# The Mathematical Analysis of the New Fractional Order Ebola Model

Authors: Faiz Muhammad Khan; Amjad Ali; Ebenezer Bonyah; Zia Ullah Khan
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

This research study focuses on the analytical behavior and numerical computation of the fractional order Ebola model. In this study we have calculated the conditions for the existence, uniqueness, and stability of the solution with the help of the fixed point results. In addition to this, we calculated the numerical solution of the fractional order smoke model with the help two-step fractional Adam’s Bashforth method using the Caputo’s fractional derivative of orderμ. Furthermore, the results obtained for different orders of the fractional derivative μ have been shown graphically with the help of Matlab.


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## Body

## 1. Introduction

The concept of fractional calculus (FC) was raised in seventeen century from famous correspondence between Leibniz and L’Hoptial. In the consequences of aforementioned correspondence, Leibniz wrote a letter to Guillaume de L’Hôpital that what will be the half order derivative of dependent variable y w.r.t x, i.e,d1/2x/dy1/2. In the response, he wrote that this will bear some useful consequence in near future. Later on, it was traced that fractional calculus was introduced by Abel in one of his papers, where the author discussed the idea of fractional-order derivatives (FOD), fractional-order integration (FOI), and the mutual inverse relationship between them [1]. In 1832, one of the greatest French mathematicians (of his era) Liouville presented the definitions for the fractional derivative and fractional integration named as Riemann-Liouville fractional derivative and integration [2]. Later, on in 1890, Heaviside practically used the fractional differential operator in electrical transmission line analysis circa [3]. Recently, the researchers of the 19th and 20th century have made their significant contributions to introduce new definitions of fractional differential and integral operators and in the study of the practical applications of FC [4].In modern era, the uses of FC in various engineering problems have been raised [5–7] 2014). For instance, FC has various applications in different diffusion phenomenon including heat transfer, gaseous exchange, and water transfer through permeable materials [8–11]. Bagley and Torvik presented FC as an instrument for displaying tissue viscoelasticity during the 1980s (Uchaikin, 2013). Study of intricacy gives another view to a few genuine wonders which appeared to be odd, and during the most recent years, new strategies have been utilized to separate secret properties of complex frameworks [12]. Further, a variety of FC tools have been widely used in several complex phenomena [13–15]. In some circumstances, FC has been perceived for taking care of issues in viscoelasticity, electrochemistry, and dispersion [16–19]. A few analysts featured FC as a tool for examination of complex phenomenon by bringing the techniques of FC and its applications to a more extensive crowd [20, 21] .The technique by which a real world problem is described in mathematical concepts or language is known as mathematical modeling [22]. Mathematical modeling of infectious diseases has been the main focus for the scientists and researchers over the last two decades. Mathematical modelers used to model the infectious diseases in the form of mathematical models consist of classical differential equations (CDEs). Recently, the researchers have diverted their focus to model the diseases in the form of fractional differential equations (FDEs) which has the potential to describe the real world phenomena more accurate and considered reliable as compared to the conventional derivatives. FDEs are global in nature, more realistic, and give great degree of freedom to modelers as compared to the CDEs. Modeling via FDEs has produced highly influential results in the investigation of transmission of the infectious diseases models [23, 24].In the year 1976, a flare-up occurred in African nation of the Democratic Republic of Congo (DRC), which was then termed after the name of the lake “Ebola” flows near to the DRC. The infection has five sorts, four out of these five spread illnesses in people. The infection use to attack on the immune system which then cause internal bleeding and affect each organ of the individuals. This terrifying infection spread by contacting directly with the tainted individuals either via body fluids or direct skin contact. The infection can also be pass through connection with the creatures like monkeys, etc. Nonetheless, the infection cannot be transmit through air and food. Later on, in 2013, the infection arose in Guckduo and Guinea, where 28,616 casualties were reported, and out of these casualties, 11,310 lost their lives. Today, where the advanced world is confronting another pandemic flare-up as COVID-19, the investigation of such irresistible sicknesses is still a center of focus for the researchers [25].


## 2. Model Formulation

In this section of the article, we have presented the formulation of the model, which we will be studying in this paper. For this, we have considered a population and divided it into five different compartments with some assumptions. The assumptions considered for the formulation of the model are stated below(i)
S: the first class of the model has been named as susceptible class. This class contains individuals who have no symptoms or any infection of the disease but can be attacked by the virus(ii)
E: the second class of the population has been named as exposed class. This class contains individuals who have been attacked by the virus but not yet shown the symptoms of the infection or not yet infectious(iii)
I: this is the third class of the population containing individuals who have been attacked by the virus and are being able to transfer the disease to not yet attacked individuals of the populations(iv)
V: this class has been named as vaccinated class containing those individuals of the susceptible class who have been vaccinated against the virus(v)
ℝ: this class is the recovered class which contains those individuals who have survived the diseaseThe transition or transfer among the compartments has been considered in the following manner(i)
S⟶τE: an individual of the population S move to the population E through the rate τ1(ii)
S⟶βI: an individual of the class S joins the class I with the rate β after getting infectious(iii)
S⟶ψV: the given parameter is used for the rate of the vaccination which transfer an individual from S to V(iv)
I⟶ξℝ: the rate of transfer of the individuals from I to ℝ after surviving the disease
(1)dSdt=Λ−d0St−τ1StEt−βStIt−ψStdEdt=τ1StEt−d0+d1+κEtdIdt=βStIt+κEt−ξIt−d0+d2ItdVdt=ψSt−d0Vtdℝdt=ξIt−d0ℝtAnd the corresponding fractional form of the system (1) is
(2)cDμSt=Λ−d0St−τStEt−βStIt−ψStcDμEt=τStEt−d0+d1+κEtcDμIt=βStIt+κEt−ξt−d0+d2ItcDμVt=ψSt−d0VtcDμℝt=ξt−d0ℝtThe paper has been organized as follows: the first section of the paper contains introduction. The second section has been restricted to the formulation of the model, while the third section has been devoted to the preliminaries. The fourth section of the paper contains the existence and uniqueness of the solution of the model. The fifth section of the paper includes the stability of the solution, while the sixth section contains qualitative study where we formulated the disease free, disease endemic, and basic reproduction numberR0 and then test the stability of the R0 locally with the help of theorems. In the seventh section, we have formulated the numerical solution of the model via Adam’s Bashforth scheme, and the eighth section contains the numerical simulation of the results obtained in the section seventh. At last, we have concluded our work in the conclusion section.


## 3. Preliminaries

In this section of the present article, we provide some basic definitions, theorems, and results that will be used and fruitful in understanding the rest of the article.Definition 1. (see [26]).
The Caputo’s fractional differential operator of any arbitrary orderμ>0 is defined as
(3)cDμθt=1Γn−μ∫0tfs,θsnt−sn−μ−1ds.Lemma 2. (see [27]).
“The following result holds for fractional differential equations(4)IμcDμθt=θt+α0+α1t+α2t2+⋯+αm−1tm−1,
for arbitraryαi∈R, i=0,1,2,3,⋯,m−1, where m=μ+1 and μ symbolizes the integer part of μ”.Lemma 3. (see [28]).
Letθ∈ACn0,T,μ>0, and n=μ, then the following result holds
(5)IμcDμθϑ=θϑ−∑j=0n−1Djθaj!t−aj.Lemma 4. (see [28]).
In view of Lemma3, the solution of Dμθt=yt, n−1<μ<n is given by
(6)θt=Iμyt+c0+c1t+c2t2++cn−1tn−1,
wherecj∈R.Definition 5. (see [26]).
Suppose we have Caputo’s fractional differential equation of orderμ(7)cDμθt=ft,θt,
then the solution is given as(8)θtn+=θtn+ftn,θnhΓμ2hμtn+1μ−tn+1μ+1μ+1+hμtnμ−tμ+1μftn−1,θn−1hΓμhμtn+1μ−tn+1μ+1μ+1+tnμμ+1+Rnμt,
whereRnμt represent the remainder term. For the study of convergence and uniqueness of the solution of the scheme, we refer to ([26]).Theorem 6. (see [29]).
“LetX be a Banach space and P:X⟶X is compact and continuous, if the set,
(9)E=θ∈X:θ=mPθ,m∈0,1,
is bounded, thenP has a unique fixed point.”


## 4. Existence of the Solution

In this section of the paper, we construct the conditions for the existence and uniqueness of the solution, and to get the desired results, we construct the following function.(10)ϑ1t,S,E,I,V,ℝ=Λ−d0St−τStEt−βStIt−ψSt,ϑ2t,S,E,I,V,ℝ=τStEt−d0+d1+κEt,ϑ3t,S,E,I,V,ℝ=βStIt+κEt−ξIt−d0+d2It,ϑ4t,S,E,I,V,ℝ=ψSt−d0Vt,ϑ5t,S,E,I,V,ℝ=ξIt−d0ℝt.Suppose that the considered spaceℂ0,T=B be a Banach space with norm
(11)θt=supt∈0,TSt+Et+It+Vt+ℝt,where
(12)θt=StEtItVtℝt,θ0t=S0E0I0V0ℝ0,Tt,θt=ϑ1t,S,E,I,V,ℝϑ2t,S,E,I,V,ℝϑ3t,S,E,I,V,ℝϑ4t,S,E,I,V,ℝϑ5t,S,E,I,V,ℝ.With the help of (12), the system (1) can be written in as
(13)cDμθt=Tt,θt,t∈0,T,θ0=θ0,By Lemma2, equation (13) converts into the following form
(14)θt=θ0+∫0tt−sμ−1ΓμTs,θsds,t∈J=0,T.To prove the existence of the solution, we make the following assumptions:P1∃ constants K1∗,M1∗∋(15)Tt,θt≤K1∗θq+M1∗.P2∃L∗>0, ∋ for each θ,θ¯(16)Tt,θ−Tt,θ¯≤L∗θ−θ¯.And letP:B⟶B be an operator as
(17)Pθt=θ0+∫0tt−sμ−1ΓμTs,θsds.Theorem 7.
When the assumptionsP1 and P2 are true, it verifies that the problem (13) has at least of one fixed point which also implies that the problem of our study has also at least one solution.Proof.
Furthermore we proceed as.Step 1.
First, we have to show thatP is continuous. To acquire the results, we suppose that Tj is continuous for j=1,2,3,4,5,6. Which implies that Ts,θs is also continuous. Assume θn.θ∈X∋θn⟶θ, we must have Pθn⟶Pθ.
For this, we consider(18)Pθn−Pθ=maxt∈J=0,T∫0tt−sμ−1ΓμTns,θnsds−1Γμ∫0tt−sμ−1Ts,θsds≤maxt∈J=0,T∫0tt−sμ−1ΓμTns,θns−Ts,θsds,≤TtΓμ+1Tn−T⟶0asn⟶∞.
AsT is continuous, therefore Pθn⟶Pθ, yields that P is continuous.Step 2.
Now, to prove thatP is bounded for any θ∈X, we make of the supposition that P satisfies the growth condition:
(19)Pθ=maxt∈0,Tθ0+1Γμ∫0tt−sμ−1Ts,θsds,≤θ0+maxt∈0,T1Γμ∫0tt−sμ−1Ts,θsds,≤θ0+TμΓμ+1K1∗θq+M1∗.
Here, we assume aS, the subset of X with the property of boundedness, and we need to prove that PS is also bounded. To reach our destination, we assume that for any θ∈S, now as S is bounded, so ∃Kq≥0∋(20)θ≤Kq,∀θ∈S.
Further, for anyθ∈S by using the growth condition, we have
(21)Pθ≤θ0+TμΓμ+1K1∗θq+M1∗≤θ0+TμΓμ+1K1∗Kq+M1∗.
Therefore,PS is bounded.Step 3.
Here, we attempt to prove that the operator we defined is equicontinuous, for this we assume thatt2≤t1∈J=0,T, then
(22)Pθt1−Pθt2=1Γμ∫0tt1−sμ−1Ts,θsds−1Γμ∫0tt2−sμ−1Ts,θsds,≤1Γμ∫0tt1−sμ−1−1Γμ∫0tt2−sμ−1Ts,θsds,≤TμΓμ+1K1∗θq+M1∗t1−t2.
By taking advantage of Arzelà-Ascoli theorem, we can say thatPS is relative compact.Step 4.
In this step, we need to prove that the set defined below is bounded(23)E=θ∈X:θ=mPθ,m∈0,1.
To prove this, we suppose thatθ∈E, ∋ for each t∈J, where J=0,T we have
(24)θ=mPθ≤mθ0+TμΓμ+1K1∗θq+M1∗.From here, we can claim that the set defined above is bounded. By using Schaefer’s FPT, the operator we defined, i.e.,P has atleast one fixed point, and hence, the model we studied in this paper has at least one solution.Theorem 8.
The problem (13) is unique solution, if TμK1∗/Γμ+1<1.Proof.
Letθ,θ¯∈X, then
(25)Pθ−Pθ¯≤maxt∈J=0,T∫0tt−sμ−1ΓμTs,θs−Ts,θ¯sds,≤TμLTΓμ+1θ−θ¯.Hence, we can say that the fixed point is unique, and therefore, our solution is unique.


## 5. Stability Results

To prove that the solution of the considered model is stable, we use the concept of Ulam and Ulam Hyer stability. To get the desired results we proceed asAssumeL:X⟶X be an operator which satisfy
(26)θ=Lθ,whereθ∈X.Definition 9. (see [27]).
Equation (26) has UH-stability, if for ς1>0 and assume any solution θ∈L for the inequality given by
(27)θ−Lθ≤ς1,∀t∈J=0,T,and the unique solution θ¯ for (25) with Cq∗>0, such that
(28)θ¯−θ≤Cq∗ς1,∀t∈0,T.Definition 5.2.
“If∃ϑ∈CR,R with ϑ0=0,” for unique result θ¯ and any solution of equation (26) ∋(29)θ¯−θ≤ϑς1,
then equation (26) has GUH-stability.Remark 10. (see [27]).
“If∋ζ1∗∗t∈C0,T,R, then θ¯∈X satisfies (27) if
(i)ζ1∗∗t≤ς1,∀t∈0,T
(ii)Lθ¯t=θ¯+ζ1∗∗t,∀t∈0,T”
For further analysis, we suppose that the following is the solution of the perturbed problem of (13)
(30)CD+0μθt=Tt,θt+ζ1∗∗t,θ0=θ0.Lemma 11.
The result stated below holds true for equation (30),
(31)θt−Pθt≤aς1,wherea=TμΓμ+1Theorem 12.
By making use of Lemma11, the solution of the problem (13) is UH-stable as well as GUH-stable, if TμLω/Γμ+1<1.Proof.
Assumeθ,θ¯∈X be any and unique solutions, respectively, problem (13), then
(32)θt−θ¯t=θt−Pθ¯t,≤θt−Pθt+Pθt+Pθ¯t,≤aς1+TμLθΓμ+1θt−θ¯t,≤aς11−TμLϑ/Γμ+1.
From here, we claim that the solution of (13) is UH and GUH stability if
(33)Yς1=aς11−TμLϑ/Γμ+1.
Such thatY0=0.Definition 13.
The UHR-Stability of (26) is ensured for g∗∈C0,T,R, if for ς1>0 and assume θ∈X be any solution of the inequality expressed by
(34)θ−Hθ≤gtς1.
∃ a unique solution θ¯ of (26) with Kq′>0∋(35)θ¯−θ≤Kq′g∗tς1,∀t∈0,T.Definition 14. (see [27]).
“Forg∗∈C0,T,R, if ∃Kq,g′ and for ς1>0, consider that θ be any solution of (34) and θ¯ be any solution of (26) ∋(36)θ¯−θ≤Kq,g′g∗t,∀t∈J=0,T,then equation (26) is generalized UHR stable.”Remark 15.
If∃ζ1∗∗t∈CJ,R, then for θ¯∈X (27) holds, if
(i)
ζ1∗∗t≤ς1ωt,∀t∈J(ii)
Lθ¯t=θ¯+ζ1∗∗t,∀t∈JLemma 16.
The stated result below holds true for (30)
(37)θt−Pθt≤aωtς1,a=TμΓμ+1Proof.
The proof has been left for the readers.Theorem 17.
With the help of Lemma16, our solution is Ulam and Generalized Ulam stable if TμLθ/Γμ+1<1.Proof.
. Assumeθ,θ¯∈X be two solutions such that θ is any and any and θ¯ is the unique solution of our problem, then
(38)θt−θ¯t=θt−Pθ¯t,≤θt−Pθt+Pθt+Pθ¯t,≤aωtς1+TμLθΓμ+1θt−θ¯t,≤aωtς11−TμLϑ/Γμ+1.Hence, the solution possesses both type of the stabilities.


## 6. Qualitative Study

In this section of the article, we present disease free equilibrium, disease endemic equilibrium, the basic reproduction numberR0, and the local assymptotical stability of the R0. To proceed, we first find the disease free equilibrium and disease endemic equilibrium of the model. The disease free equilibrium is given as S0=Λ/d0+ψ,0,0,0,0, while the endemic equilibrium is given below.


### 6.1. Endemic Equilibrium

The endemic equilibrium of the model is given as(39)S∗=d0+d1+κτ,E∗=1τκτξ+d0+d2−βd0+d1+κI∗,I∗=Λτd0+d1−κ−d0+ψκτξ+d0+d2−βd0+d1+κ−κβ,V∗=ψd0d0+d1−κτ,R∗=ξd0R∗.


### 6.2. The Basic Reproduction Number

To findR0, we construct two vectors such as
(40)F=f1f2=τStEtβStIt,(41)V=v1v2=d0+d1+κEtξ+d0+d2It−κEt.Now, we present the Jacobian of both the matrix, i.e.,(42)JF=τSt00βSt,(43)JV=d0+d1+κ0−κξ+d0+d2.with(44)JV−1=1d0+d1+κ0κξ+d0+d2d0+d1+κ1ξ+d0+d2.Now, to find the next generation matrix (NGM), we find the product ofJF and JV−1, i.e.,
(45)NGM=τStd0+d1+κ0βStκξ+d0+d2d0+d1κβStξ+d0+d2.Clearly, the eigen values are (say)λ1 and λ2 which are given as λ1=τSt/d0+d1+κ and λ2=βSt/ξ+d0+d2. Therefore, the basic reproduction number R0=maxλ1,λ2.Theorem 18.
The basic reproduction numberR0 is locally assymptotically stable at the disease free equilibrium point that is stable if R0<1.Proof.
For this purpose, we construct the following Jacobian.(46)Aj=−d0+ψ−τS0−βS0000τS0−d0+d1+κ000βS0κ−ξ−d0+d200ψ00−d0000ξ0−d0.
Now, let the eigen values are (say)λ1,λ2,λ3,λ4,λ5. Clearly λ1=λ2=−d0, λ3=τS0−d0+d1+κ, λ4=−d0+ψ, and λ5=−β2S2−d0+ψξ+d0+d2. From λ3, we have τS0/d0+d1+κ<1, and from λ5, we have d0+ψ/βS0<βS0/ξ+d0+d−2<1. Therefore, R0=maxλ3,λ5<1 and hence is locally assymptotically stable at the disease free equilibrium point.Theorem 6.2.
The basic reproduction numberR0 is locally assymptotically stable at the endemic equilibrium point if R0>1.Proof.
The proof of this result can be obtained on the same manner as the proof in [30].


## 6.1. Endemic Equilibrium

The endemic equilibrium of the model is given as(39)S∗=d0+d1+κτ,E∗=1τκτξ+d0+d2−βd0+d1+κI∗,I∗=Λτd0+d1−κ−d0+ψκτξ+d0+d2−βd0+d1+κ−κβ,V∗=ψd0d0+d1−κτ,R∗=ξd0R∗.


## 6.2. The Basic Reproduction Number

To findR0, we construct two vectors such as
(40)F=f1f2=τStEtβStIt,(41)V=v1v2=d0+d1+κEtξ+d0+d2It−κEt.Now, we present the Jacobian of both the matrix, i.e.,(42)JF=τSt00βSt,(43)JV=d0+d1+κ0−κξ+d0+d2.with(44)JV−1=1d0+d1+κ0κξ+d0+d2d0+d1+κ1ξ+d0+d2.Now, to find the next generation matrix (NGM), we find the product ofJF and JV−1, i.e.,
(45)NGM=τStd0+d1+κ0βStκξ+d0+d2d0+d1κβStξ+d0+d2.Clearly, the eigen values are (say)λ1 and λ2 which are given as λ1=τSt/d0+d1+κ and λ2=βSt/ξ+d0+d2. Therefore, the basic reproduction number R0=maxλ1,λ2.Theorem 18.
The basic reproduction numberR0 is locally assymptotically stable at the disease free equilibrium point that is stable if R0<1.Proof.
For this purpose, we construct the following Jacobian.(46)Aj=−d0+ψ−τS0−βS0000τS0−d0+d1+κ000βS0κ−ξ−d0+d200ψ00−d0000ξ0−d0.
Now, let the eigen values are (say)λ1,λ2,λ3,λ4,λ5. Clearly λ1=λ2=−d0, λ3=τS0−d0+d1+κ, λ4=−d0+ψ, and λ5=−β2S2−d0+ψξ+d0+d2. From λ3, we have τS0/d0+d1+κ<1, and from λ5, we have d0+ψ/βS0<βS0/ξ+d0+d−2<1. Therefore, R0=maxλ3,λ5<1 and hence is locally assymptotically stable at the disease free equilibrium point.Theorem 6.2.
The basic reproduction numberR0 is locally assymptotically stable at the endemic equilibrium point if R0>1.Proof.
The proof of this result can be obtained on the same manner as the proof in [30].


## 7. Numerical Solution

This section of the article is devoted to the numerical solution of the considered model. For this, we will use the well-known two-step fractional order Adam’s Bashforth method. The considered model is given as(47)cDμSt=Λ−d0St−τStEt−βStIt−ψSt,cDμEt=τStEt−d0+d1Et,cDμIt=βStIt+KappaEt−XiIt−d0+d2It,cDμIt=ψSt−d0Vt,cDμℝt=XiIt−d0ℝt.To obtain the desired results, we apply the fundamental theorem of fractional calculus to system (3) gives
(48)St=S0+1Γμ∫0tA1β,Sβt−βμ−1dβ,Et=E0+1Γμ∫0tA2β,Eβt−βμ−1dβ,It=I0+1Γμ∫0tA3β,Iβt−βμ−1dβ,Vt=V0+1Γμ∫0tA4β,Vβt−βμ−1dβ,ℝt=ℝ0+1Γμ∫0tA5β,ℝβt−βμ−1dβ.The unknown termsA1,A2,A3,A4,A5 are given below. Now, for t=tn+1, we get
(49)Stn+1=S0+1Γμ∫0tn+1A1t,Sttn+1−tμ−1dt,Etn+1=E0+1Γμ∫0tn+1A2t,Ettn+1−tμ−1dt,Itn+1=I0+1Γμ∫0tn+1A3t,Ittn+1−tμ−1dt,Vtn+1=V0+1Γμ∫0tn+1A4t,Vttn+1−tμ−1dt,ℝtn+1=ℝ0+1Γμ∫0tn+1A5t,ℝttn+1−tμ−1dt.Fort=tn, we get the following
(50)Stn=S0+1Γμ∫0tnA1t,Sttn−tμ−1dt,Etn=E0+1Γμ∫0tnA2t,Ettn−tμ−1dt,Itn=I0+1Γμ∫0tnA3t,Ittn−tμ−1dt,Vtn=V0+1Γμ∫0tnA4t,Vttn−tμ−1dt,ℝtn=ℝ0+1Γμ∫0tnA5t,ℝttn−tμ−1dt.ByStn+1−Stn, Etn+1−Etn, Itn+1−Itn, Vtn+1−Vtn, and ℝtn+1−ℝtn in (49) and (50), we obtain
(51)Stn+1=Stn+Aμ,11+Aeta,21,Etn+1=Etn+Aμ,12+Aeta,22,Itn+1=Itn+Aμ,13+Aeta,23,Vtn+1=Vtn+Aμ,14+Aeta,24,ℝtn+1=ℝtn+Aμ,15+Aeta,25,where(52)Aμ,11=1Γμ∫0tn+1A1t,Sttn+1−tμ−1dt,Aμ,12=1Γμ∫0tn+1A2t,Ettn+1−tμ−1dt,Aμ,13=1Γμ∫0tn+1A3t,Ittn+1−tμ−1dt,Aμ,14=1Γμ∫0tn+1A4t,Vttn+1−tμ−1dt,Aμ,15=1Γμ∫0tn+1A5t,ℝttn+1−tμ−1dt.(53)Aμ,21=1Γμ∫0tnA1t,Sttn−tμ−1dt,Aμ,22=1Γμ∫0tnA2t,Ettn−tμ−1dt,Aμ,23=1Γμ∫0tnA3t,Ittn−tμ−1dt,Aμ,24=1Γμ∫0tnA4t,Vttn−tμ−1dt,Aμ,25=1Γμ∫0tnA5t,ℝttn−tμ−1dt.By approximatingAμ,11, Aμ,21, Aμ,12, Aμ,22, Aμ,13, Aμ,23, Aμ,14, Aμ,24, Aμ,15, and Aμ,25 with the help of Lagrange’s polynomials and the plugging back in (51), we get the following solution
(54)Stn+1=Stn+A1tn,StnℏΓμ2ℏμtn+1μ−tn+1μ+1μ+1+ℏμtnμ−tμ+1μ+A1tn−1,Sn−1ℏΓμℏμtn+1μ−tn+1μ+1μ+1+tnμμ+1+ℝ1,nμt,Etn+1=Etn+A2tn,EtnℏΓμ2ℏμtn+1μ−tn+1μ+1μ+1+ℏμtnμ−tμ+1μ+A2tn−1,En−1ℏΓμℏμtn+1μ−tn+1μ+1μ+1+tnμμ+1+ℝ2,nμt,Itn+1=Itn+A3tn,IℏtnℏΓμ2ℏμtn+1μ−tn+1μ+1μ+1+ℏμtnμ−tμ+1μ+A3tn−1,In−1ℏΓμℏμtn+1μ−tn+1μ+1μ+1+tnμμ+1+ℝ3,nμt,Vtn+1=Vtn+A4tn,VtnℏΓμ2ℏμtn+1μ−tn+1μ+1μ+1+ℏμtnμ−tμ+1μ+A4tn−1,Vn−1ℏΓμℏμtn+1μ−tn+1μ+1μ+1+tnμμ+1+ℝ4,nμt,ℝtn+1=ℝtn+A5tn,ℝtnℏΓμ2ℏμtn+1μ−tn+1μ+1μ+1+ℏμtnμ−tμ+1μ+A5tn−1,Sn−1ℏΓμℏμtn+1μ−tn+1μ+1μ+1+tnμμ+1+ℝ5,nμt,where(55)A1=Λ−d0St−τStEt−βStIt−ψSt,A2=τStEt−d0+d1+κEt,A3=βStIt+κEt−ξIt−d0+d2It,A4=ψSt−d0Vt,A5=ξIt−d0ℝt.Andℝ1,nμt, ℝ2,nμt, ℝ3,nμt, ℝ4,nμt, and ℝ5,nμt are the remainder’s terms.


## 8. Numerical Simulation

In this section of the article, we present the graphical results of the solution obtained in (54). For this purpose we have simulated the results via Matlab by assigning the values given in (Table 1) to the parameters and classes of the model. The graphical results are shown in the following.Table 1
The physical interpretation and numerical values of the parameters.
ParametersPhysical descriptionNumerical valueSourceΛThe birth rate0.4Assumedd0Natural death rate0.7Assumedd1Disease death rate inEt0.075Assumedd2Disease death rate inIt0.35AssumedτThe contact rate ofSt and Et0.14280AssumedκThe transference rate fromEt to It0.048AssumedβThe contact rate ofSt and It0.35AssumedξRecovery rate0.53AssumedψVaccination rate0.00493Assumed


## 9. Discussion

Figure1 describes the dynamics of susceptible population for different values of the order of fractional derivatives. Each curve tends to the equilibrium solution irrespective of the value of μ. As we increase the value of μ, the rate of convergence to the stated equilibrium increases. Figure 2 represents the behavior of E(t) along the time direction, and the figure shows that for describing the slow evolution of disease, one might assume small values of μ. Infection from the community could be rapidly eliminated by increasing the order of the derivative as shown in Figure 3. A similar conclusion could be drawn from Figures 4 and 5, i.e., to capture the realistic scenario of slowly spreading diseases, one must consider the tools of fractional order derivative while modeling such epidemics.Figure 1
The behavior of susceptible population.Figure 2
The behavior of exposed population.Figure 3
The behavior of infected population.Figure 4
The behavior of vaccinated population.Figure 5
The behavior of recovered population.


## 10. Conclusion

In this paper, we have studied the fractional order Ebola model containing Caputo’s fractional derivative of orderμ. The paper contains the study related to the existence of the solution performed by using theorems of fixed point theory for the existence of fixed point. In addition, we have proved that the solution of the system is unique as well as Ulam stable. Apart from this, we have find the numerical solution of the studied model with the help of two-point fractional order Adam’s Bashforth method presented for the approximation of the fractional differential equations containing the Caputo’s fractional derivative. In addition, we have visualized the results graphically with the help of Matlab. At last, we have discussed the dynamical behavior of the obtained solution for all classes of the said model.



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