Abstract
This manuscript is devoted to investigate qualitative theory of existence and uniqueness of the solution to a dynamical system of an infectious disease known as measles. For the respective theory, we utilize fixed point theory to construct sufficient conditions for existence and uniqueness of the solution. Some results corresponding to Hyers–Ulam stability are also investigated. Furthermore, some semianalytical results are computed for the considered system by using integral transform due to the Laplace and decomposition technique of Adomian. The obtained results are presented by graphs also.
1. Introduction
In recent time, the subject of fractional calculus has got much attention from the researcher. This is due to large numbers of applications in various disciplines of science and engineering where the concept of derivatives and integrals is frequently used. Normally, the integer-order derivative does not explore the dynamics of real-world problems more comprehensively as compared to the fractional derivative. Also, the fractional differential operator is global and poses greater degree of freedom, while the ordinary differential operator is local and often cannot explain the memory and hereditary process of the real-world problem more efficiently. In fact, fractional derivative is definite integral over a domain; therefore, it has been defined in many ways. In this regard, a variety of definitions have been introduced by different researchers in the literature. Some famous definitions are those given by Riemann and Liouville, Caputo, Hadamard, and so on (see [1]). The definitions of Riemann–Liouville and Caputo have been very well used in applied problems. These definitions involve singular kernel which often causes difficulty in dealing some problems. Therefore, Caputo and Fabrizio in 2015 introduced a new concept about fractional-order derivatives based on nonsingular kernel (see [2–4]). Some remarkable merits of Caputo–Fabrizio fractional derivative and integral and their applications were given by many researchers in previous few years (see, for detail, [5–13]). The concerned fractional integral of a function is the fractional average of the function itself and its fractional integral in Riemann–Liouville sense. Moreover, in some articles, it has been displayed that the derivative has some constructive applications in thermal science, material sciences, and so on (see, for detail, [11, 12, 14, 15]). Since differential operators have greater degree of freedom, therefore, to find the exact solution to each and every problem is quite difficult. In this regard, great motivation has been observed in the last two decades to establish best tools to handle such problems. One of the important techniques is to find analytical approximate solutions to many nonlinear problems of FODEs. For this purpose, the usual decompositions, perturbation, and integral transform methods were greatly utilized to investigate ordinary differential and integral equations. The mentioned techniques have been very well explored for fractional differential equations (see [16–25]). One of the powerful methods which has been used very frequently in the past is due to the Laplace Adomian decomposition method. For usual FODEs, the mentioned method has been used very regularly in the literature. However, to the best of our information, the aforesaid method is very rarely used for FODEs involving nonsingular kernel, see [26, 27].
Here, we remark that mathematical models are the powerful tools to study various dynamical problems of physical and biological sciences. The concept was initiated by Bernoulli in 1776. However, a formal mathematical model of three compartments was constructed in 1927 by McKendrick and his co-author called the SIR mode. Later on, the subject of mathematical modeling was extended to infectious diseases. Because mathematical models of the biological problem have become powerful tools to understand various infectious diseases, the proper method has to be developed to control the disease or minimize its transmission in the society. From ancient time, measles disease is one of the most threatful diseases. It was a big reason for children death in the past. This dangerous disease was caused by germs called morbilliform. Measles disease spreeds, when an infected person coughs or sneezes, because its virus can live for up to two hours in an airspace where the infected person coughed or sneezed. Measles-infected individuals can transfer their germs to the other people 4–8 days before and after the skin eruptions start. It transmits a disease in young babies up to 30–40 million every year. Measles appears once and is present for a long time in life for immunity. Symptoms of the disease include runny nose, high temperature, coughing, and spots on the whole body and in highly complicated cases, ear infections, diarrhea, and pneumonia. The vaccination of measles has been used to control disease in kids. Some vaccinated individuals could remain susceptible individuals when vaccination gets failed. Worldwide vaccination reduced death caused by measles between 2000 and 2017. However, measles disease is still familiar disease in highly developing countries of Asia and Africa particularly due to the lack of proper treatment of this infectious disease. For this purpose, a massive research has been carried out to enhance the understanding of the virus of measles dynamics in various areas. For example, the authors in [28–33] discussed the global stability of the model with five compartments aswhere represents the susceptible, represents the vaccinated, represents the exposed, represents the infected, and represents the recovered individuals. The description of the parameters is given in the analytical section. These models have been investigated corresponding to ordinary and usual fractional-order derivatives. Furthermore, the researchers have given the global and local dynamics by computing the basic reproductive numbers. Here, we investigate the given model under the nonsingular fractional derivative of Caputo and Fabrizio (CFFD) from other perspectives including the qualitative analysis by using fixed point approach. Further stability is a required aspect in the dynamical problem. Since we are going to derive approximate solutions, therefore, in this regard, Hyers–Ulam-type stability results are investigated. The mentioned stability has been very well studied for the general problem of FODEs, see, for detail, [5, 34–36]. Also, the analytical results are investigated through the Laplace Adomian decomposition method. We considered model (1) under the CFFD with fractional order asunder the initial conditions
We obtain the solution in the form of series for the considered problem. Also, we display the results against different values of fractional order . Also, we provide results about the existence and uniqueness of the solution for the concerned model by using fixed point theorems due to Schauder and Banach. Here, we remark that we use the Laplace Adomian decomposition method because this method is easy and efficient and less expensive. Furthermore, the mentioned method does not require any predefine step size or controlling parameter which are needed by RK methods or the homotopy method, respectively, for detail, see [37–39]. Furthermore, the convergence of the method has been showed in many papers of the proposed method, for instance, see [27].
2. Preliminaries
Definition 1. (see [3]). Let , , and ; then, the CFFD is recalled as is the normalization function defined as and satisfies the conditions . If the function fails to exist in , then the derivative can be redefined as
Definition 2. (see [3]). The integral of fractional order of a function is defined by
Lemma 1. (see [40]). The solution of the differential equation involving CFFD for asis given by
Definition 3. (see [34, 35]). The solution of the differential equationis Hyers–Ulam stable; if there exists such that, for the inequalitythere exists a unique solution with a constant such thatFurthermore, if there exists nondecreasing function such that (11) may be written asthen the concerned solution of problem (9) is generalized Hyers–Ulam stable.
Theorem 1 (see [40]). Let be a convex subset of Banach space , with operators and with(1) for all (2) is the condensing operator(3) is continues and compactThen, there exists at least one solution such that
Definition 4. (see [4, 14]). The Laplace transform of CFFD is given as
3. Existence and Stability Results for the Considered Model
In this part of the manuscript, we determine existence results for model (2) using the fixed point theorem due to Banach. In this regard, we first define the following functions:
Then, we write some notions for easiness asand
Using (16), system (2) can be written as
In view of Lemma 1, problem (18) can be converted to the given integral equation as
Now, let , and let ; we define the Banach space under the norm
The assumptions given in the following hold true: Under the continuity of , there exists such that There exists positive constant such that .
Theorem 2. Under the assumptions , problem (19) has at least one solution if ; consequently, the considered system (2) has at least one solution.
Proof. Let be a closed convex set. Now, we define the operators from (19) asTo derive the required results, we first prove that operator is a contraction.
For any , one hasThis shows that is a contraction. Now, to show that is bounded and equicontinuous, the continuity of implies that is continuous. For any , we haveThis shows that is bounded; for equicontinuity, let , and we havewhich implies that as . So, is uniformly continuous and bounded. Thus, by Arzelá–Ascoli theorem, is relatively compact and so is completely continuous. Thus, by Theorem 1, problem (18) has at least one solution. Consequently, the considered model (2) has at least one solution.
Next, we define the operator by
Theorem 3. Under assumption , the operator as defined in (26) is a contraction; then, problem (18) has a unique solution with the condition , and consequently, our proposed system (2) has a unique solution.
Proof. Let ; then, from (26), one hasThis shows that is a contraction. Therefore, problem (18) has a unique solution. Hence, our considered system (2) has a unique solution.
Remark 1. Next, for stability analysis, we consider a small perturbation , such that , depends only on the solution.(i) for (ii)
Lemma 2. The solution of the perturbed problemsatisfies the given relation:where .
Proof. In view of Lemma 1, the solution of perturb problem (28) is given byFrom (30), on using Remark 1, we have
Theorem 4. Under assumption , together with Lemma 2, problem (18) is Hyers–Ulam stable if , which yields that our considered system (2) is Hyers–Ulam stable.
Proof. Let be any solution, and is a unique solution; then,This implies thatTherefore, the solution of (18) is Hyers–Ulam stable. Hence, the solution of the proposed system (2) is Hyers–Ulam stable.
Remark 2. In the same line, we can also develop the results of generalized Hyers–Ulam, Rassias–Hyers–Ulam stability. The aforementioned stability analysis has been studied for simple mathematical models of biology and physics in [41–43].
4. Derivation of the General Semianalytical Solution to the Considered Model (2)
Here, in this section, we are going to compute series solution for the suggested problem. To receive this goal, taking Laplace transform of (16), we have
Now, assuming the solution in the series form,
Further decomposing the nonlinear terms , etc. in terms of Adomian polynomials,where the Adomian polynomial can be defined as
In the same way, the other polynomials can be defined.
Hence, in view of (35) and (36), system (34) becomes
Now, equating terms on both sides of (38), we have
Evaluating the Laplace transform in (39), we getand so on. Therefore, we get the required solution as given by
Note. For the convergence of the proposed method, see the paper [27].
5. Results and Discussion
This part of the manuscript is related to provide numerical results and some discussion about the approximate solution of the considered problem. For this purpose, we apply the Laplace Adomian decomposition methods for the solution. We select appropriate values for parameters, as we take [3] , , , , , , , , , , , and . In view of these values, we draw the graph of approximate solutions (40) for first ten terms by using Matlab in Figures 1–5 against various fractional-order derivatives.
From Figures 1–5, we see that the susceptible class population is decreasing with different rates. It decreases with faster speed when the order is smaller as compared with the larger order. Also, the dynamics of the vaccinated class is increasing with different rates due to the fractional order. Also, the exposed population is increasing up, and hence, on using the vaccine, the density of the population of the infected class is decreasing, while the density of the recovered class increases with the same scenario. From all these figures, we concluded that the fractional derivative with exponential kernel can also be used to provide the global dynamics of the considered model.
6. Conclusion
By fixed point approach, we have investigated the existence of the considered model under CFFD. Also, by using the tools of nonlinear functional analysis, we have established sufficient conditions for Hyers–Ulam-type stability of the approximate solutions of the considered model. Also, we have provided the semianalytical solution to the considered model by the Laplace Adomian decomposition method. The concerned method needs no discretization of data nor extra axillary parameter as needed by homotopy methods on which these methods depend. The proposed method has been utilized extensively for usual fractional differential equations, but in the case of new fractional differential operators, this method has not been properly used. Hence, we concluded that CFFD can also be used as powerful tools to investigate biological models of infectious diseases. Also, the concerned models involving CFFD can be handled easily by using the Laplace Adomian decomposition method.
Data Availability
No data were used to support this study.
Conflicts of Interest
There exist no conflicts of interest regarding this research work.
Authors’ Contributions
All authors have equal contribution to this work.
Acknowledgments
The authors are thankful to the HED for supporting this work under grant no. HEREF-46 and HEC of Pakistan under grant no. NRPU-10039. This work was supported financially by the National Natural Science Foundation of China (11971493).