola
virus in the environment
𝜉Shedding rate of infectious
individuals
𝛼Shedding rate of deceased
individuals
𝜂Decay rate of Ebola virus in
the environment
4.1 Existence and uniqueness result
In this subsection, by making use of
the fixed point theory, the existence and
uniqueness of solution for model (3.1) were
investigated. Let denote B(𝐽)the Banach
space of all continuous real-valued function
defined on 𝐽=[0, 𝑇 ]with sub norm and
Q=B(𝐽) × B(𝐽) × B(𝐽) × B(𝐽) × B(𝐽)
with the norm ∥ (𝑆, 𝐼 , 𝑅, 𝐷, 𝑃) ∥ =∥𝑆∥ +
∥𝐼∥+∥𝑅∥+∥𝐷∥+∥𝑃∥,∥𝑆∥=sup
𝑡∈𝐽
|𝑆(𝑡)|,
∥𝐼∥=sup
𝑡∈𝐽
|𝐼(𝑡)|,∥𝑅∥=sup
𝑡∈𝐽
|𝑅(𝑡)|,∥𝐷∥=
sup
𝑡∈𝐽
|𝐷(𝑡)|,∥𝑃∥=sup
𝑡∈𝐽
|𝑃(𝑡)|.Applying the
Caputo operator to the Ebola virus model
(3.1) gives
𝑆(𝑡) − 𝑆(0)=𝐶𝐷𝑟[𝜋− ( 𝛽1𝐼+𝛽2𝐷
+𝜆𝑃)𝑆−𝜇𝑆],
𝐼(𝑡) − 𝐼(0)=𝐶𝐷𝑟[(𝛽1𝐼+𝛽2𝐷+𝜆𝑃)𝑆
−( 𝜇+𝛿+𝛾)𝐼],
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I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
𝑅(𝑡) − 𝑅(0)=𝐶𝐷𝑟[𝛾𝐼 −𝜇𝑅],
𝐷(𝑡) − 𝐷(0)=𝐶𝐷𝑟[(𝜇+𝛿)𝐼−𝑏𝐷],
𝑃(𝑡) − 𝑃(0)=𝐶𝐷𝑟[𝜎+𝜉 𝐼 +𝛼 𝐷 −𝜂𝑃].
(4.1)
Let us denote
𝑓1=𝜋− (𝛽1𝐼+𝛽2𝐷+𝜆𝑃)𝑆−𝜇𝑆,
𝑓2=(𝛽1𝐼+𝛽2𝐷+𝜆𝑃)𝑆− (𝜇+𝛿+𝛾)𝐼,
𝑓3=𝛾𝐼 −𝜇𝑅,
𝑓4=(𝜇+𝛿)𝐼−𝑏𝐷,
𝑓5=𝜎+𝜉𝐼 +𝛼𝐷 −𝜂𝑃.
(4.2)
By means of the Caputo fractional operator,
systems (4.1) can be express as
𝑆(𝑡) − 𝑆(0)=M (𝑟)𝑡
0
𝑓1(𝑟, 𝑧, 𝑆 (𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝐼(𝑡) − 𝐼(0)=M (𝑟)𝑡
0
𝑓2(𝑟 , 𝑧, 𝐼 (𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝑅(𝑡) − 𝑅(0)=M (𝑟)𝑡
0
𝑓3(𝑟, 𝑧, 𝑅 (𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝐷(𝑡) − 𝐷(0)=M (𝑟)𝑡
0
𝑓4(𝑟, 𝑧, 𝐷 (𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝑃(𝑡) − 𝑃(0)=M (𝑟)𝑡
0
𝑓5(𝑟, 𝑧, 𝑃 (𝑡))
(𝑡−𝑧)𝑟𝑑𝑧.
(4.3)
It should be mentioned that
𝑓1(𝑆, 𝑧), 𝑓2(𝐼, 𝑧), 𝑓3(𝑅, 𝑧), 𝑓4(𝐷, 𝑧)
and 𝑓5(𝑃, 𝑧)obeys the Lipschitz condition
if and only if 𝑆(𝑡), 𝐼 (𝑡), 𝑅 (𝑡), 𝐷(𝑡)and
𝑃(𝑡)have an upper bound. Let 𝑆(𝑡)and
𝑆∗(𝑡)be two functions, then
∥𝑓1(𝑟, 𝑧, 𝑆 (𝑡)) − 𝑓1(𝑟, 𝑧, 𝑆∗(𝑡))∥
=∥ − (( 𝛽1𝐼+𝛽2𝐷+𝜆𝑃)
−𝜇)(𝑆(𝑡) − 𝑆∗(𝑡)) ∥ .
(4.4)
For 𝐿1=∥ − (( 𝛽1𝐼+𝛽2𝐷+𝜆𝑃) − 𝜇) ∥,we
get
∥𝑓1(𝑟, 𝑧, 𝑆(𝑡)) − 𝑓1(𝑟, 𝑧, 𝑆∗(𝑡)) ∥ ≤ 𝐿1∥𝑆(𝑡) − 𝑆∗(𝑡) ∥,
(4.5)
and likewise when 𝐿2=∥ − ( 𝜇+𝛿+
𝛾)∥, 𝐿3=−𝜇, 𝐿4=−𝑏and 𝐿5=−𝜂,
we attain as follows:
∥𝑓2(𝑟 , 𝑧, 𝐼 (𝑡)) − 𝑓1(𝑟, 𝑧, 𝐼 ∗(𝑡)) ∥
≤𝐿2∥𝐼(𝑡) − 𝐼∗(𝑡)∥,
∥𝑓3(𝑟, 𝑧, 𝑅 (𝑡)) − 𝑓1(𝑟 , 𝑧, 𝑅∗(𝑡))∥
≤𝐿3∥𝑅(𝑡) − 𝑅∗(𝑡)∥,
∥𝑓4(𝑟, 𝑧, 𝐷 (𝑡)) − 𝑓1(𝑟, 𝑧, 𝐷∗(𝑡)) ∥
≤𝐿4∥𝐷(𝑡) − 𝐷∗(𝑡)∥,
∥𝑓5(𝑟, 𝑧, 𝑃 (𝑡)) − 𝑓1(𝑟, 𝑧, 𝑃∗(𝑡))∥
≤𝐿5∥𝑃(𝑡) − 𝑃∗(𝑡) ∥,(4.6)
thus, the Lipschitz condition is achieved.
Re-written Eq. (4.3) in recursive form
yields:
𝑆𝑛(𝑡)=M(𝑟)𝑡
0
𝑓1(𝑟, 𝑧, 𝑆𝑛−1(𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝐼𝑛(𝑡)=M(𝑟)𝑡
0
𝑓2(𝑟, 𝑧, 𝐼𝑛−1(𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝑅𝑛(𝑡)=M(𝑟)𝑡
0
𝑓3(𝑟, 𝑧, 𝑅𝑛−1(𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝐷𝑛(𝑡)=M(𝑟)𝑡
0
𝑓4(𝑟, 𝑧, 𝐷 𝑛−1(𝑡))
(𝑡−𝑧)𝑟𝑑𝑧,
𝑃𝑛(𝑡)=M(𝑟)𝑡
0
𝑓5(𝑟, 𝑧, 𝑃𝑛−1(𝑡) )
(𝑡−𝑧)𝑟𝑑𝑧,
(4.7)
associated with the initial conditions
𝑆0(𝑡)=𝑆(0), 𝐼0(𝑡)=𝐼(0), 𝑅0(𝑡)=
𝑅(0), 𝐷0(𝑡)=𝐷(0),and 𝑃0(𝑡)=𝑃(0).
By subtracting the successive terms, we get
Φ𝑆, 𝑛 (𝑡)=𝑆𝑛(𝑡) − 𝑆𝑛−1(𝑡)
=M(𝑟)𝑡
0
𝑓1(𝑟, 𝑧, 𝑆𝑛−1(𝑧))
(𝑡−𝑧)𝑟
−𝑓1(𝑟, 𝑧, 𝑆𝑛−2(𝑧))
(𝑡−𝑧)𝑟𝑑𝑧,
Φ𝐼 ,𝑛 (𝑡)=𝐼𝑛(𝑡) − 𝐼𝑛−1(𝑡)
=M(𝑟)𝑡
0
𝑓2(𝑟, 𝑧, 𝐼𝑛−1(𝑧))
(𝑡−𝑧)𝑟
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I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
−𝑓2(𝑟, 𝑧, 𝐼𝑛−2(𝑧))
(𝑡−𝑧)𝑟𝑑𝑧,
Φ𝑅, 𝑛 (𝑡)=𝑅𝑛(𝑡) − 𝑅𝑛−1(𝑡)
=M(𝑟)𝑡
0
𝑓3(𝑟, 𝑧, 𝑅𝑛−1(𝑧))
(𝑡−𝑧)𝑟
−𝑓3(𝑟, 𝑧, 𝑅𝑛−2(𝑧))
(𝑡−𝑧)𝑟𝑑𝑧,
Φ𝐷, 𝑛 (𝑡)=𝐷𝑛(𝑡) − 𝐷𝑛−1(𝑡)
=M(𝑟)𝑡
0
𝑓4(𝑟, 𝑧, 𝐷 𝑛−1(𝑧))
(𝑡−𝑧)𝑟
−𝑓4(𝑟, 𝑧, 𝐷 𝑛−2(𝑧))
(𝑡−𝑧)𝑟𝑑𝑧,
Φ𝑃, 𝑛 (𝑡)=𝑃𝑛(𝑡) − 𝑃𝑛−1(𝑡)
=M(𝑟)𝑡
0
𝑓5(𝑟, 𝑧, 𝑃𝑛−1(𝑧) )
(𝑡−𝑧)𝑟
−𝑓5(𝑟, 𝑧, 𝑃𝑛−2(𝑧))
(𝑡−𝑧)𝑟𝑑𝑧.
(4.8)
If we consider as
𝑆𝑛(𝑡)=
𝑛

𝑘=0
Φ𝑆𝑛,𝑘 (𝑡),
𝐼𝑛(𝑡)=
𝑛

𝑘=0
Φ𝐼𝑛,𝑘 (𝑡),
𝑅𝑛(𝑡)=
𝑛

𝑘=0
Φ𝑅𝑛,𝑘 (𝑡),
𝐷𝑛(𝑡)=
𝑛

𝑘=0
Φ𝐷𝑛,𝑘 (𝑡),
𝑃𝑛(𝑡)=
𝑛

𝑘=0
Φ𝑃𝑛,𝑘 (𝑡),
(4.9)
and employing Eqs. (4.5)-(4.6) and consid-
ering:
Φ𝑆, 𝑛−1(𝑡)=𝑆𝑛−1(𝑡) − 𝑆𝑛−2(𝑡),
Φ𝐼 ,𝑛−1(𝑡)=𝐼𝑛−1(𝑡) − 𝐼𝑛−2(𝑡),
Φ𝑅,𝑛−1(𝑡)=𝑅𝑛−1(𝑡) − 𝑅𝑛−2(𝑡),
Φ𝐷, 𝑛−1(𝑡)=𝐷𝑛−1(𝑡) − 𝐷𝑛−2(𝑡),
Φ𝑃, 𝑛−1(𝑡)=𝑃𝑛−1(𝑡) − 𝑃𝑛−2(𝑡),
we obtain the followings:
∥Φ𝑆, 𝑛 (𝑡)∥ =M (𝑟)𝑡
0
∥Φ𝑆, 𝑛−1(𝑧) ∥
(𝑡−𝑧)𝑟𝑑𝑧,
∥Φ𝐼 ,𝑛 (𝑡) ∥ =M (𝑟)𝑡
0
∥Φ𝐼 ,𝑛−1(𝑧) ∥
(𝑡−𝑧)𝑟𝑑𝑧,
∥Φ𝑅, 𝑛 (𝑡)∥ =M (𝑟)𝑡
0
∥Φ𝑅,𝑛−1(𝑧) ∥
(𝑡−𝑧)𝑟𝑑𝑧,
∥Φ𝐷, 𝑛 (𝑡)∥ =M (𝑟)𝑡
0
∥Φ𝐷, 𝑛−1(𝑧) ∥
(𝑡−𝑧)𝑟𝑑𝑧,
∥Φ𝑃, 𝑛 (𝑡)∥ =M (𝑟)𝑡
0
∥Φ𝑃, 𝑛−1(𝑧) ∥
(𝑡−𝑧)𝑟𝑑𝑧.
(4.10)
From the results above, we prove the fol-
lowing theorem.
Theorem 4.1. The Caputo fractional-order
Ebola virus model (3.1) has a unique solu-
tion if
M(𝑟)
𝑟𝑇𝑟𝐿𝑗<1, 𝑗 =1, . . . , 5,(4.11)
is true when 𝑡∈ [0, 𝑇 ].
Proof. As we can see from the above, the
functions 𝑆(𝑡), 𝐼 (𝑡), 𝑅 (𝑡), 𝐷(𝑡)and 𝑃(𝑡)
are bounded and 𝑓1, 𝑓2, 𝑓3, 𝑓4, 𝑓5obeys the
Lipschitz condition. So, with the help of the
recursive techniques and Eq. (4.10), gives
∥Φ𝑆, 𝑛 (𝑡)∥ ≤ ∥𝑆0(𝑡) ∥ M (𝑟)
𝑟𝑇𝑟𝐿1𝑛
,
∥Φ𝐼 ,𝑛 (𝑡) ∥ ≤ ∥𝐼0(𝑡) ∥ M (𝑟)
𝑟𝑇𝑟𝐿2𝑛
,
∥Φ𝑅, 𝑛 (𝑡)∥ ≤ ∥𝑅0(𝑡) ∥ M (𝑟)
𝑟𝑇𝑟𝐿3𝑛
,
∥Φ𝐷, 𝑛 (𝑡)∥ ≤ ∥𝐷0(𝑡) ∥ M (𝑟)
𝑟𝑇𝑟𝐿4𝑛
,
∥Φ𝑃,𝑛(𝑡) ∥ ≤ ∥𝑃0(𝑡) ∥ M (𝑟)
𝑟𝑇𝑟𝐿5𝑛
.
(4.12)
Thereby, it can be considered that for
𝑛→ ∞,∥Φ𝑆, 𝑛 (𝑡)∥ → 0,∥Φ𝐼 , 𝑛 (𝑡)∥ →
31

I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
0,∥Φ𝑅, 𝑛 (𝑡)∥ → 0,∥Φ𝐷 ,𝑛 (𝑡) ∥ → 0and
∥Φ𝑃,𝑛(𝑡) ∥ → 0.Moreover, by triangle in-
equality and the system (4.12) for any 𝑝, we
have
∥𝑆𝑛+𝑝(𝑡) − 𝑆𝑛∥ ≤
𝑛+𝑝

𝑘=𝑛+1
𝑗𝑘
1=𝑗𝑛+1
1−𝑗𝑛+𝑝+1
1
1−𝑗1
,
∥𝐼𝑛+𝑝(𝑡) − 𝐼𝑛∥ ≤
𝑛+𝑝

𝑘=𝑛+1
𝑗𝑘
2=𝑗𝑛+1
2−𝑗𝑛+𝑝+1
2
1−𝑗2
,
∥𝑅𝑛+𝑝(𝑡) − 𝑅𝑛∥ ≤
𝑛+𝑝

𝑘=𝑛+1
𝑗𝑘
3=𝑗𝑛+1
3−𝑗𝑛+𝑝+1
3
1−𝑗3
,
∥𝐷𝑛+𝑝(𝑡) − 𝐷𝑛∥ ≤
𝑛+𝑝

𝑘=𝑛+1
𝑗𝑘
4=𝑗𝑛+1
4−𝑗𝑛+𝑝+1
4
1−𝑗4
,
∥𝑃𝑛+𝑝(𝑡) − 𝑃𝑛∥ ≤
𝑛+𝑝

𝑘=𝑛+1
𝑗𝑘
5=𝑗𝑛+1
5−𝑗𝑛+𝑝+1
5
1−𝑗5
,
(4.13)
such that M ( 𝑟)
𝑟𝑇𝑟𝐿𝑗≤1.Thus,
𝑆𝑛, 𝐼𝑛, 𝑅𝑛, 𝐷 𝑛, 𝑃𝑛are Cauchy se-
quences in B(𝐽)and hence uniformly
convergent. Thus, by the hypothesis of the
limit theorem, we conclude that the limit of
the sequences (4.7) is the unique solution
of the fractional-order Ebola virus model
(3.1). □
4.2 Positivity and boundedness of solu-
tion
Positivity and the boundedness of so-
lutions are important features of epidemio-
logical models. To do so, it is enough to
show that all state variables are nonnega-
tive for any 𝑡 > 0, which means that for any
𝑡 > 0, any trajectory that begins with a pos-
itive initial condition will remain positive.
Now, systems (3.1), gives
𝐶𝐷𝛼𝑆(𝑡)|𝑆=0=𝜋≥0,
𝐶𝐷𝛼𝐼(𝑡)|𝐼=0=𝛽2𝐷𝑆 +𝜆𝑃𝑆 ≥0,
𝐶𝐷𝛼𝑅(𝑡)|𝑅=0=𝛾𝐼 ≥0,
𝐶𝐷𝛼𝐷(𝑡)|𝐷=0=(𝜇+𝛿)𝐼≥0,
𝐶𝐷𝛼𝑃(𝑡)|𝐷=0=𝜎+𝜉 𝐼 +𝛼𝐷 ≥0.
(4.14)
Moreover, since 𝑁(𝑡)=𝑆(𝑡) + 𝐼(𝑡) + 𝑅(𝑡),
by adding the first three equations of the
model (3.1)gives
𝐶𝐷𝑟𝑁(𝑡)=𝜋−𝜇𝑁 −𝛿 𝐼 , (4.15)
then
𝑁(𝑡) ≤ 𝑁(0) − 𝜋
𝜇𝐸𝑟(−𝜇𝑡𝑟) + 𝜋
𝜇.
Therefore,
Ω = (𝑆(𝑡), 𝐼 (𝑡), 𝑅(𝑡)) ∈ R3
+: 0 ≤𝑁(𝑡) ≤ 𝜋
𝜇,
(4.16)
is the feasible region for the Caputo
fractional-order model (3.1) which is pos-
itively invariant. Thus proposed fractional-
order model (3.1) is both mathematically
and epidemiologically well-posed.
5. Sensitivity Analysis in Respect to
𝑅0
In this section, we use the forward
sensitivity index to analyze the sensitiv-
ity of the biological parameters in the
fractional-order Ebola virus model with re-
spect to the basic reproduction number 𝑅0,
which is an important factor in determining
the spread of the Ebola virus, and reducing
it to less than one is critical in controlling
the infection.
By using sensitivity analysis, we can
identify the parameters that have the great-
est impact on 𝑅0. Parameters with a posi-
tive sensitivity index are considered highly
sensitive and will increase 𝑅0if their values
increase. Parameters with a negative sen-
sitivity index are considered sensitive for
decreasing 𝑅0if their values decrease. Pa-
rameters with a sensitivity index of zero are
considered neutral.
32

I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
The goal of this analysis is to deter-
mine the sensitivity status of each parame-
ter and optimize the model’s output. This
will allow us to identify the most critical
parameters and develop effective strategies
for controlling the spread of the Ebola virus.
The relation
𝛤𝑅0
𝛽1
=𝛽1
𝑅0
×𝜕𝑅0
𝜕𝛽1
,(5.1)
denote sensitivity index of 𝑅0with respect
to a parameter 𝛽1where
𝑅0=𝜋(𝑏𝜂 𝛽1+𝜂 𝛽2(𝜇+𝛿) + 𝜆(𝑏 𝜉 +𝛼𝛿 +𝛼𝜇)
𝑏𝜂 𝜇(𝜇+𝛿+𝛾).
(5.2)
The results from the sensitivity analysis
show that the effective contact rate (𝜆),
shedding rate of deceased human individ-
uals (𝛼),shedding rate of infectious human
individuals (𝜉),effective contact rate of de-
ceased human individuals 𝛽2and effective
contact rate of infectious human individu-
als (𝛽1), respectively, are the most sensitive
parameters that lead to the increase of the
basic reproduction number 𝑅0. This means
that an increase or decrease in these parame-
ters will increase or decrease 𝑅0.Therefore,
finding the optimal strategies for decreasing
these parameters will help in controlling the
spread of the virus (see Table 3).
Fig. 4. Ebola-transmission-medium.
Table 3. Forward normalized sensitivity in-
dices.
Parameters Value Elasticity index
𝜋20 1
𝛽10.0007 0.1734
𝛽20.0013 0.2277
𝜆0.001 0.5989
𝜇0.49 -1.3890
𝛿0.86 -0.0365
𝛾0.056 -0.0946
𝑏0.758 -0.4799
𝜉0.035 0.2477
𝛼0.036 0.2522
𝜂0.025 -0.5989
6. Numerical Simulations and Dis-
cussions
In order to capture the paths that so-
lutions take, both classical and fractional-
order models require the use of numerical
schemes. The purpose of using this nu-
merical scheme was to gain insight into the
trajectories of the solutions. For more in-
formation about the accuracy, stability, and
convergence of this method, see [33].
It is worth noting that the numerical
scheme we have utilized in our simulations,
as mentioned earlier, is not only effective,
but also has several desirable properties.
Specifically, it is a convergent scheme, con-
ditionally stable, and comes equipped with
error bounds. The presence of these fea-
tures ensures the safe and reliable use of
this method in our simulations. For conve-
niences, we have compiled Table 4 which
presents the values of parameters used in
our numerical simulations of the proposed
model. This information may be useful for
reproducing our results or for conducting
further research using this model.
Using the numerical scheme de-
scribed earlier, we obtained the dynamic
behavior of each compartment in the pro-
33

I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
Table 4. Parameters value of model (3.1).
Parameters Parameters value Source
𝜋20 (Variable) Assume
𝛽10.0007 (Variable) [32]
𝛽20.0013 (Variable) [32]
𝜆0.001 (Variable) Assume
𝜇0.49 [32]
𝛿0.86 [0.4, 0.9] [32]
𝛾0.056 (0, 1) [32]
𝑏0.758 (0, 1) [32]
𝜎0.001 (variable) Assume
𝜉0.035 (0, ∞) Assumed
𝛼0.036 (0, ∞) Assume
𝜂0.025 (0, ∞) [32]
posed model (3.1), for the case where the
fractional-order was set to 𝑟=1, as shown
in Fig. 5. To further analyze the dynamics
of the model and gain insight into its behav-
ior, we varied the fractional-order value 𝑟=
1.0,0.8,0.6,0.4while keeping the model
parameters fixed (as listed in Table 4).
Fig. 5. Simulations of the for model (3.1) of
each state variable for the classical version.
The resulting numerical simulations,
shown in Fig. 6, indicate a reduction in
the number of susceptible individuals and
Ebola virus pathogen compartments, sug-
gesting a decrease in the transmission of the
virus. This reduction in the number of sus-
ceptible individuals can be attributed to the
effectiveness of measures such as vaccina-
tion, quarantine, and contact tracing in con-
trolling the spread of the virus.
Fig. 6. Simulations of the for model (3.1) of
each state variable for the fractional-order ver-
sion.
In addition, the decrease in the num-
ber of individuals in the susceptible com-
partment can result in an increase in the
number of individuals in the infected, re-
covered, and dead compartments, as shown
in Fig. 6. This suggests that some individ-
uals who were initially susceptible have be-
come infected and subsequently either re-
covered from or died from the virus.
Therefore, the proposed fractional-
order Ebola virus model (3.1), provides
valuable insight into the complex dynam-
ics of virus transmission and allows us to
visualize the memory effect when varying
the order of the derivative. This information
can be useful in developing more effective
strategies for controlling and preventing the
spread of the virus in the future.
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I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
6.1 Effects of sensitive parameters
Fig. 7. Impact of sensitive parameters.
Fig. 7 shows the different dynami-
cal phenomena when varying the most sen-
sitive parameters that have an impact on the
basic reproduction number 𝑅0(see Table 3
and Fig. 4), respectively. Because we are
interested in the infected individuals com-
partment, we first vary for different values
of the recruitment rate (𝜋), and we observed
that the more individuals are recruiting into
the susceptible population, the number of
infected individuals is getting higher. This
scenario happens throughout the infected
compartment when varying the remaining
sensitive parameters, that is, the effective
contact rate (𝜆),the shedding rate of de-
ceased human individuals (𝛼),the shedding
rate of infectious human individuals (xi),
the effective contact rate of deceased human
individuals (𝛽2),and the effective contact
rate of infectious human individuals (𝛽1),
respectively. The analysis shows that con-
trolling these sensitive parameters will lead
to a decrease in the number of infected in-
dividuals, which means that there is a need
for policymakers to adopt some strategies in
order to control these parameters.
7. Conclusions
In this paper, we have formulated and
analyzed a mathematical model based on
a system of Caputo fractional-order differ-
ential equations to investigate the effect of
sensitive parameters as a disease control
strategy. We were able to establish a region
in such a way that the model is mathemati-
cally and epidemiologically well-posed due
to the fact that its solutions are positive and
bounded. Fixed point results were utilized
to establish the existence and uniqueness of
the proposed model.
The sensitivity analysis revealed that
reducing the rate of recruitment of suscep-
tible individuals (𝜋), effective contact rate
(𝜆), shedding rate of deceased human in-
dividuals (𝛼), shedding rate of infectious
human individuals (𝜉), effective contact
rate of deceased human individuals (𝛽2),
and effective contact rate of infectious hu-
man individuals (𝛽1)(as seen in Fig. 4
and Fig. 7) is crucial for reducing the ba-
sic reproduction number and mitigating dis-
ease spread. Furthermore, numerical results
show the advantages of using a fractional-
order model with memory effects over a
classical-order model (as illustrated in Figs.
5-6).
In light of the results, we recom-
mend that policymakers and health prac-
titioners prioritize using effective media
coverage to conduct widespread awareness
campaigns on preventive measures, regard-
less of whether there is an ongoing epidemic
or not.
Acknowledgment
The first author acknowledge the fi-
nancial support provided by the TertiaryEd-
ucation Trust Fund (TetFund). In addition,
the authors express their gratitude for the
positive comments received by anonymous
reviewers and the editors which have im-
35

I. Ahmed et al. | Science & Technology Asia | Vol.28 No.4 October - December 2023
proved the readability and correctness of the
manuscripts.
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