# Study of a Fractional-Order Epidemic Model of Childhood Diseases

Authors: Aman Ullah; Thabet Abdeljawad; Shabir Ahmad; Kamal Shah
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we discuss the existence and uniqueness of the solution of the fractional-order epidemic model of childhood diseases by using fixed point theory. The technique of natural transform coupled with the Adomian decomposition is used to find the solution of the proposed model. At the end of the article, the model is demonstrated with appropriate numerical and graphical description.


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## Body

## 1. Introduction

The immunity of the children is very weak; therefore, all the children are exposed to various infectious and noninfectious diseases during childhood [1]. The most common childhood diseases are either viral or bacterial infections or allergic and immunologic diseases [2]. “Respiratory Syncytial Virus” (RSV) causes bronchiolitis, which affects baby children up to 1 year. “Coxsackievirus” causes hand, foot, and mouth disease, which has no proper treatment for the infection and needs proper care. The infection due to “rotavirus” causes vomiting, high fever, and diarrhea in children, often leading to critical problems with dehydration. The “varicella” virus causes “chickenpox.” The “rubeola virus” causes measles, also known as “German measles,” leading to serious complications, even death. Mumps is another viral disease which initially starts with flu and then causes parotitis. Viruses or bacteria cause meningitis, which have symptoms like headache, stiff neck, fever, and malaise. It is an inflammation of the tissue surrounding the brain and spinal cord “meninges.” A strep infection causes “scarlet fever,” which commonly appears after the throat infection. There are many other diseases that are found in children in which it is not possible to list them all. The term modeling is derived from the Latin word “modulus” [3] which means that it is without restraint an available application that permits using mathematics to create model interactivity. We use the concept of mathematical modeling [4], for this divided the population into two major classes. The first one is a mature class while the other is a premature class. The premature class is then converted to a mature class with a constant time, known as maturation delay. To analyze the dynamics of childhood diseases, many of the diseases cannot be spread rapidly in the human body; for a certain disease, it will take a particular time called “latent period.” All over the world, proper vaccination is suggested to control all the mentioned childhood diseases. For this purpose, WHO launched the immunization program all over the world in 1974 [5]. Singh et al. [2] provided a mathematical model to formulate the procedure and present a numerical solution of vaccination in various diseases during childhood. The SIR model is presented by Makinde [6], as given below:
(1)Ṡt=1−pω−βsin−ωs,İt=βsin−γ+ωi,Ṙt=pω+γi−ωr.Arafa et al. rearranged the model (1) using the following relations, s/n=S,i/n=I,r/n=R, and modified it as follows:
(2)Ṡt=1−pω−βSI−ωS,İt=βSI−σ+ωI,Ṙt=pω+σI−ωR.The model shows 100 percent efficiency of the vaccination. The natural death rate and the birth rate are denoted byσ and β; however, the rate of mortality of the childhood disease is very low. A fraction of the vaccinated population at birth is represented by the parameter p, where 0<p<1. Assume that the rest of the population is susceptible. A susceptible individual suffers from the disease through a contact with infected individuals at rate ω.In the last few decades, the research in applied mathematics particularly in modern calculus and fractional calculus has attracted scientists and researchers [7–10]. It should be kept in mind that in classical calculus, DEs have been extended to fill the research gap in many branches of pure and applied mathematics. Therefore, classical calculus is extended to modern calculus, and ODEs are extended to fractional-order DEs (FODEs). Fractional calculus has many applications in physical and natural sciences [11–18]. Different real-world problems have been modeled by FODEs and the system of FODEs [19–21]. Due to the degree of freedom, fractional calculus is used to solve nonlinear problems [22–28]. Various researchers and scientists discussed the existence and uniqueness of the solutions of different phenomena [12, 23, 29, 30] and applied different techniques to solve these problems [17, 20, 21, 31]. Haq et al. presented model (2) in a fractional order as
(3)DtγUt=1−pω−βUI−ωU,DtγIt=βUI−σ+ωI,DtγRt=pω+σI−ωR.It is very essential to find the solution of (3) with an efficient technique. Many researchers suggested different analytical and numerical methods for finding the solution of nonlinear problems. In this paper, we will use natural transform coupled with the Adomian decomposition method (NTADM) for finding an approximate solution of a nonlinear problem. This method can also be used for solving PDEs in which no perturbation or linearization is required. This method does not need any extra memory or any additional parameters.


## 2. Preliminaries

Definition 1 (see [32]).
LetΦ be a continuous function on L10,T,ℝ, a fractional integral of order κ∈0,1 in the Riemann-Liouville sense corresponding to t defined by
(4)IκΦt=1Γκ∫0tt−ζκ−1Φζdζ,where integral on right side is pointwise defined on 0,∞.Definition 2 (see [32]).
LetΦ be a continuous function on 0,T. The Caputo fractional derivative may be expressed as
(5)DβΦt=1Γn−β∫0tt−ζn−β−1dndζnΦtζdζ,where n=β+1 and β represents the integer part of β.Definition 3 (see [33]).
The natural transform (abbreviated as NT) of the functionΦt is presented as ℕΦt and with variables s and W is defined as
(6)ℕΦt=Rs,ϖ=∫−∞∞e−stΦWtdt,s,ϖ∈−∞,∞.
Now, the NT with the Heaviside functionHt is defined as
(7)ℕΦtHt=ℕ+Φt=R+s,ϖ=∫0∞e−stΦϖtdt,s,ϖ∈0,∞.Definition 4 (see [34]).
The NTRβs,W of the arbitrary derivative in Caputo’s sense of Φt is defined as
(8)ℕ+DβΦt=Rβs,ϖ=sβϖβRs,ϖ−∑k=0n−1sβ−k+1ϖβ−kDkΦtt=0.


## 3. Existence and Uniqueness

In this section, to address the existence and uniqueness of the proposed model, fixed-point theory is used. The existence of the model is guaranteed by Schauder’s fixed-point theorem, while the uniqueness is guaranteed by Banach’s contraction theorem.System (2) can be generalized by involving a fractional derivative in the Caputo sense for 0<γ≤1 as follows:
(9)DtγUt=1−pω−βUI−ωU,DtγIt=βUI−σ+ωI,DtγRt=pω+σI−ωR,with initial conditions U0=k1,I0=k2,R0=k3, where ω,p,σ,β are positive parameters. Assume that
(10)Δ1t,U,I,R=1−pω−βUI−ωU,Δ2t,U,I,R=βUI−σ+ωI,Δ3t,U,I,R=pω+σI−ωR.System (9) becomes
(11)DtγUt=Δ1t,U,I,R,DtγIt=Δ2t,U,I,R,DtγRt=Δ3t,U,I,R.Applying a fractional integral and using initial conditions, we have(12)Ut=k1+1Γγ∫0tt−ζκ−1Δ1t,U,I,Rdζ,It=k2+1Γγ∫0tt−ζκ−1Δ2t,U,I,Rdζ,Rt=k3+1Γγ∫0tt−ζκ−1Δ3t,U,I,Rdζ.Let(13)Xt=Ut,It,Rt,K=k1,k2,k3,Ht,Xt=Δ1t,U,I,R,Δ2t,U,I,R,Δ3t,U,I,R.So, system (12) becomes
(14)Xt=K+1Γγ∫0tt−ζκ−1Hζ,Xζdζ.Consider a Banach spaceΩ=C0,T×C0,T, with a norm
(15)U,I,R=maxt∈0,TU+I+R.Letd:Ω⟶Ω be a mapping defined as
(16)dXt=K+1Γγ∫0tt−ζκ−1Hζ,Xζdζ.Further, we impose some conditions on a nonlinear function as follows:(C1). There exist constantsCH>0 and QH>0 such that
(17)Ht,Xt≤CHXt+QH.(C2). There exists constantLH>0 such that for each X~t, X~1t∈Ω such that
(18)Ht,X~t−Ht,X~1t≤LHX~t−X~1t.Theorem 5.
Suppose that condition (C1) holds. Then, the system (11) has at least one solution.Proof.
To show that operatord is bounded, let Ψ=X∈Ψρ≥X, where
(19)ρ≥maxt∈0,TK+QHTγ/Γγ+11−CHTγ/Γγ+1is a closed and convex subset of Ω. Now, take
(20)dX=maxt∈0,TK+1Γγ∫0tt−ζγ−1Hζ,Xζdζ≤K+1Γγ∫0tt−ζγ−1Hζ,Xζdζ≤K+1Γγ∫0tt−ζγ−1CHXt+QHdζ≤K+CHX+QHTγΓγ+1≤ρ.
It means thatX∈Ψ⇒dΨ⊆Ψ, which shows that d is bounded. Next, to show that d is completely continuous, let t1<t2∈0,T and take
(21)dXt2−dXt1=1Γγ∫0t2t2−ζγ−1Hζ,Xζdζ−1Γγ∫0t1t1−ζγ−1Hζ,Xζdζ≤t2γ−t1γCHX+QHΓγ+1.
This shows thatdXt2−dXt1⟶0 as t2→t1. Hence, operator d is completely continuous by the Arzelá-Ascoli theorem. Thus, the given system (11) has at least one solution by Schauder’s fixed-point theorem.Next, the Banach fixed-point theorem was used to show that the system (11) has unique solution.Theorem 6.
Suppose that condition (C2) holds. Then, the system (11) has unique solution.Proof.
LetX~,X~1∈Ω. Take
(22)dX~−dX~1=maxt∈0,T1Γγ∫0tt−ζγ−1Hζ,X~ζdζ−1Γγ∫0tt−ζγ−1Hζ,X~1ζdζ≤TγΓγ+1LHX~−X~1≤TγΓγ+1LHX~−X~1.
Hence,d is the contraction. Using the Banach contraction theorem, the system (11) has unique solution.


## 4. Approximate Solution

In this part, the natural transform method is used to find the approximate solution of the system (9). By using NT and initial conditions, we get
(23)ℕ+DtγUt=ℕ+1−pω−βUI−ωU,ℕ+DtγIt=ℕ+βUI−σ+ωI,ℕ+DtγRt=ℕ+pω+σI−ωR,sγϖγℕ+Ut−∑k=0n−1sγ−k+1ϖγ−kDkUtt=0=ℕ+1−pω−βUI−ωU,sγϖγℕ+It−∑k=0n−1sγ−k+1ϖγ−kDkItt=0=ℕ+βUI−σ+ωI,sγϖγℕ+Rt−∑k=0n−1sγ−k+1ϖγ−kDkRtt=0=ℕ+pω+σI−ωR,ℕ+Ut=1sk1+ϖγsγℕ+1−pω−βUI−ωU,ℕ+It=1sk2+ϖγsγℕ+βUI−σ+ωI,ℕ+Rt=1sk3+ϖγsγℕ+pω+σI−ωR.Applying inverse NT, we get(24)Ut=k1+ℕ−1ϖγsγℕ+1−pω−βUI−ωU,It=k2+ℕ−1ϖγsγℕ+βUI−σ+ωI,Rt=k3+ℕ−1ϖγsγℕ+pω+σI−ωR.Consider the series solution of the system (9) to be
(25)Ut=∑n=0∞Unt,It=∑n=0∞Int,Rt=∑n=0∞Rnt,while the nonlinear term is represented as
(26)UtIt=∑n=0∞Ant,where each An is the Adomian polynomial and is defined as
(27)An=1Γn+1dndλn∑i=0nλiUit∑i=0nλiIitλ=0.The first three terms ofAn are given below:
(28)A0=U0tI0t,A1t=U0tI1t+I0tU1t,A2t=2U0tI2t+2U1tI1t+2U2tI0t.Therefore, equation (24) can be written as
(29)∑n=0∞Unt=k1+ℕ−1ϖγsγℕ+1−pω−β∑n=0∞Ant−ω∑n=0∞Unt,∑n=0∞Int=k2+ℕ−1ϖγsγℕ+β∑n=0∞Ant−σ+ω∑n=0∞Int,∑n=0∞Rnt=k3+ℕ−1ϖγsγℕ+pω+σ∑n=0∞Int−ω∑n=0∞Rnt.Thus, we establish the series solution as(30)Ut=U0t+U1t+⋯,It=I0t+I1t+⋯,Rt=R0t+R1t+⋯,where U0t=k1, I0t=k2, and R0t=k3.
(31)U1t=1−pω−βU0I0−ωU0tγΓγ+1,I1t=βU0A0−σ+ωI0tγΓγ+1,R1t=pω+σI0−ωR0tγΓγ+1.U2t=1−pωtγΓγ+1−βk1σ+ω−β2k1k2t2γΓ2γ+1−βk2+ω1−pω−βk1k2−ωk1t2γΓ2γ+1,I2t=βk1−σ−ωβk1k2−σ+ωk2t2γΓ2γ+1+βk2ω−pω−βk1k2−ωk1t2γΓ2γ+1,R2t=pωtγΓγ+1−σβk1k2−σ+ωk2t2γΓ2γ+1−ωpω−σk2−ωk3t2γΓ2γ+1.Similarly, we can calculate the other terms.


## 5. Numerical Results

Usingk1=1, k2=0.5, k3=0,p=0.9,ω=0.4, β=0.8, and σ=0.03, corresponding to these numerical values, we present the approximate solution (30) through MATLAB for ten terms in Figures 1–3.Figure 1
Graphical representation of the approximate solution of a susceptible class at three different fractionals.Figure 2
Graphical representation of the approximate solution of an infected class at three different fractionals.Figure 3
Graphical representation of the approximate solution of a recovered class at three different fractionals.In Figures1–3, we have presented the approximate solutions of different compartments against the given data and corresponding to different fractional orders. Further, the obtained results show that the solution is continuously dependent on the time-fractional derivative and values of the parameters. Since the population of the susceptible class decreases, however, that of the recovered class gradually increases with time due to vaccination. As shown in Figure 1, the susceptibility is decreasing for some initial time which results in infection decrease as in shown Figure 2. This is because of vaccination which reduces infection, and hence, the recovery from disease is increasing as shown in Figure 3. The increase and decrease of different compartments are different corresponding to the fractional-order derivative. The concerned process for some initial time is faster on a smaller fractional order, and then, the process reverses and becomes slow at the same smaller fractional order. Hence, the fractional calculus presents global dynamics of the decay and the growth process in the biological models. Here, it is interesting that the number of peaks and their respective amplitudes are similar between models; however, there are variations in the timing of these peaks. The transient oscillations of the considered model under fractional orders are more stretched out than those of its corresponding integer-order model whose solutions experience longer interepidemic times. Hence, we concluded that depending on the particular values of the parameters, the fractional-order model either converges to the classical model and fits data, similarly, or fits the data better and outperforms the classical model correspondingly.


## 6. Conclusion

In this paper, we show the existence and uniqueness of the fractional-order childhood disease model applying metric fixed-point theory. We also develop a proper procedure of natural transform, coupled with the Adomian decomposition method, and obtained an approximate solution of the proposed model. In the end, some numeric interpretation and their graphical presentation have been provided.



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