Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 4057089, 7 pages

https://doi.org/10.1155/2017/4057089

## Numerical Analysis of Fractional Order Epidemic Model of Childhood Diseases

^{1}Department of Mathematics, Hazara University, Mansehra, Khyber Pakhtunkhwa, Pakistan^{2}Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan^{3}Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan

Correspondence should be addressed to Fazal Haq

Received 26 July 2017; Revised 4 October 2017; Accepted 5 November 2017; Published 3 December 2017

Academic Editor: Stefania Tomasiello

Copyright © 2017 Fazal Haq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The fractional order Susceptible-Infected-Recovered (SIR) epidemic model of childhood disease is considered. Laplace–Adomian Decomposition Method is used to compute an approximate solution of the system of nonlinear fractional differential equations. We obtain the solutions of fractional differential equations in the form of infinite series. The series solution of the proposed model converges rapidly to its exact value. The obtained results are compared with the classical case.

#### 1. Introduction

Childhood diseases are most serious infectious diseases. Measles, poliomyelitis, and rubella are famous among them [1, 2]. Measles is a highly infectious disease, caused by respiratory infection by a* Morbillivirus*. These diseases normally affect the children, because child population is more prone to the disease as compared to the adults [1]. Therefore, the population can be divided into two major classes: premature and mature populations. Premature population takes a constant time to become mature, which is known as maturation delay. In disease dynamics, a disease cannot spread instantaneously but rather it will take some time in the body which is called latent period for the particular disease. For the control of childhood disease vaccination is a significant strategy being used all over the world. A universal effort to extend vaccination range to all children began in 1974, when the World Health Organization (WHO) founded the Expanded Program on Immunization [3]. Mathematical model plays an important role to comprehend the process of transmission of a disease and provides different techniques to control its propagation. Many mathematicians investigated childhood disease; for instance, Singh et al. [2] studied about vaccination of the childhood diseases. In addition, the authors presented numerical solution of the childhood disease model. Makinde [4] presented a Susceptible-Infected-Recovered modelThe authors [1] rearranged model (1) using the relations , , and and obtained a new (SIR) model in the following form:The description of the preceding model is given below.

The model shows that vaccination is 100 percent efficient and the natural death rate is unequal. Therefore the total population size is not constant. The birth rate is represented by while the rate of mortality of the childhood disease is very low. The parameter, , represents a fraction of vaccinated population at birth, where , considering that the rest of population is susceptible. A susceptible individual suffers from the disease through a contact with infected individuals at rate . Infected individuals recover at a rate . There are many applications of fractional calculus [5, 6]. Mathematicians and researchers used fractional calculus to model real world problems. Fractional calculus has greater degree of freedom; therefore it helps to solve nonlinear problems [7, 8]. Also with the help of fractional derivative interdisciplinary applications can be studied. The nonlinear oscillation of earthquake can be modeled with the help of fractional derivative.

In 1980, Laplace–Adomian Decomposition Method (LADM) was introduced by Adomian, which is an effective method for finding the numerical and explicit solution of a system of differential equations representing physical problems. This method works efficiently for solving several kinds of differential equations. It includes nonlinear boundary value/initial value problems. Moreover, we can use it for the partial differential equations; in addition, it can be used to solve a system of stochastic differential equations also. In this method, no perturbation or linearization is required. The advantage of LADM is that it needs no extra memory and does not require any additional parameters, which wastes time and memory.

Studying the literature review of childhood disease model it has been found that Arafa et al. [1] presented the fractional order model for the childhood disease and provided numerical solution by using homotopy analysis method. The reason behind the use of fractional order differential equations (FDEs) is that FDEs are naturally related to systems that involve memory, which can be found in many biological systems. Also, they show the realistic behavior of infection of disease but at a slower rate. Motivated by the applications of fractional calculus and LADM we explore numerical solution of childhood disease model. The Caputo derivative, which is a modification of the Riemann-Liouville definition [9, 10], is considered as a differential operator in our model.

We now gather some well-known definitions and results from the literature, which we will use throughout this paper. For more details, we refer the reader to [1, 7–12].

*Definition 1. *The Caputo fractional order derivative of a function on the interval is defined bywhere and represents the integer part of .

*Definition 2. *We recall the definition of Laplace transform of Caputo derivative as

#### 2. The Laplace–Adomian Decomposition Method

Using the Caputo fractional derivative system (2) gets the following form:where while , , , and are positive parameters and the given initial conditions are , , and . To solve system (5), we use Laplace–Adomian Decomposition Method (LADM) [5, 13]. Moreover, the obtained solution will be compared with the integer order derivative case. Furthermore, we use Laplace transformation to convert the system of differential equations into a system of algebraic equations. Then, the algebraic equations are used to obtain the required solution in form of series. We will discuss the procedure for solving model (5) with given initial conditions. Applying Laplace transform on both sides of model (5), we obtain the following system:orUsing the initial conditions (7), we obtain the formAssume that the solutions , , and in the form of infinite series are given bywhile the nonlinear term is decomposed as follows:where each is the Adomian polynomials defined asThe first three polynomials are given by

Substituting (9) and (10) into (8) results in

Matching the two sides of (13) yields the following iterative algorithm:Taking Laplace inverse of (14) and considering first three terms, we get

#### 3. Numerical Results and Discussion

Using , , , , , , , and , the LADM provides us with an approximate solution in the form of infinite series. Thus we calculate the first four terms of (5) to obtainFor , (16) attains the formSimilarly, we get the following system for :Now, for , one can obtain the following system:Similarly, the solution after three terms for is calculated as follows:

From the graphical results it is clear that the result obtained by using LADM is very efficient. It also shows that the presented method can predict the behavior of the variables accurately for the region under consideration. It is also clear that the efficiency of this method can be dramatically increased by increasing the terms. Fractional order derivative provides a greater degree of freedom as compared to integer order derivative. The dynamics of various compartments have been shown in Figure 1, Figure 2, and Figure 3, respectively. In addition, we give a comparison of and LADM in Tables 1 and 2 for , which shows that both the methods agree for short interval of time. The proposed method is better than method as it needs no extra predefined parameter which controls the method.