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.
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:
Arafa et al. rearranged the model (1) using the following relations, , and modified it as follows:
The model shows 100 percent efficiency of the vaccination. The natural death rate and the birth rate are denoted by and ; 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 , where . Assume that the rest of the population is susceptible. A susceptible individual suffers from the disease through a contact with infected individuals at rate .
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
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]). Let be a continuous function on , a fractional integral of order in the Riemann-Liouville sense corresponding to defined by where integral on right side is pointwise defined on .
Definition 2 (see [32]). Let be a continuous function on . The Caputo fractional derivative may be expressed as where and represents the integer part of .
Definition 3 (see [33]). The natural transform (abbreviated as NT) of the function is presented as and with variables and is defined as Now, the NT with the Heaviside function is defined as
Definition 4 (see [34]). The NT of the arbitrary derivative in Caputo’s sense of is defined as
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 as follows: with initial conditions , where are positive parameters. Assume that
System (9) becomes
Applying a fractional integral and using initial conditions, we have
Let
So, system (12) becomes
Consider a Banach space , with a norm
Let be a mapping defined as
Further, we impose some conditions on a nonlinear function as follows:
(C1). There exist constants and such that
(C2). There exists constant such that for each , such that
Theorem 5. Suppose that condition (C1) holds. Then, the system (11) has at least one solution.
Proof. To show that operator is bounded, let , where is a closed and convex subset of . Now, take It means that , which shows that is bounded. Next, to show that is completely continuous, let and take This shows that as . Hence, operator 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. Let . Take Hence, 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
Applying inverse NT, we get
Consider the series solution of the system (9) to be while the nonlinear term is represented as where each is the Adomian polynomial and is defined as
The first three terms of are given below:
Therefore, equation (24) can be written as
Thus, we establish the series solution as where , , and .
Similarly, we can calculate the other terms.
5. Numerical Results
Using , , , , and , corresponding to these numerical values, we present the approximate solution (30) through MATLAB for ten terms in Figures 1–3.
In Figures 1–3, 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
There does not exist any conflict of interest regarding this paper.
Authors’ Contributions
All authors have equal contribution in this research paper; particularly, the second author has significant contributions to revising the manuscript. He also polished the paper and its language style.
Acknowledgments
The second author would like to thank Prince Sultan University for funding this work through the research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (Group Number RG-DES-2017-01-17).