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Fractals, Vol. 30, No. 1 (2022) 2240046 (16 pages)
c
The Author(s)
DOI: 10.1142/S0218348X22400461
MODELING FRACTIONAL-ORDER DYNAMICS
OF MERS-COV VIA MITTAG-LEFFLER LAW
QU HAIDONG
Department of Mathematics
Hanshan Normal University
Chaozhou 515041,P. R. China
qhaidong@163.com
MATI UR RAHMAN∗
Department of Mathematics
Shanghai Jiao Tong University
800 Dongchuan Road,Shanghai,P. R. China
mati-maths 374@sjtu.edu.cn
YE WANG∗
Department of Mathematics
Huzhou University
Huzhou 313000,P. R. China
03019@zjhu.edu.cn
MUHAMMAD ARFAN†and ADNAN‡
Department of Mathematics
University of Malakand,Chakdara Dir(L)
18000,Khyber Pakhtunkhwa Pakistan
†arfan uom@yahoo.com
‡adnanckd@uom.edu.pk
∗Corresponding authors.
This is an Open Access article in the “Special Issue Section on Fractal AI-Based Analyses and Applications to Complex
Systems: Part II”, edited by Yeliz Karaca (University of Massachusetts Medical School, USA), Dumitru Baleanu (Cankaya
University, Turkey), Majaz Moonis (University of Massachusetts Medical School, USA), Khan Muhammad (Sejong University,
South Korea), Yu-Dong Zhang (University of Leicester, UK) & Osvaldo Gervasi (Perugia University, Italy) published by World
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which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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Q. Haidong et al.
Received May 27, 2021
Accepted August 22, 2021
Published November 9, 2021
Abstract
This paper considers an arbitrary-order mathematical model that analyzes the syndrome type
Middle Eastern Coronavirus (MERS-CoV) under the nonsingular Mittag-Leffler derivative.
Such types of viruses were transferred from camels to the population of humans in the Arabian
deserts. We investigate the said problem for theoretical results and determine the existence of a
solution by using the fixed point theory concept. For the approximate solution, the well-known
method of iteration arbitrary-order Adams–Bashforth (AB) technique has been used. A numer-
ical scheme is developed for the analyzed problem in the continuation of simulations at different
noninteger and natural order for the interval (0,1]. The graphical representation shows that all
classes show convergence and achieve a stable position with growing time. A good compara-
tive result has been given by the analysis of various noninteger orders with natural orders and
achieves stability quickly at fewer non-integer orders.
Keywords: Fractional Mathematical Model; MERS-Cov Model; Existence and Uniqueness
Results; Adams–Bashforth Method; Numerical Simulations.
1. INTRODUCTION
Several types of coronaviruses have been tested by
medical laboratories in various countries. Among
them is the Middle East respiration syndrome coro-
navirus (MERS-CoV) found in December 2012 in
Saudi Arabia.1–3 Mostly, the said virus is tested
in animals from Middle Eastern countries. It was
also discovered that the same virus was also found
in human beings.4This virus caused infection and
transmission through small drops of saliva or cough-
ing. The MERS-CoV infection also spreads through
close contact with infected peoples.5For t he la s t
decades, MERS-CoV has caused 150 death cases
and about 550 infected cases. The death percent-
age is about 30% which is very high as compared to
the new COVID-19 which mostly occurs in Middle
East countries of Saudi Arabia, Syria and Qatar.6
The transmission is of the zoonotic type caused by
cross spreading across camels, bats and the human
population.
Researchers and scholars tested (MERS-CoV)
virus mostly in the trained riding camels as a reser-
voir source for spreading. The transmission rate
from animals to humans is very low as compared
to the rate of spreading from human to human,
which is nearly 25% from animals to humans and
75% from human to human. Scientists have been
working hard to collect data for some predictions
of the present and future to cure the infection rate.
For this, the simplest way is the conversion of the
said problem to a mathematical formulation in the
differential or integral equation. From the analy-
sis of these equations, we can easily gain a lot of
information. The largest MERS-CoV transmission
from human to human was first time formulated
by Assire et al.7The animal to human transmis-
sion due to the indirect exposition was described
by Zaman et al .8They also analyzed the said trans-
mission caused by consuming nonpasteurized milk
of the camels in the Middle Eastern countries. Some
scientists like Poletto et al.9claim the reason for
the said infection is overcrowding of pilgrims in the
Manasike Hajj and Umrah. He also reported the
other reasons like contacts with camels in the mar-
kets, Eid festivals, racing festivals, closing and open-
ing ceremonies.
In our research paper, we will investigate the
human and camel population interaction integer-
order model10 to the fractional-order (FO) model
in sense of ABC derivative. We construct an agents
model for the transmission dynamics of MERS-
CoV. The model is divided into different human
populations that is Susceptible human class Sh(t),
Exposed or latent human class Eh(t), Infected
and Symptomatical class of human Ih(t), Infected
but Asymptomatic human class Ah(t), Hospitalize
human class Hh(t) and the recovered human class
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Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
Rh(t). The Camel classes of the population are the
Susceptible Camel class Sc(t), Asymptomatic infec-
tious Camel class Xc(t) and Symptomatical infec-
tious Camel class Yc(t) given as follows:
dSh
dt =kh−(ζ1Xc+ζ2Yc+ζ3Ih+ζ4bHh)Sh
Nh
−π0Sh,
dEh
dt =(ζ1Xc+ζ2Yc+ζ3Ih+ζ4bHh)Sh
Nh
−(λ+π0)Eh,
dIh
dt =σλEh−(θa+θ1)Ih−(π0+π1)Ih,
dAh
dt =λ(1 −σ)Eh−(π0+π2)Ah,
dHh
dt =θaIh−θθHh−π0Hh,
dRh
dt =θ1Ih+θθHh−π0Rh,
dSc
dt =kc−(ζ5Xc+ζ6Yc)Sc
Nc
−α1Sc,
dXc
dt =(ζ5Xc+ζ6Yc)
Nc
Sc−(α2+λc)Xc,
dYc
dt =λcXc−(α3+ηc)Yc.
(1)
Here, the parameter khis the birth rate of the
human population, ζ1,ζ
2,ζ
3,ζ
4,ζ
5,ζ
6show the rate
of transmission per unit time, bis the transmission
rate of a hospitalized person, λis the rate of a per-
son who the leaves the exposed class and becomes
infected, σis the progression rate from class Ehto
Ih,(1−σ) is the progression rate to class Ah,θais
the average rate at which the class Ahhospitalized,
θ1is the recovery rate without being admitted, θθis
the recovery rate of the admitted persons, the natu-
ral death rate is denoted by π0and π1,π2are rates
of death due to the disease.
Modern calculus has gained much attention
from scholars and has been used in several fields.
Researchers have established mathematical mod-
els for different types of diseases, see Refs. 11–13.
Most mathematical models are based upon dif-
ferential and integral equations of integer order.
Though for the past few decades, to develop real
phenomena by a better degree of precision and accu-
racy, noninteger-order fractional differential equa-
tions (FDEs) have been used. Many researchers
have used several methods for the qualitative study
of FO mathematical models.14,15 Generally, it is not
easy for nonlinear FDEs to find their general solu-
tion. To overcome this situation, many researchers
defined various methods for the determination of
approximate solutions for nonlinear systems. Few
of such approaches are analytical and a few of them
are semi-analytical.16–20 For instance, Adomian in
1980 formulated a useful decomposition method
to solve nonlinear systems analytically. Later on,
the above-mentioned method was enforced as an
actual tool for finding semi-analytical or estimated
results of several systems in applied sciences. A
variety of mathematical models have been exam-
ined widely using the Homotopy method, LADM
and method of difference, see Refs. 16 and 17.
These methods handle both linear and nonlinear
FODEs. Currently, several techniques are used to
investigate systems based on DEs of fractional and
classical orders. Some of them are the residual
power series technique, double Laplace transform
technique and many other forms of computational
approaches.21–23
Different types of FO nonlocal derivatives
have been developed to fulfill the deficiencies
of IOD (integer-order derivatives). For example,
Riemann–Liouville introduced the concept of frac-
tional derivative, which Caputo later modified and
improved. Analysis of practical problems through
fractional derivatives often produces singularities
that are impractical for the dynamics of mathemat-
ical models. To avoid such a situation, a novel frac-
tional derivative, known as Caputo–Fabrizio (CF)
operator was recommended.24 Using the CF oper-
ator yields, in several problems, the locality of the
kernel.25–27 To handle such issues, a novel type of
fractional derivative along with the application of
imported MLF (Mittag-Leffler function) as a non-
singular and nonlocal was suggested by Atangana
and Baleanu (AB).28 Mittag-Leffler derivative oper-
ator has also been used which guarantees no singu-
larity and no locality in the kernel. This new deriva-
tive is based on the MLF, which is more suitable
for describing real-world problems. Analytical and
numerical techniques are very helpful because they
can play a very important role in describing the
behavior of the solution of the fractional differen-
tial equations as revealed in Refs. 29–38.
Inspired by the above discussion, in this paper,
we analyze a MERS-Cov model (1), under the
ABC fractional derivative operator. This novel
work includes theoretical, analytical and numerical
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2240046
Q. Haidong et al.
results regarding the dynamics of an infectious dis-
ease in terms of different paths of transmission
under the effect of various infections in humans and
camels to fill the gap. Our considered problem in
sense of ABC derivative of order 0 <ν≤1is
expressed as follows:
ABC Dν
tSh=k
h−(ζ1Xc+ζ2Yc+ζ3Ih+ζ4bHh)Sh
Nh
−π0Sh,
ABC Dν
tEh=(ζ1Xc+ζ2Yc+ζ3Ih+ζ4bHh)Sh
Nh
−(λ+π0)Eh,
ABC Dν
tIh=σλEh−(θa+θ1)Ih−(π0+π1)Ih,
ABC Dν
tAh=λ(1 −σ)Eh−(π0+π2)Ah,
ABC Dν
tHh=θaIh−θθHh−π0Hh,
ABC Dν
tRh=θ1Ih+θθHh−π0Rh,
ABC Dν
tSc=bc−(ζ5Xc+ζ6Yc)Sc
Nc
−α1Sc,
ABC Dν
tXc=(ζ5Xc+ζ6Yc)
Nc
Sc−(α2+λc)Xc,
ABC Dν
tYc=λcXc−(α3+ηc)Yc.
(2)
To analyze (2), we use the following subsidiary con-
ditions
Sh(0) = Sh(0) ≥0,Eh(0) = Eh(0) ≥0,
Ih(0) = Ih(0) ≥0,Ah(0) = Ah(0) ≥0,
Hh(0) = Hh(0) ≥0,Rh(0) = Rh(0) ≥0,(3)
Sc(0) = Sc(0) ≥0,Xc(0) = Xc(0) ≥0,
Yc(0) = Yc(0) ≥0.
The FO mathematical model has been analyzed for
different arbitrary-order derivatives having an extra
degree of freedom of choice. It provides the whole
spectrum for each compartment and a more real-
istic result than that of the natural-order model.
At least one unique solution of the proposed prob-
lem is derived with the help of fixed point the-
ory. Ulam–Hyer’s (UH) stability is also established.
The FO modified Euler technique is applied to
the considered model for an approximate solution
having FO terms involved in the analysis. Academ-
ically integer-order models analysis for an approxi-
mate solution did not have the FO term, therefore,
neglecting the extra degree of selection.
The paper is organized as follows: In Sec. 2,
we recall some basic preliminaries and notations
from fractional calculus. Some useful results are
developed for the qualitative analysis of the given
problem by exploiting the approach of fixed point
theorem along with UH stability in Sec. 3. In Sec. 4,
with the aid of the Adams–Bashforth technique, we
computed the numerical approximation of the sys-
tem under consideration. After that, we establish
different simulations to verify and support the ana-
lytical findings of the previous sections and briefly
discussed the obtained results in the same section
as well. We conclude our work in Sec. 5.
2. FUNDAMENTAL RESULTS
To understand the paper, we have considered some
significant and beneficial results of modern calcu-
lus and the nonlinear dynamics from the litera-
ture.28,33,39
Definition 1. Let the ABC noninteger derivative
in the sense of Caputo with ν∈(0,1] and for a
mapping X(t)∈H
1[0,T] is defined as follows:
ABC Dν
t(X(t)) = N(ν)
1−νt
0
Eν−ν
1−ν(t−ß)νd
dß
×X(ß)dß,(4)
where N(ν) is the normalization constant such that
N(0) = N(1) = 1, and Eνis the ML function given
by
Eν(y)=
∞
k=0
yk
ν(νk +1).
Definition 2. Let X(t)∈L1(0,T) be a function
then the LHS of arbitrary-order integration of order
ν∈(0,1] in the approach of ABC is given as follows:
ABC Iν
tX(t)=1−ν
N(ν)X(t)+ ν
N(ν)
1
λ(ν)
×t
0
(t−ß)(ν−1)X(ß)dß,t>0.
(5)
Lemma 3 (Ref. 39). For ν∈(0,1],then the fol-
lowing problem solution can be defined as follows :
ABC Dν
tX(t)=Ψ(t),
X(0) = X0,(6)
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2240046
Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
and can be assumed as follows:
X(t)=X0+1−ν
N(ν)Ψ(t)+ ν
N(ν)
1
ν(ν)
×t
0
(t−ß)ν−1Ψ(ß)dß.(7)
Theorem 2.1. Assume Pbeaclosednormspace
and D⊂Pbe a convex,bounded and closed set. If
a continuous mapping ψ:D→Dsuch that ψD⊂P
and ψDis relatively compact,then the operator has
at least one fixed point in D.
3. QUALITATIVE ANALYSIS OF
THE CONSIDERED MODEL (2)
This portion of the paper deals with the existence
result of the considered model (2)–(3) using the
approach of fixed point theorem. Fixed point the-
ory is a powerful tool and gives information that
the considered model has a solution or not. This
existence theory guaranteed to check whether the
modeling physical phenomena exist or not. Let
us take the Banach space Υ = C([0,T],R)with
the norm W=sup
t∈[0,T ]|W(t)|. We define the
Banach space =(Υ
9,W)withnormW=
supt∈[0,T ](|Sh|+|Eh|+|Ih|+|Ah|+|Hh|+|Rh|+
|Sc|+|Xc|+|Yc|).To get the desired results, we
define the model in the following form:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
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⎪
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⎪
⎪
⎪
⎪
⎪
⎪
⎪
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⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
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⎪
⎪
⎪
⎪
⎪
⎪
⎪
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⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ABC Dν
tSh(t)
=G1(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=bh−(ζ1Xc+ζ2Yc+ζ3Ih+ζ4qHh)Sh
Nh
−π0Sh,
ABC Dν
tEh(t)
=G2(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=(ζ1Xc+ζ2Yc+ζ3Ih+ζ4qHh)Sh
Nh
−(λ+π0)Eh,
ABC Dν
tIh(t)
=G3(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=σλEh−(θa+θ1)Ih−(π0+π1)Ih,
ABC Dν
tAh(t)
=G4(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=λ(1 −σ)Eh−(π0+π2)Ah,
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
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⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ABC Dν
tHh(t)
=G5(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=θ1Ih+θθHh−π0Rh,
ABC Dν
tRh(t)
=G6(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=θaIh−θθHh−π0Hh,
ABC Dν
tSc(t)
=G7(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=bc−(ζ5Xc+ζ6Yc)Sc
Nc
−α1Sc,
ABC Dν
tXc(t)
=G8(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=(ζ5Xc+ζ6Yc)
Nc
Sc−(α2+λc)Xc,
ABC Dν
tYc(t)
=G9(t, Sh,Eh,Ih,Ah,Hh,Rh,Sc,Xc,Yc)
=λcXc−(α3+αc)Yc.
(8)
Next, the model (2)–(3) can be defined in the fol-
lowing form
ABC Dν
tW(t)=Ω(t, W(t)),
W(0) = W0,(9)
where
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
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⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
W:= (Sh,Eh,Ih,Ah,Hh,Rh,Sc,
Xc,Yc)T,
W(0) := (Sh(0),Eh(0),Ih(0),Ah(0),Hh(0),
Rh(0),Sc(0),Xc(0),Yc(0))T,
Ω(t, W(t)) := Gi(t, Sh,Eh,Ih,Ah,Hh,Rh,
Sc,Xc,Yc)T,i=1,2,3,...,9.
(10)
The transpose of a vector is defined as (.)T.Using
Lemma 3, the problem (9) is equivalent to the fol-
lowing noninteger integral equation:
W(t)=W0+1−ν
N(ν)Ω(t, W(t)) + ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1Ω(ß,W(ß))dß.(11)
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Q. Haidong et al.
In the next theorem, the “Schauder’s fixed point
theorem” is applied and the existence results for
the considered problem (2)–(3) are developed.
Theorem 3.1. Let Ω∈be a continuous function
and ∃a constant M>0,|Ω(t, W(t))|≤M(1 +
|W|),∀t∈[0,T]and W∈.∃at least one solution
for the proposed problem (2)–(3) as follows:
∇1=(1 −ν)λ(ν)M+MTν
N(ν)λ(ν)<1.(12)
Proof. It is straightforward that root of the pro-
posed problem (2)–(3) is same to fractional integral
equation (11). Suppose the operator Y:→is
defined by
(YW)(t)=W0+1−ν
N(ν)Ω(t, W(t)) + ν
W(ν)λ(ν)
×t
0
(t−ß)ν−1Ω(ß,W(ß))dß.(13)
Let us define a closed convex bounded ball as B=
{W∈Ω:W≤, > 0} with ≥∇2
1−∇1,
where
∇2=|W0|+1−ν
N(ν)M+Tν
N(ν)λ(ν)M.(14)
Now, we need to prove that (YB)⊂B,∀t∈
[0,T]. So going ahead
|(YW)(t)|≤|W0|+1−ν
N(ν)|Ω(t, W(t))|+ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1|Ω(ß,W(ß))|dß,
≤|W0|+1−ν
N(ν)N(1 + |W(t)|)
+ν
N(ν)λ(ν)t
0
(t−ß)ν−1
×N(1 + |W(t)|)dß.(15)
Again since W∈B,one may write
(YW)(t)≤|W0|+1−ν
N(ν)M(1 + W(t))
+Tν
N(ν)λ(ν)M(1 + W(t)),
≤|W0|+1−ν
N(ν)M+Tν
N(ν)λ(ν)M
+1−ν
N(ν)M+Tν
N(ν)λ(ν)Mσ,
≤∇
2+∇1≤.
Thus it is shown that (YB)⊂B.In the next
step, we will derive that the mapping Yis defined
on their domain, let us consider a sequence {Wn}
Wn→Win Bas n→∞.Thisimpliesthatforall
t∈[0,T], we obtain
|(YWn)(t)−(YW)(t)|
≤1−ν
N(ν)|Ω(t, Wn(t)) −Ω(t, W(t))|
+ν
N(ν)λ(ν)t
0
(t−y)ν−1
×|Ω(ß,Wn(ß)) −Ω(ß,W(ß))|dß
≤1−ν
N(ν)Ω(t, Wn(t)) −Ω(t, W(t))
+Tν
N(ν)λ(ν)Ω(ß,Wn(ß)) −Ω(y, W(ß)).
We may write from the continuous function Ω as
follows:
(YWn)−(YW)→0asn→0,
which shows that the operator Yis defined on their
domain, hence continuous on B.
In the next step, we show that the operator
(YB) is relatively compact. Since (YB)⊂B,
it is clear that the operator (YB) is bounded uni-
formly.
In the next step, we show that the operator Y
is equi-continuous on B. For this let W∈Band
there is t1,t
2∈[0,T]witht1<t
2.Then we have
(YW)(t2)−(YW)(t1)
≤1−ν
N(ν)|Ω(t2,W(t2)) −Ω(t1,W(t1))|
+ν
N(ν)λ(ν)t2
0
(t2−ß)ν−1
−t1
0
(t1−ß)ν−1Ω(ß,W(ß))dß
≤1−ν
N(ν)|Ω(t2,W(t2)) −Ω(t1,W(t1))|
+ν
N(ν)
M(1 + W)
λ(ν+1) (tν
2−tν
1).
Apparently, the right side YW(t2)−YW(t1)goes
to zero as t2→t1. By using the “Arzela–Ascoli
theorem”, the mapping (YB) is compact relatively
and thus Yis completely continuous. From 2.1, we
deduce that the considered model (2)–(3) has their
solution. Thus, the derivation is finished.
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Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
Here, we will show the results for stability of the
proposed system (2)–(3) due to a very little change
in the subsidiary conditions.
Theorem 3.2. Let a function be defined on their
interval Ω∈and ∃some constant J>0
|Ω(t, W)−Ω(t,
W)|≤J|W−
W|,∀t∈[0,T]and
W∈having
1>(1 −ν)λ(ν)J+JTν
N(ν)λ(ν).
let Wand
Wbe the solution for problem (9) and
ABC Dν
W(t)=Ω(t,
W(t)),
W(0) = W0+≥0.
(16)
Here,
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W=(
Sh,
Eh,
Ih,
Ah,
Hh,
Rh,
Sc,
Xc,
Yc)T
W0+=(Sh(0) + ε, Eh(0) + ε, Ih(0) + ε,
Ah(0) + ε, Hh(0) + ε, Rh(0) + ε,
Sc(0) + ε, Xc(0) + ε, Yc(0) + ε)T
Ω(t,
W(t)) = Gi(
Sh,
Eh,
Ih,
Ah,
Hh,
Rh,
Sc,
Xc,
Yc)T,i=1,2,3,...,9,
(17)
the following holds :
W−
W≤1−(1 −ν)λ(ν)J+JTν
N(ν)λ(ν)−1
|ε|.
(18)
Proof. We see that the solutions of the systems (9)
and (16) are equivalent to the integral equation (11)
and
W(t)=W0++1−ν
N(ν)Ω(t,
W(t)) + ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1Ω(ß,
W(ß))dß.(19)
This implies that for each t∈[0,T]wemayhave
|W(t)−
W(t)|
≤|ε|+1−ν
N(ν)|Ω(t, W(t)) −Ω(t,
W(t))|
+ν
N(ν)λ(ν)t
0
(t−ß)ν−1
×|Ω(ß,W(ß)) −Ω(ß,
W(ß))|dß,
≤|ε|+1−ν
N(ν)J|W(t)−
W(t)|+ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1J|W(ß) −
W(ß)|dß,
≤|ε|+1−ν
N(ν)+Tν
N(ν)λ(ν)JW−
W.
One may then write
W−
W≤|ε|+(1 −ν)λ(ν)+Tν
N(ν)λ(ν)GW−
W.
As a result we have
W−
W≤1−(1 −ν)λ(ν)J+JTν
N(ν)λ(ν)−1
|ε|.
(20)
Hence, the theorem is proved.
Remark 4. If we include ε= 0 in the above theo-
rem, then we will have the unique solution for the
proposed system (2)–(3).
4. NUMERICAL SCHEME
Here, in this section, we will study the approx-
imate solution of the considered system (2)–(3),
by using the Adam–Bashforth iterative technique29
and obtain the numerical simulation. With the help
of subsidiary conditions along with the operator
ABC Iν
0, further we set up the model (2)–(3) into frac-
tional integral equations as follows:
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Sh(t)−Sh(0) =AB Iν
0K1(t, Sh(t)),
Eh(t)−Eh(0) =AB Iν
0K2(t, Eh(t)),
Ih(t)−Ih(0) =AB Iν
0K3(t, Ih(t)),
Ah(t)−Ah(0) =AB Iν
0K4(t, Ah(t)),
Hh(t)−Hh(0) =AB Iν
0K5(t, Hh(t)),
Rh(t)−Rh(0) =AB Iν
0K6(t, Rh(t)),
Sc(t)−Sc(0) =AB Iν
0K7(t, Sc(t)),
Xc(t)−Xc(0) =AB Iν
0K8(t, Xc(t)),
Yc(t)−Yc(0) =AB Iν
0K9(t, Yc(t)).
(21)
This further gives
Sh(t)−Sh(0) = 1−ν
N(ν)K1(Sh(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K1(Sh(ß),ß)dß,
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Q. Haidong et al.
Eh(t)−Eh(0) = 1−ν
N(ν)K2(Eh(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K2(Eh(ß),ß)dß,
Ih(t)−Ih(0) = 1−ν
N(ν)K3(Ih(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K3(Ih(ß),ß)dß,
Ah(t)−Ah(0) = 1−ν
N(ν)K4(Ah(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K4(Ah(ß),ß)dß,
Hh(t)−Hh(0) = 1−ν
N(ν)K5(Hh(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K5(Hh(ß),ß)dß,
Rh(t)−Rh(0) = 1−ν
N(ν)K6(Rh(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K6(Rh(ß),ß)dß,
Sc(t)−Sc(0) = 1−ν
N(ν)K7(Sc(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K7(Sc(ß),ß)dß,
Xc(t)−Xc(0) = 1−ν
N(ν)K8(Xc(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K8(Xc(ß),ß)dß,
Yc(t)−Yc(0) = 1−ν
N(ν)K9(Yc(t),t)+ ν
N(ν)λ(ν)
×t
0
(t−ß)ν−1K9(Yc(ß),ß)dß.
(22)
On setting t=tν+1 for ν=0,1,2,..., into the
system (22), to produce an iterative scheme gives
the following:
Sh(tν+1)−Sh(0)
=1−ν
N(ν)K1(Sh(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K1(Sh(ß),ß)dß,
Eh(tν+1)−Eh(0)
=1−ν
N(ν)K2(Eh(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K2(Eh(ß),ß)dß,
Ih(tν+1)−Ih(0)
=1−ν
N(ν)K3(Ih(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K3(Ih(ß),ß)dß,
Ah(tν+1)−Ah(0)
=1−ν
N(ν)K4(Ah(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K4(Ah(ß),ß)dß,
Hh(tν+1)−Hh(0)
=1−ν
N(ν)K5(Hh(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K5(Hh(ß),ß)dß,
Rh(tν+1)−Rh(0)
=1−ν
N(ν)K6(Rh(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K6(Rh(ß),ß)dß,
Sc(tν+1)−Sc(0)
=1−ν
N(ν)K7(Sc(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K7(Sc(ß),ß)dß,
Xc(tν+1)−Xc(0)
=1−ν
N(ν)K8(Xc(tν),tν)+ ν
N(ν)λ(ν)
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K8(Xc(ß),ß)dß,
Yc(tν+1)−Yc(0)
=1−ν
N(ν)K9(Yc(tν),tν)+ ν
N(ν)λ(ν)
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Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
×
ν
i=0 ti+1
ti
(tν+1 −ß)ν−1K9(Yc(ß),ß)dß.
(23)
On applying the bi-step interpolating algebraic
expression for the given mapping K1(Sh(ß),ß),
K2(Eh(ß),ß),K3(Ih(ß),ß), K4(Ah(ß),ß),K5(Hh(ß),
ß),K6(Rh(ß),ß), K7(Sc(ß),ß),K8(Xc(ß),ß) and
K9(Yc(ß),ß) which are inside in the integral (23)
on the interval [ti,ti+1], we obtain
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K1(Sh(ß),ß) ∼
=K1(Sh(ti),ti)
(t−ti−1)
+K1(Sh(ti−1),ti−1)
(t−ti),
K2(Eh(ß),ß) ∼
=K2(Eh(ti),ti)
(t−ti−1)
+K2(Eh(ti−1),ti−1)
(t−ti),
K3(Ih(ß),ß) ∼
=K3(Ih(ti),ti)
(t−ti−1)
+K3(Ih(ti−1),ti−1)
(t−ti),
K4(Ah(ß),ß) ∼
=K4(Ah(ti),ti)
(t−ti−1)
+K4(Ah(ti−1),ti−1)
(t−ti),
K5(Hh(ß),ß) ∼
=K5(Hh(ti),ti)
(t−ti−1)
+K5(Hh(ti−1),ti−1)
(t−ti),
K6(Rh(ß),ß) ∼
=K6(Rh(ti),ti)
(t−ti−1)
+K6(Rh(ti−1),ti−1)
(t−ti),
K7(Sc(ß),ß) ∼
=K7(Sc(ti),ti)
(t−ti−1)
+K7(Sc(ti−1),ti−1)
(t−ti),
K8(Xc(ß),ß) ∼
=K8(Xc(ti),ti)
(t−ti−1)
+K8(Xc(ti−1),ti−1)
(t−ti),
K9(Yc(ß),ß) ∼
=K9(Yc(ti),ti)
(t−ti−1)
+K9(Yc(ti−1),ti−1)
(t−ti).
(24)
This system further gives
Sh(tz+1)=Sh(0) + 1−ν
N(ν)K1(Sh(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K1(Sh(ti),ti)
Ii−1,ν
+K1(Sh(ti−1),ti−1)
Ii,ν ,
Eh(tz+1)=Eh(0) + 1−ν
N(ν)K2(Eh(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K2(Eh(ti),ti)
Ii−1,ν
+K2(Eh(ti−1),ti−1)
Ii,ν ,
Ih(tz+1)=Ih(0) + 1−ν
N(ν)K3(Ih(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K3(Ih(ti),ti)
Ii−1,ν
+K3(Ih(ti−1),ti−1)
Ii,ν ,
Ah(tz+1)=Ah(0) + 1−ν
N(ν)K4(Ah(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K4(Ah(ti),ti)
Ii−1,ν
+K4(Ah(ti−1),ti−1)
Ii,ν ,
Hh(tz+1)=Hh(0) + 1−ν
N(ν)K5(Hh(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K5(Hh(ti),ti)
Ii−1,ν
+K5(Hh(ti−1),ti−1)
Ii,ν ,
Rh(tz+1)=Rh(0) + 1−ν
N(ν)K6(Rh(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K6(Rh(ti),ti)
Ii−1,ν
+K6(Rh(ti−1),ti−1)
Ii,ν ,
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Sc(tz+1)=Sc(0) + 1−ν
N(ν)K7(Sc(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K7(Sc(ti),ti)
Ii−1,ν
+K7(Sc(ti−1),ti−1)
Ii,ν ,
Xc(tz+1)=Xc(0) + 1−ν
N(ν)K8(Xc(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K8(Xc(ti),ti)
Ii−1,ν
+K8(Xc(ti−1),ti−1)
Ii,ν ,
Yc(tz+1)=Yc(0) + 1−ν
N(ν)K9(Yc(tz),tz)
+ν
N(ν)λ(ν)
z
i=0 K9(Yc(ti),ti)
Ii−1,ν
+K9(Yc(ti−1),ti−1)
Ii,ν ,
(25)
where
Ii−1,ν =ti+1
ti
(t−ti−1)(tz+1 −t)ν−1dt
and
Ii,ν =ti+1
ti
(t−ti)(tz+1 −t)ν−1dt.
Now, we simplify the integrals Ii−1,ν and Ii,ν as fol-
lows:
Ii−1,ν =−1
ν[(ti+1 −ti−1)(tz+1 −ti+1)ν
−(ti−ti−1)(tz+1 −ti)ν]−1
ν(ν−1)
×[(tz+1 −ti+1)ν+1 −(tv+1 −ti)ν+1],
Ii,ν =−1
ν[(ti+1 −ti)(tz+1 −ti+1)ν]−1
ν(ν−1)
×[(tz+1 −ti+1)ν+1 −(tz+1 −ti)ν+1].
By putting ti=ione can easily deduce that
Ii−1,ν =−ν+1
ν(ν+1)[(z+1−i)ν(s−i+2+ν)
−(z−i)ν(z−i+2+2ν)] (26)
and
Ii,ν =ν+1
ν(ν+1)[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)].(27)
Substituting Eqs. (26) and (27) into (25), we get
Sh(tz+1)=Sh(t0)+ (1 −ν)
N(ν)[K1(Sh(tz),tz)]
+ν
N(ν)
z
i=0 K1(Sh(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K1(Sh(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν
×(z−i+1+ν)],(28)
Eh(tz+1)=Eh(t0)+ (1 −ν)
N(ν)[K2(Eh(tz),tz)]
+ν
N(ν)
z
i=0 K2(Eh(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K2(Eh(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(29)
Ih(tz+1)=Ih(t0)+ (1 −ν)
N(ν)[K3(Ih(tz),tz)]
+ν
N(ν)
z
i=0 K3(Ih(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K3(Ih(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(30)
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Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
Ah(tz+1)=Ah(t0)+ (1 −ν)
N(ν)[K4(Ah(tz),tz)]
+ν
N(ν)
z
i=0 K1(Ah(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K4(Ah(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(31)
Hh(tz+1)=Hh(t0)+ (1 −ν)
N(ν)[K5(Hh(tz),tz)]
+ν
N(ν)
z
i=0 K5(Hh(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K5(Hh(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(32)
Rh(tz+1)=Rh(t0)+(1 −ν)
N(ν)[K6(Rh(tz),tz)]
+ν
N(ν)
z
i=0 K6(Rh(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K6(Hh(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(33)
Sc(tz+1)=Sc(t0)+ (1 −ν)
N(ν)[K7(Sc(tz),tz)]
+ν
N(ν)
z
i=0 K7(Sc(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K7(Sc(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(34)
Xc(tz+1)=Xc(t0)+ (1 −ν)
N(ν)[K8(Xc(tz),tz)]
+ν
N(ν)
z
i=0 K8(Xc(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K8(Xc(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)],(35)
Yc(tz+1)=Yc(t0)+ (1 −ν)
N(ν)[K9(Yc(tz),tz)]
+ν
N(ν)
z
i=0 K9(Yc(tz),tz)
ν(ν+2)
×ν[(z+1−i)ν(z−i+2+ν)
−(z−i)ν(z−i+2+2ν)]
−K9(Yc(tz−1),tz−1)
ν(ν+2) ν[(z+1−i)ν+1
−(z−i)ν(z−i+1+ν)].(36)
4.1. Numerical Simulations and
Discussion
In this section, we provide numerical simulation for
validation of our established iterative scheme. For
this, we take two sets of initial values for all the
nine compartments of our proposed FO model (2)–
(3). All the compartments have been simulated at
different fractional orders of νfrom Figs. 1 to 3
against the available data given in Table 1.
Figures 1 and 2 are the representation of the sus-
ceptible human class Sh(t)andtheexposedhuman
class Eh(t) for two sets of initial values Sh(t)=
400,300, Eh(t) = 300,200. Both the classes con-
verge to their equilibrium point at two different ini-
tial values. Also, we have simulated the said classes
for four different fractional orders which converge
quickly at low order.
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Q. Haidong et al.
(a) (b)
Fig. 1 Plots of Sh(t) having two different initial values for (2) at various fractional orders ν.
(a) (b)
Fig. 2 Plots of Eh(t) having two different initial values for (2) at various fractional orders ν.
(a) (b)
Fig. 3 Plots of Ih(t) having Two different initial values for (2) at various fractional orders ν.
Table 1 Parametric Values for Model (1).
Notation Value Notation Value
kh0.09 b0.004
ζ10.09 σ0.065
ζ20.022 θa0.09
ζ30.09 λ0.026
ζ40.022 π00.09
ζ50.0001 θθ0.014
ζ60.008 π10.0.01
kc0.022 λc0.022
α10.022 α20.0002
α30.05 ηc0.04
θ10.09
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Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
(a) (b)
Fig. 4 Plots of Ah(t) having two different initial values for (2) at various fractional orders ν.
(a) (b)
Fig. 5 Plots of Hh(t) having two different initial values for (2) at various fractional orders ν.
Figures 3 and 4 are the representation of the
Infected human class Ih(t) and asymptomatic
human class Ah(t) for two sets of initial values
Ih(t) = 200,100, Ah(t) = 100,50. Both the human
classes converge to their equilibrium point at two
different initial values. Also, we have simulated the
said classes for four different fractional orders which
converge quickly at lower order and slowly at higher
order.
Figures 5 and 6 show the hospitalized human
class Hh(t) and the recovered human class Rh(t)
for two different sets of initial values Hh(t)=70,35,
Rh(t)=50,25. Both the human classes converge to
the same point at two different initial values. Also,
we have simulated the said classes for four different
fractional orders which converge quickly at lower
order and slowly at higher order. Both the classes
decline after the small growth.
(a) (b)
Fig. 6 Plots of Rh(t) having two different initial values for (2) at various fractional orders ν.
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Q. Haidong et al.
(a) (b)
Fig. 7 Plots of Sc(t) having two different initial values for (2) at various fractional orders ν.
(a) (b)
Fig. 8 Plots of Xc(t) having two different initial values for (2) at various fractional orders ν.
Figures 7 and 8 show the susceptible camel class
Sc(t) and the asymptomatic infectious Camel class
Xc(t) for two different sets of initial values Sc(t)=
40,20, Xc(t)=30,15. Both the human classes con-
verge to their equilibrium point at two different ini-
tial values. Also we have simulated the said classes
for four different fractional orders which converge
quickly at lower order and slowly at higher order.
Both the classes show small increase in their behav-
ior. It means that the small amount of infection may
lie in the camels but they have a strong immune sys-
tem, therefore cannot be affected by large degrees.
Figures 9a and 9b show the symptomatical infec-
tious Camel class Yc(t) for two different sets of
initial values Yc(t)=15,7. This camel class also
converges to their equilibrium point at two different
initial values. Also, we have simulated the said class
for four different fractional orders which converge
quickly at low fractional order. This class shows
decay in their behavior.
(a) (b)
Fig. 9 Plots of Yc(t) having two different initial values for (2) at various fractional orders ν.
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Modeling FO Dynamics of MERS-CoV via Mittag-Leffler Law
5. CONCLUSION
In this paper, we have studied the transmission
dynamics of a MERS-Cov with the help of a frac-
tional model under the ABC derivative. This type
of derivative operator has replaced the singular and
local kernel with the nonsingular and non-local ker-
nel. We have exploited the Banach fixed point the-
orems of non-linear functional analysis to derive
the existence of a solution to the problem under
consideration. The stability and convergence have
been achieved by all compartments of the proposed
model. We have performed numerical simulations
for two different sets of fractional orders and initial
sizes of the compartments to verify our analytical
investigations. In the process of decay, the stability
occurs quickly at the lowest fractional orders while
in the growth process the stability has attained
quickly at higher orders. Much better and compara-
ble results have been shown which emphasized the
importance of fractional analysis of dynamical sys-
tems. Such types of techniques may be applied to
different problems of mathematical problems, biol-
ogy problems, society problems, IT problems, nat-
ural sciences problem, infectious diseases problems
to check the inside characteristics between any two
different integers for the said problems.
REFERENCES
1. E. I. Azhar, S. A. El-Kafrawy, S. A. Farraj, A. M.
Hassan, M. S. Al-Saeed, A. M. Hashem and T. A.
Madani, Evidence for camel-to-human transmission
of MERS coronavirus, New Engl. J. Med. 370(26)
(2014) 2499–2505.
2. H. R. Thieme, Mathematics in Population Biology
(Princeton University Press, 2018).
3. K. Yunhwan, L. Sunmi, C. Chaeshin, C. Seoyun,
H. Saeme and S. Youngseo, The characteristics of
Middle Eastern respiratory syndrome coronavirus
transmission dynamics in South Korea, Osong Pub-
lic Health Res. Perspect. 7(1) (2016) 49–55.
4. J. A. AlTawfiq, K. Hinedi and J. Ghandour, Mid-
dle East respiratory syndrome coronavirus: A case-
control study of hospitalized patients, Clin. Infect.
Dis. 59(2) (2014) 160–165.
5. Y. M. Arabi, A. A. Arifi and H. H. Balkhy, Clini-
cal course and outcomes of critically ill patients with
Middle East respiratory syndrome coronavirus infec-
tion, Ann. Int. Med. 160(6) (2014) 389–397.
6. A. N. Alagaili, T. Briese and N. Mishra, Mid-
dle East respiratory syndrome coronavirus infection
in dromedary camels in Saudi Arabia, mBio 5(2)
(2014) e00884-14.
7. A. Assiri et al., Epidemiological, demographic, and
clinical characteristics of 47 cases of Middle East res-
piratory syndrome coronavirus disease from Saudi
Arabia: a descriptive study, Lancet Infect. Dis. 13(9)
(2013) 752–761.
8. G. Zaman, Y. H. Kang and I. H. Jung, Optimal
treatment of an SIR epidemic model with time delay,
BioSystems 98(1) (2009) 43–50.
9. C. Poletto, C. Pelat and D. Levy-Bruhl, Assess-
ment of the Middle East respiratory syndrome coro-
navirus (MERS-CoV) epidemic in the Middle East
and risk of international spread using a novel max-
imum likelihood analysis approach, Euro Surveill.
19(23) (2014) 20824.
10. B. Fatima, G. Zaman and F. Jarad, Transmis-
sion dynamic and backward bifurcation of Middle
Eastern respiratory syndrome coronavirus, Nonlin-
ear An al. , accepted (2021).
11. M. Goyal, H. M. Baskonus and A. Prakash, An effi-
cient technique for a time fractional model of lassa
hemorrhagic fever spreading in pregnant women,
Eur. Phys. J. Plus 134 (2019) 1–10.
12. W. Gao, P. Veeresha, D. G. Prakasha, H. M.
Baskonus and G. Yel, New approach for the model
describing the deathly disease in pregnant women
using Mittag-Leffler function, Chaos Solitons Frac-
tals 134 (2020) 109696.
13. K. Shah, M. A. Alqudah, F. Jarad and T. Abdel-
jawad, Semi-analytical study of Pine Wilt disease
model with convex rate under Caputo–Fabrizio frac-
tional order derivative, Chaos Solitons Fractals 135
(2020) 109754.
14. H. Khan, Y. Li, A. Khan and A. Khan, Existence
of solution for a fractional order Lotka–Volterra
reaction diffusion model with Mittag-Leffler kernel,
Math. Methods Appl. Sci. 42(9) (2019) 3377–3387.
15. R. A. Khan and K. Shah, Existence and uniqueness
of solutions to fractional order multi-point bound-
ary value problems, Commun. Appl. Anal.19(3–4)
(2015) 515–526.
16. F. Awawdeh, A. Adawi and Z. Mustafa, Solutions
of the SIR models of epidemics using HAM, Chaos
Solitons Fractals 42 (2009) 3047–3052.
17. J. Biazar, Solution of the epidemic model by ado-
mian decomposition method, Appl. Math. Comput.
173 (2006) 1101–1106.
18. A. Abdilraze, Admoian decomposition method:
Convergence analysis and numerical approxima-
tions, M.sc. Dissertation, McMaster University
Hamilton, Canada (2008).
19. F. Haq, K. Shah, G. Rahman and M. Shahzad,
Numerical solution of fractional order smoking
model via laplace adomian decomposition method,
Alex.Eng.J.57(2) (2018) 1061–1069.
2240046-15
Fractals Downloaded from www.worldscientific.com
by 39.60.111.9 on 11/19/21. Re-use and distribution is strictly not permitted, except for Open Access articles.
2nd Reading
November 8, 2021 13:12 0218-348X
2240046
Q. Haidong et al.
20. F. Haq, K. Shah, G. Rahman and M. Shahzad,
Numerical analysis of fractional order model of HIV-
1 infection of CD4+ t-cells, Comput. Methods Differ.
Equ. 5(1) (2017) 1–11.
21. A. Khan, T. S. Khan, M. I. Syam and H. Khan, Ana-
lytical solutions of time-fractional wave equation by
double laplace transform method, Eur. Phys. J. Plus
134(4) (2019) 163.
22. A. Kumar, S. Kumar and S. Yan, Residual power
series method for fractional diffusion equations, Fu n -
dam. Inform.151(1–4) (2017) 213–230.
23. M. Kaplan and B. Ahmet, A novel analytical
method for time-fractional differential equations,
Optik 127(20) (2016) 8209–8214.
24. M. Caputo and M. Fabrizio, A new definition of
fractional derivative without singular kernel, Prog.
Fract. Differ. Appl.1(2) (2015) 1–13.
25. S. Ahmad, M. ur Rahman and M. Arfan, On the
analysis of semi-analytical solutions of Hepatitis B
epidemic model under the Caputo–Fabrizio opera-
tor, Chaos Solitons Fractals 146 (2021) 110892.
26. S. Qureshi, N. A. Rangaig and D. Baleanu, New
numerical aspects of Caputo–Fabrizio fractional
derivative operator, Mathematics 7(4) (2019) 374.
27. K. M. Owolabi and A. Atangana, Analysis and
application of new fractional Adams–Bashforth
scheme with Caputo–Fabrizio derivative, Chaos
Solitons Fractals 105 (2017) 111–119.
28. A. Atangana and D. Baleanu, New fractional deriva-
tives with non-local and non-singular kernel: Theory
and application to heat transfer model, Therm. Sci.
20(2) (2016) 763–785.
29. M. Toufik and A. Atangana, New numerical approx-
imation of fractional derivative with non-local and
non-singular kernel: Application to chaotic models,
Eur. Phys. J. Plus 132(10) (2017) 444.
30. A. Atangana, Non validity of index law in fractional
calculus: A fractional differential operator with
markovian and non-markovian properties, Physica
A505 (2018) 688–706.
31. M. ur Rahman, M. Arfan, K. Shah and J. F. G´omez-
Aguilar, Investigating a nonlinear dynamical model
of COVID-19 disease under fuzzy Caputo, random
and ABC fractional order derivative, Chaos Solitons
Fractals 140 (2020) 110232.
32. S. Ahmad, A. Ullah, M. Arfan and K. Shah,
On analysis of the fractional mathematical model
of rotavirus epidemic with the effects of breast-
feeding and vaccination under Atangana–Baleanu
(AB) derivative, Chaos Solitons Fractals 140 (2020)
110233.
33. A. Atangana and J. F. G´omez-Aguilar, Fractional
derivatives with no-index law property: Application
to chaos and statistics, Chaos Solitons Fractals 114
(2018) 516–535.
34. J. E. Sol´ıs-P´erez, J. F. G´omez-Aguilar and A. Atan-
gana, Novel numerical method for solving variable-
order fractional differential equations with power,
exponential and Mittag-Leffler laws, Chaos Solitons
Fractals 114 (2018) 175–185.
35. D. Kumar, J. Singh and D. Baleanu, A new analysis
of the Fornberg–Whitham equation pertaining to a
fractional derivative with Mittag-Leffler-type kernel,
Eur. Phys. J. Plus 133(2) (2018) 1–10.
36. D. Baleanu, A. Jajarmi and M. Hajipour, On the
nonlinear dynamical systems within the general-
ized fractional derivatives with Mittag-Leffler kernel,
Nonlinear Dyn. 94(1) (2018) 397–414.
37. D. Baleanu, A. Jajarmi and M. Hajipour, A new
formulation of the fractional optimal control prob-
lems involving Mittag-Leffler nonsingular kernel,
J. Optim. Theory Appl. 175(3) (2017) 718–737.
38. M. Arfan, K. Shah, A. Ullah and T. Abdeljawad,
Study of fuzzy fractional order diffusion problem
under the Mittag-Leffler Kernel Law, Phys. Scr.
96(7) (2021) 074002.
39. T. Abdeljawad and D. Baleanu, Discrete fractional
differences with nonsingular discrete Mittag-Leffler
kernels, Adv. Differ. Equ. 2016(1) (2016) 232.
2240046-16
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by 39.60.111.9 on 11/19/21. Re-use and distribution is strictly not permitted, except for Open Access articles.
