# A Fractional-Order Investigation of Vaccinated SARS-CoV-2 Epidemic Model with Caputo Fractional Derivative

Authors: Badr Saad T. Alkahtani
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we consider a fractional-order mathematical system comprising four different compartments for the recent pandemic of SARS-CoV-2 with regard to global and singular kernels of Caputo fractional operator. The SARS-CoV-2 fractional mathematical model is analyzed for series-type solution by Laplace–Adomian decomposition techniques (LADM) and homotopy perturbation method (HPM). The whole quantity of each compartment is divided into small parts, and then the sum of these all parts is written as a series solution for each agent of the system, while the nonlinear part is decomposed using the Adomian polynomial. The model is also checked for approximate solution by HPM through a comparison of the parameter power,p, for each equation. The numerical simulation for both methods is provided in different fractional orders along with comparison with each other as well as with natural order 1.


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## Body

## 1. Introduction

The novel coronavirus (SARS-CoV-2), considered the most dangerous virus of this decade, belong to family of severe acute respiratory syndrome (SARS) [1, 2]. Therefore, this new virus is related to the viruses associated with the syndrome. It has become a new novel strain of the SARS family, which was recognised in humans before [3, 4]. SARS-CoV-2 affected not only humans but also several animals. This virus has been transmitted from human to human, and it occurs similarly in animals. But many times, it has become a mystery that how the animals have been affected by it in certain security. Infected humans and different species of various animals are also recognised as an active cause of the spreading of the virus [5]. In the past, some similar viruses like the Middle East respiratory syndrome coronavirus (MERS-CoV) were spread from camels to the human population, and for SARS-CoV-1, the civet cats were recognised as the source of spreading into humans [6].Mathematical models concerning infections have vastly been used since the last century to study the dynamics and transmission of various pandemics and epidemics and to apply valuable techniques for their control or minimization. Scholars investigating pandemics in the various areas of sciences are working to control these epidemics or reduce these negative impacts to a stable situation [7–9]. They also give the concept of the globalization of the ODEs that have the natural-order differentiation providing more degrees of freedom at any order. The equations which contain the FO differential equation δ, with FO 0<δ≤1, may be studied in [10–12]. Mostly, in epidemiological problems, FDEs refer to models with memory effects [12]. Next, the fractional-order derivative has the term of integration providing the knowledge of the past spreading for an infection. We can expect the behavior of the transmission based on the previous results and studies. The hereditary and historical characteristics point to the past transmission of diseases, which is very beneficial for making predictions. Therefore, these characteristics can be tested by the application of noninteger order derivatives, and it impacts the transmission of an epidemic [10–12].Modern calculus goods production arbitrary order may be used in different fields of clinical and physical sciences, such as goods production by engineering, control theory, economics, financing, and infectious disease conditions. The extensive huge study of FODEs in modelling global phenomena is because of more attracting characteristics which are not explained in the natural order derivatives. The natural order differential equations are local in nature, while FDEs are concerned with nonlocality, which provides more globalization of their dynamics. Usage of integer operator is hot area and recently caught the researchers interest, whereas the noninteger order operators have been studied and used intensively. Infectious problems for the endemic are the most realistic area for the researchers as a research gap and are applied to test them in recent times. Moreover, the analysis for the mathematical models referring to the real-globe situation is made using the theory of stability, existence of solution, and with the optimal problem as in [13–15].We proposed a new vaccinated SARS-CoV-2 epidemic model that has four quantities including the susceptible classWt, the acutely infected class Xt, chronically infected class Yt, and the recovered class ℤt, which takes the following form:(1)dWtdt=Λ−βWtXt−ρ+ξWt,dXtdt=βWtXt−ξ+γ+λXt,dYtdt=λXt−ξ+δ+κYt,dℤtdt=γXt+κYt+ρWt−ξℤt.The parameters used in system (1) are described in Table 1:Table 1
The description of different parameters given in model (1).
Parameter symbolsDescriptionΠRate of new birth or recruitmentνRate of vaccination for SARS-CoV-2μRate of natural deathsαRate of death due to SARS-CoV-2ηInteraction rate of infectious and healthy populationγRecovered rate for acute infectionκRecovered rate for chronic infectionλTransferring rate acute class to chronic infectionThe analysis of model (1) under fractional-order derivative with regard to the Caputo operator is given as(2)DℂtζWt=Λ−βWtXt−ρ+ξWt,DℂtζXt=βWtXt−ξ+γ+λXt,DℂtζYt=λXt−ξ+δ+κYt,Dℂtζℤt=γXt+κYt+ρWt−ξℤt,with general initial approximation W0=N1, X0=N2, Y0=N3, and ℤ0=N4.In the seventeenth century, many researchers Euler, L'Hôpital, Fourier, Abels, Riemann–Liouville, etc., made fundamental contributions in this field of modern calculus and were known as the pioneers of fractional calculus. Besides them, many other scientists have made significant findings and discovered some fractional models as seen in [10–12]. Most of the basic properties needed for real phenomena such as memory, globality, and hereditary are involved in various fractional operators, while the integer-order differential operator have no such properties; therefore, modern calculus gives more realistic result. Modern calculus refers to the appliances of biological, physical, and engineering as in [11, 16–18]. Other properties of fractional operators like nonlocality, globality, singularity, and nonsingularity also attract the interest of many researchers. These properties are more applicable to most real-world problems.It should be kept in mind that fractional operators do have not a unique definition and are formulated by different formulae. Most of the operators have the definite integral with singular and nonsingular kernels. These kernels can also be found in the various fractional integration formulae. These kernels are mostly taken from the Cauchy integral formula. These integral kernels present in the fractional operator may be analyzed by various techniques. Some researchers have used Laplace–Adomian decomposition techniques for both linear and nonlinear fractional differential models. For the terms of noninearity, Adomian polynomial is applied, which decomposes that term into a small one (see [19, 20]). Similarly, the homotopy perturbation techniques are also applied for a series solution, which can be obtained by perturbed by a small factor and then comparing the power of parameter ζ∈0,1 on both sides of the equation. For the iterative numerical solution of integer-order models, mostly Runge–Kutta techniques (RK4 and RK2) were used. They can be used for fractional-order equations by involving the fractional terms. In this article, we will apply Laplace transformation along with Adomian decomposition techniques by converting the while quantity into small ones. We will also apply the homotopy perturbation techniques (HPM) for the series solution of the said problem [21, 22].Different global real-life situations may be represented by noninteger differential equationsFDEs such as physics problems, problems of control theory, chemistry problems (polymerization, rheology, and electronics), physical biology, heat, aerodynamics, infectious diseases, electro-statics, electrical circuits, and blood flow [11, 23–25]. In the last few years, the interesting description of the existence of solutions of FDEs by changing the boundary conditions counting integer order phenomena, nonlocal, nonsingular kernel, periodic/antiperiodic, and multipoints has been investigated and has provided good supposition, as can be seen in [26–28]. Different researchers have tested the solution of the coupled models by differential equations of noninteger orders which gives the key role in applied fields of mathematics. This is the establishment for the model developed from biochemistry, biology, ecology, and other areas of engineering and physical sciences such as in [29].


## 2. Basic Definition

In this section, some definitions are written from [19, 20, 30].Definition 1.
ConsiderY∈L10,∞ℤ to be a mapping; then, the Riemann–Liouville noninteger order integral of order ζ is as follows:(3)ItζYt=1Γζ∫0tYβt−β1−ζdβ,ζ>0,where the right integral must exist.Definition 2.
LetY be an operator, then the Caputo noninteger order derivative can be defined as(4)DℂtζYt=1Γn−ζ∫0tt−βn−ζ−1Ynβdβ,and the R.H.S of the integration must exist and n=ζ+1. If ζ∈0,1, then someone has(5)DℂtζYt=1Γ1−ζ∫0tY′βt−βζdβ.Lemma 1.
From the noninteger order differential equation, the following is satisfied:(6)IζDℂζpt=pt+γ0+γ1t+γ2t2+⋯+γn−1tn−1.Definition 3.
With regard to the Caputo operator, the Laplace transformation can be written as(7)ℒDℂtζpt=sζPs−∑j=0m−1sζ−j−1pk0,m−1<ζ<m,m∈ℕ.Definition 4.
On application of the homotopy perturbation techniques to an equation having linearL and nonlinear N classes, we may write a homotopy vr,g:Ω×0×1⟶ℤ.(8)Hv,g=1−gLv−Lu0+gLv+Nv−fr=0,where r∈Ω and q∈0,1 is the embidding parameter.


## 3. Series Solution of Problem (2) via LADM

This section is devoted to the analysis of general-typed series techniques for the considered model (2) along with some starting conditions. On the application of the Laplace transformation to the considered problem (2), we get the following:(9)ℒDCtζWt=ℒΛ−βSA−ρ+ξW,ℒDℂtζXt=ℒβSA−ξ+γ+λX,ℒDℂtζYt=ℒλX−ξ+δ+κY,ℒDℂtζℤt=ℒγX+κY+ρW−ξℤ.Applying the initial approximation, problem (9) becomes(10)ℒWt=N1s+1sζℒΛ−βSA−ρ+ξW,ℒXt=N2s+1sζℒβSA−ξ+γ+λX,ℒYt=N3s+1sζℒλX−ξ+δ+κY,ℒℤt=N5s+1sζℒγX+κY+ρW−ξℤ.Taking the infinite series solution forW,X,Y,andℤ as(11)Wt=∑n=0∞Wnt,Xt=∑n=0∞Xnt,Yt=∑n=0∞Ynt,ℤt=∑n=0∞ℤnt,the nonlinear term WtXt=∑n=0∞Xnt can be decomposed as(12)Xnt=1n!dndλn∑k=0nλkWkt∑k=0nλkXktλ=0.Substitute equations (11) and (12) into equation (10), and on comparing the terms on each side, we obtain(13)ℒW0t=N1s,ℒX0t=N2s,ℒY0t=N3s,ℒℤ0t=N4s,ℒW1t=1sζℒΛ−βW0A0−ρ+ξW0,ℒX1t=1sζℒβW0A0−ξ+γ+λX0,ℒY1t=1sζℒλX0−ξ+δ+κY0,ℒℤ1t=1sζℒγX0+κY0+ρW0−ξℤ0,⋮ℒWn+1t=1sζℒΛ−βWnAn−ρ+ξWn,ℒXn+1t=1sζℒβWnAn−ξ+γ+λXn,ℒYn+1t=1sζℒλXn−ξ+δ+κYn,ℒℤn+1t=1sζℒγXn+κYn+ρWn−ξℤn.Upon using the inverse Laplace transform in expression (14), we get(14)W0t=ℒ−1N1s=N1,X0t=ℒ−1N2s=N2,Y0t=ℒ−1N3s=N3,,ℤ0t=ℒ−1N4s=N4,W1t=Λ−βN1N2−ρ+ξN1tζΓζ+1,X1t=βN1N2−ξ+γ+λN2tζΓζ+1,Y1t=λN2−ξ+δ+κN3tζΓζ+1,ℤ1t=γN2+κN3+ρN1−ξN4tζΓζ+1,W2t=ΛtζΓζ+1−βN1G11+N2K11−ρ+ξG11t2ζΓ2ζ+1,X2t=βN1G11+N2K11−ξ+γ+λK11t2ζΓ2ζ+1,Y2t=λK11−ξ+δ+κL11tζ2Γ2ζ+1,ℤ2t=γK11+κL11+ρG11−ξM11t2ζΓ2ζ+1.In the same way, the remaining terms can be obtained, and the terms given in equation (14) are as follows:(15)G11=Λ−βN1N2−ρ+ξN1,K11=βN1N2−ξ+γ+λN2,L11=λN2−ξ+δ+κN3,M11=γN2+κN3+ρN1−ξN4.


## 4. Approximate Solution for Problem (2) via HPM

Furthermore, we apply the HPM to obtain the approximate solution for the proposed problem (2), according to [20, 21] as follows:(16)1−qDℂtζWt−DℂtζW0t+qDℂtζWt−Λ+βSA+ρ+ξW=0,1−qDℂtζXt−DℂtζE0t+qDℂtζXt−βSA+ξ+γ+λX=0,1−qDℂtζYt−DℂtζI0t+qDℂtζYt−λX+ξ+δ+κY=0,1−qDℂtζℤt−Dℂtζℤ0t+qDℂtζℤt−γX−κY−ρW+ξℤ=0.By applyingq=0 in equation: (16), a model of arbitrary differential equations can be obtained:(17)DℂtζWt−DYtζW0t=0,DℂtζXt−DYtζE0t=0,DℂtζYt−DYtζI0t=0,Dℂtζℤt−DYtζℤ0t=0.Solutions for the equation (17) are straightforward. If we use q=1 in equation (16), we will obtain the same system as (2). The infinite series solution can be write in the following form as(18)Wt=∑n=0∞qnSnt,Xt=∑n=0∞qnAnt,Yt=∑n=0∞qnCnt,ℤt=∑n=0∞qnRnt.So, by comparing each termq=1, in equation (18), we get the original model. Plugging equation (18) in to equation (16) and on comparison of each terms with the same power of q, we get(19)q0:W0t=N1,X0t=N2,Y0t=N3,ℤ0t=N4..Similarly,(20)q1:W1=Λ−βW0X0−ρ+ξW0tζΓζ+1,X1=βW0X0−ξ+γ+λX0tζΓζ+1,Y1=λX0−ξ+δ+κY0tζΓζ+1,ℤ1=γX0+κY0+ρW0−ξℤ0tζΓζ+1,(21)q2:W2=Λ−βW0X0βW0X0−ξ+γ+λX0−βX0+ρ+ξΛ−βW0X0−ρ+ξW0t2ζΓ2ζ+1,X2=W0−ξ+γ+λβW0X0−ξ+γ+λX0+X0Λ−βW0X0−ρ+ξW0t2ζΓ2ζ+1,Y2=λβW0X0−ξ+γ+λX0−ξ+δ+κλX0−ξ+δ+κY0t2ζΓ2ζ+1,ℤ2=γβW0X0−ξ+γ+λX0+κλX0−ξ+δ+κY0+ρΛ−βW0X0−ρ+ξW0−ξγX0+κY0+ρW0−ξℤ0t2ζΓ2ζ+1.Similarly, the high terms can be obtained, and the required terms are given in the part above. Hence, we obtain the same high terms as we obtained using LADM. Both the methods are applied as strong techniques for nonlinear fractional-order equations.


## 5. Discussion along with Numerical Simulation for the Proposed System (2)

Next, we compute the numerical solution for the considered system (2), by assigning different values for the parameter given in Table 1 used in problem (2).The first four terms of the considered model (2), by using values from the table are given as follows:(22)W0t=6,X0t=3,Y0t=2,ℤ0t=0,W1t=−5.2Γζ+1tζ,X1t=,5.3Γζ+1tζ,Y1t=0.8Γζ+1tζ,ℤ1t=0,W2t=0.4Γζ+1tζ+4.7Γ2ζ+1t2ζ,X2t=−5.7Γ2ζ+1t2ζ,Y2t=2.25Γ2ζ+1t2ζ,ℤ2t=0.15Γ2ζ+1t2ζ.The solution for the first few terms are the following:(23)Wt=6−5.2Γζ+1tζ+4.876Γ2ζ+1t2ζ+0.3687Γ2ζ+1Γ2ζ+1Γ3ζ+1t3ζ…,Xt=3+5.3Γζ+1tζ−5.765Γ2ζ+1t2ζ−0.3465Γ2ζ+1Γ2ζ+1Γ3ζ+1t3ζ…,Yt=2+0.8Γζ+1tζ+2.7654Γ2ζ+1t2ζ−0.6753Γ3ζ+1t3ζ+…,ℤt=0.065Γ2ζ+1t2ζ−0.0076Γ3ζ+1t3ζ+….Now, evaluating equation (23) with ζ=0.8, we get(24)Wt=6−5.765470670t0.9+3.876379584t1.8−0.507094532t2.7+…,Xt=2−4.543942654t0.9−2.655544765t1.8−0.5464427072t2.7+…,Yt=2+0.08768770670t0.9+0.876594565t1.8−0.8766862398t2.7+…,ℤt=0.87656354145t1.8−0.00872947616t2.7+….Similarly, forζ=0.6, the approximate solution for (23) is(25)Wt=6−5.654737025t0.7+2.8759942t1.4−1.324229934t2.1+…,Xt=3−4.765353842t0.7−2.654389698t1.4+0.654351204t2.1…,Yt=2+0.05502737025t0.7+0.8473079812t1.4−0.4321037668t2.1+…,ℤt=0.098769077782t1.4−0.98767203392t2.1+…,and for ζ=0.4, the approximate solution for (23) is(26)Wt=6−5.765495834t0.5+2.645t1.0−2.8763389788t1.5…,Xt=3−4.987430418t0.5−3.343t1.0+2.765458202t1.5…,Yt=2+0.32421895835t0.5+2.2345t1.0−0.7643392382t1.5+…,ℤt=0.098t1.0−0.764062165752t1.5+….For the verification of our semianalytical solution by both methods, we provide numerical simulations for problem (2). Our simulation pertains to the qualitative point analysis, and the parameters are considered through a biological feasibility approach. We take parameter numerical value and then different initial class sizes for each of the compartment, namely, susceptible Wt, acutely infectious Xt, chronic carries, Yt, and recovered ℤt, from Table 2. Figure 1(a) shows a a quick decline in the starting in the quantity of susceptible class at at different fractional orders. The occurred decline took much more time at low nonnatural order and very slow at high noninteger order. We observe that, as the arbitrary order values increased, the curve of simulation goes, converging to the order 1. Besides this, after few days, the said individuals show low growth and then show converging towards stability to the equilibrium point. Figure 1(b) represents the acute infectious class, Xt increases for some beginning days at various fractional orders. Beside that, the curve declines and stay stable through all arbitrary orders and converges to an integer order. In Figure 1(c), one may see the chronical infectious cases. Yt decreases at the starting of few days and then moves up to the maximum value of 1.5 at different noninteger order as no treatment and precautionary measures are done. But after keeping the precautions, the infectious case declines to 0.9 and then their after show stability and convergency. From Figure 1(d) we see that the recovery class goes up to 2.7 at the starting at various fractional orders and by keeping precautions and treatment the recovered class also stabilized.Table 2
Numerical values for the parameters given in model (2).
ParametersValueΛ0.8ρ0.5ξ0.2δ0.5β0.4γ0.04κ0.05λ0.2Figure 1
Graphical representation of the series solution ofWt,Xt,Yt,ℤt of the proposed system (2) for various arbitrary orders. (a) W(t): susceptible population. (b) X(t): acutely infected population. (c) Y(t): chronically infected population. (d) Z(t): recovered population.
(a)(b)(c)(d)Next, we have provided the comparison of different agents of the considered model by both the methods of LADM and HPM, as given from Figures2(a)–3(b). Figure 4 represents the comparison of various agents at ζ=1.Figure 2
Comparison of LADM and HPM for the series solution ofWt,Xt for various arbitrary orders of the considered model (2). (a) Comparison of LADM and HPM for the series solution of W(t) for various arbitrary orders. (b) Comparison of LADM and HPM for the series solution of X(t) for various arbitrary orders.
(a)(b)Figure 3
Comparison of LADM and HPM for the series solution ofYt,ℤt for various arbitrary orders of the considered model (2). (a) Comparison of LADM and HPM for the series solution of Y(t) for various arbitrary orders. (b) Comparison of LADM and HPM for the series solution of Z(t) for various arbitrary orders.
(a)(b)Figure 4
Graphical representation of the series solution for all the quantities of the considered model (2) with ζ=1.


## 6. Conclusions

The current investigation is the development of the four compartmental fractional-order SARS-CoV-2 model with regard to the Caputo fractional-order operative of Caputo having a singular kernel. The analysis for the series type solution of the proposed problem has been successfully achieved by two methods, one is the Laplace transform along with the Adomian polynomial (LADM) for a nonlinear term and the other one is the homotopy perturbation method (HPM). The semianalytical type solution has been obtained by both the methods which are comparable with each other. The numerical simulation for a few terms has been plotted using the available data given in Table1 for four different values of ζ. We also compare few fractional-order values with that of integer 1, and as increasing the fractional-order value, we achieved the behavior of order 1. A complete spectrum for all compartments has established, and we can use other fractional values between 0 and 1. As a result, we say that fractional-order analysis provides better results than those of the integer-order problems.



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