Abstract
In this paper, we propose a Caputo–Fabrizio fractional derivative mathematical model consisting of smoker, people exposed to secondhand smoker, people exposed to thirdhand smoker, and quitters. Secondhand smoke exposure consists of an unintentional inhalation of smoke that occurs close to people smoking and/or in indoor environments where tobacco was recently used, and thirdhand smoke consists of pollutants that remain on surfaces and in dust after tobacco has been smoked, are reemitted into the gas phase, or react with other compounds in the environment to form secondary pollutants. The solution of the proposed model, which is carried out using a fixed-point theorem and an iterative method, exists and is unique. Furthermore, the model is biologically meaningful, that is, positive and bounded. The reproduction number is determined from the model. If , the smoking-free equilibrium point is asymptotically stable, and if , the smoking-free equilibrium point is unstable. The results confirm that the smoking-free equilibrium point becomes increasingly stable as the fractional order is increased. Numerical simulations are performed using a three-step Adams-Moulton predictor-corrector method for a range of fractional orders to show the effects of varying the fractional order and to support the theoretical results.
1. Introduction
Smoking tobacco is epidemic and one of the biggest public health problems in the world nowadays [1] because rates of cigarette smoking are still too high [2]. It causes cancers and heart and lung diseases [3] and kills more than 8 million people a year [1]. The smoke emitted from the burning end of a cigarette where the smoke is exhaled by the smoker is also the cause of cancers and heart and lung disease and causes around million deaths. This exposure is referred to as secondhand smoke. It is an unintentional inhalation of smoke that occurs close to people smoking and/or in indoor environments where tobacco was recently used [1].
Smoking tobacco leaves chemical residue on surfaces including floors, carpets, furniture, and clothing where smoking has occurred. The chemicals live long after the smoke itself has been cleared from the environment. People are exposed in this phenomenon called thirdhand smoke. It is increasingly recognized as a potential danger, especially to children because they ingest residues that get on their hands after crawling on floors or touching walls and furniture [4]. A study which was conducted on mice showed that thirdhand smoke exposure has several behavioral and physical health impacts, including hyperactivity and adverse effects on the liver and lungs [5]. In this perspective, thirdhand smoke is a research agenda starting from 2011 [6]. So researches are conducted on thirdhand smoke and its health consequences [7, 8]. Due to this, we motivate to study a fractional model for the dynamics of smoking consisting people exposed to secondhand and thirdhand smoke as a compartment.
Different scholars have studied the mathematical model of smoking with integer-order derivatives [9–18] and non-integer-order derivatives [19–23]. In this article, we modify and extend the work in [9] by adding one compartment (people exposed to thirdhand smoke) and considering the new model with non-integer-order derivatives, specifically the Caputo–Fabrizio derivative.
The rest of the paper are arranged as follows. In Section 2, we discuss about the Caputo–Fabrizio fractional derivative and integral. In Section 3, we present the dynamics of the Caputo–Fabrizio fractional model of smoking. In Section 4, the existence and uniqueness of the solution of the proposed model are presented. Section 5 is devoted to the invariant region, positivity, and boundedness of the proposed model. Section 6 is devoted to equilibrium points and reproduction number. In Section 7, we present the stability of equilibrium points. Numerical methods and numerical results and discussions are included in Sections 8 and 9, respectively. Lastly, conclusions are given in Section 10.
2. Preliminary
Fractional calculus is as old as classical calculus and yet a novel topic. The origins can be traced back to the end of the seventeenth century, and it has been developed up to nowadays. It deals with the study of fractional-order integrals and derivatives and generalizes the ordinary integral and differential operators [24]. Fractional differential equations have been applied to formulate problems arising in engineering, physics, economics, and chemistry. From a modeling point of view, it has been better compared with integer-order derivatives [25, 26]. Though there are different definitions of fractional integral and derivatives of a function, in this paper, we use the Caputo–Fabrizio integral and derivative of a function. For , Caputo and Fabrizio in [27] introduced a new definition of fractional derivative with smooth kernel, that is, where is a normalization constant depending on Later, Losada and Nieto in [28] investigated a new definition of fractional integral corresponding to the Caputo–Fabrizio derivative as
By imposing an explicit formula for can be obtained, that is,
Due to this, Losada and Nieto in [28] proposed the following definitions.
Definition 1. Let . The fractional Caputo–Fabrizio integral of order of a function is defined by
Definition 2. Let . The fractional Caputo–Fabrizio derivative of order of a function is given by
Remark 3.
3. Dynamic System with the Caputo–Fabrizio Derivative
In this section, we describe the fractional smoking model described by the Caputo–Fabrizio derivative. The model will be introduced by adding the compartment people exposed to thirdhand smoker and by changing the integer-order derivative with the Caputo–Fabrizio derivative in [9]. In Table 1, we describe the variables and parameters to form the mathematical model that represents the dynamics of transmission of the habit of smoking.
We assume that an individual does not belong to compartment and at the same time. The diagram in Figure 1 describes the habit of smoking.
The new fractional smoking model using the Caputo–Fabrizio derivative can be written as with initial conditions
We note that the dimension of the left-hand side of (7)–(10) is people per unit time.
4. Existence and Uniqueness
In this section, we will show the existence and uniqueness of the system (7)–(10) with initial conditions (11). To prove it, we will use a fixed-point theory that is applied in [19, 29]. Applying Remark 3, equations (7)–(10), respectively, become
We denote by , by , by , and by .
Let
Then, (12) becomes
We now suppose for We have the following theorem.
Theorem 4. For , 2, 3, and , satisfy the Lipschitz condition in the second variable, that is, where In addition, is contraction if for
Proof. We consider only . The other can be done analogously. Let and be two functions. Then, by triangle inequality.
We now define the following recurrence formula: where
Let Then, , where
Lemma 5. Let be as defined in (17). We have
Proof. Since satisfies the Lipschitz condition with Lipschitz condition , we have It follows that Applying (22) recursively, we get We can easily show that where
For the purpose of the next theorem, we state and prove the following lemma.
Lemma 6. Let , , and be defined as in (25), (17), and (15), respectively. Then,
Proof. Applying (27) recursively, we get
Theorem 7. If , then Consequently, the solution of the system (7)–(10) exists.
Proof. Using Lemma 6, when . It follows that Hence, using (24),
Theorem 8. If , then the solution of the system (7)–(10) is unique.
Proof. Let and be solutions of the system (7)–(10). We notice that It follows that So if , then Hence,
5. Invariant Region, Positivity, and Boundedness
The dynamics of the Caputo–Fabrizio fractional model (7)–(10) is explored in a feasible region such that where and
Lemma 9. The region is positively invariant with nonnegative initial conditions for model (7)–(10) in
Proof. After adding the components of human population in model (7)–(10), we get Then, we have By applying Laplace transform and then its inverse, we obtain Thus, the solution of the model (7)–(10) with the nonnegative conditions in remains in . So, the region is positively invariant and attracts all the solutions in . Now, for the positivity of the system solution, let
Corollary 10 (see [30]). Suppose and , where . Then, (1)if , then is nondecreasing(2)if , then is nonincreasing
Theorem 11. If the initial population sizes of the model are positive, then the solution is positive and bounded at any time.
Proof. We observe that By Corollary 10, we have Since , the solution is bounded.
6. Equilibrium Points and Reproduction Number
Let us denote , , and The smoking-free equilibrium point is , where
We need to distinguish new infections from all other changes in population to compute reproduction number [31]. Let be the rate of appearance of new infections in each compartment and be the rate of transfer of individuals into and out of each compartment by all other means. It is assumed that each function is continuously differentiable at least twice in each variable. The smoking transmission model (7)–(10) can be written as where ,
The Jacobian matrices of and at the smoking-free equilibrium point are, respectively,
We see that
The reproduction number of a fractional dynamical system can be found using the next-generation matrix method [29, 32]. So, the reproduction number of the model (7)–(10) can be computed using the next-generation matrix method and is given by the spectral radius of [33], that is,
If , then
Substituting in (43), we get
Let Then, the following holds true. (1) is continuous on (2)(3)
Consequently, if then has a unique zero in Let be the unique solution of (45). The endemic equilibrium point is , where
7. Stability
Consider the following fractional-order linear system described by the Caputo–Fabrizio derivative: where , and
Definition 12 (see [34]). The characteristic equation of system (48) is
Theorem 13 (see [34]). If is invertible, then system (48) is asymptotically stable if and only if the real parts of the roots to the characteristic equation of system (48) are negative.
We next state and prove the asymptotic stability of smoking-free equilibrium point of the dynamic system (7)–(10).
Theorem 14. The smoking-free equilibrium point of the system (7)–(10) with is asymptotically stable if and only if real parts of the roots of the characteristic equation are negative.
Proof. The characteristic equation of the linearized system (7)–(10) at smoking-free equilibrium point is where is the Jacobian matrix at , that is, The roots of the characteristic equation (50) are
8. Numerical Methods
In this section, we will use the three step Adams-Moulton predictor-corrector methods to determine the unknowns since it is superior to the three-step Adams-Bashforth predictor-corrector method that was applied in [29]. The truncation error of Adams-Bashforth methods is of , and we will show in what follows the truncation error of Adams-Moulton methods is of . Let , where and , be the discretization of the interval We will define the recursive formula as follows.
We approximate by a Lagrange polynomial of degree 3 where
Let Then,
Substituting (56) into (53), we get
Equation (57) is a type of three-step Adams-Moulton method. It is an implicit method because its right-hand side contains So the left-hand side of (57) can be calculated using Adams-Bashforth methods
We now write the methods as follows: where and
In (60), and can be computed using the method
The truncation error for the three-step Adams-Moulton methods can be estimated by using the error estimate for the Lagrange interpolating polynomial, namely, where
We notice that
We denote the right-hand side of (59) by . The total truncation error of the use of formula (59) is
9. Numerical Results and Discussions
For the purpose of numerical simulations, we utilize the values of the initial conditions and parameters Thus, we have
The solutions of model (7)–(10) are computed using the corrector three-step Adams-Moulton methods (59) and predictor three-step Adams-Bashforth methods (60). We use MATLAB software to plot Figures 2 and 3. As we notice in Figure 2, the number of quitters and smokers goes to zero when Hence, the smoking-free equilibrium point is asymptotically stable for , and and eventually disappear from the system.
Remark 15. represents the standard derivative.
In Figure 3, and are plotted for different values of . We can see that as increases, and increase and converge to smoking-free equilibrium point and , respectively, whereas quitters and smokers decrease and converge to smoking-free equilibrium point and , respectively.
10. Conclusions
In this work, we investigated a Caputo–Fabrizio fractional derivative smoking model containing smoker, people exposed to secondhand smoker, people exposed to thirdhand smoker, and quitters. The existence and uniqueness of solution of the proposed model have been shown with the help of the fixed-point theorem. Moreover, the positivity and boundedness of the model were discussed. We also calculated the reproduction number of the model and showed that the smoking-free equilibrium point is asymptotically stable when . Furthermore, the three-step Adams-Moulton method was applied to compute the numerical solutions. The solution depends on the fractional order and as the solutions close to the smoking-free equilibrium point. So we recommend the Caputo–Fabrizio derivative to study a model that represents a real-world problem.
Data Availability
Data sharing is not applicable for this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors contributed equally in preparing and writing this manuscript. They read and approved the final manuscript.