# A Fractional Model for the Dynamics of Smoking Tobacco Using Caputo–Fabrizio Derivative

Authors: Belaynesh Melkamu; Benyam Mebrate
Journal: Journal of Applied Mathematics
 (2022)

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## 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 numberR0 is determined from the model. If R0<1, the smoking-free equilibrium point is asymptotically stable, and if R0>1, 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.


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## Body

## 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 1.2 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 Section2, 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 0<δ<1, Caputo and Fabrizio in [27] introduced a new definition of fractional derivative with smooth kernel, that is,
(1)CFD0δft=2−δMδ21−δ∫0tf′ξe−δy−ξ1−δdξ,t≥0,where Mδ is a normalization constant depending on δ. Later, Losada and Nieto in [28] investigated a new definition of fractional integral corresponding to the Caputo–Fabrizio derivative as
(2)CFI0δft=21−δ2−δMδft+2δ2−δMδ∫0tfξdξ,t≥0.By imposing(3)21−δ2−δMδ+2δ2−δMδ=1,an explicit formula for Mδ can be obtained, that is,
(4)Mδ=22−δ.Due to this, Losada and Nieto in [28] proposed the following definitions.Definition 1.
Let0<δ<1. The fractional Caputo–Fabrizio integral of order δ of a function f is defined by
(5)CFI0δft=1−δft+δ∫0tfξdξ,t≥0.Definition 2.
Let0<δ<1. The fractional Caputo–Fabrizio derivative of order δ of a function f is given by
(6)CFD0δft=11−δ∫0tf′ξe−δy−ξ1−δdξ,t≥0.Remark 3.
CFI0δCFD0δft=ft−f0.


## 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.Table 1
Description of variables and parameters.
Variables/parametersDescriptionValueSDensity of people exposed to secondhand smokerS≥0TDensity of people exposed to thirdhand smokerT≥0ISmokerI≥0QQuitterQ≥0π1Rate of healthy people toSπ1>0π2Rate of healthy people toTπ2>0αNatural death rate0<α<1β1The exit rate ofS to healthy people0≤β1≤1β2The exit rate ofT to healthy people0≤β2≤1β3The exit rate ofQ to healthy people0≤β3≤1γ3Death rate as a result of member ofQ0≤γ3≤1γ4Death rate as a result ofI0≤γ4≤1δ1Infection rate fromS to I0<δ1≤1δ2Infection rate fromT to I0<δ2≤1ε1Exit rate fromI to Q0≤ε1≤1ε2Infection rate fromQ to I0≤ε2≤1We assume that an individual does not belong to compartmentS and T at the same time. The diagram in Figure 1 describes the habit of smoking.Figure 1
Smoking diagram.The new fractional smoking model using the Caputo–Fabrizio derivative can be written as(7)CFD0δSt=π1−δ1I+α+β1S,(8)CFD0δTt=π2−δ2I+α+β2T,(9)CFD0δQt=ε1I−ε2I+α+β3+γ3Q,(10)CFD0δIt=δ1S+δ2T+ε2Q−ε1−α−γ4I,with initial conditions
(11)S0=S0,T0=T0,Q0=Q0,I0=I0.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
(12)St=S0+CFI0δπ1−δ1I+α+β1S,Tt=T0+CFI0δπ2−δ2I+α+β2T,Qt=Q0+CFI0δε1I−ε2I+α+β3+γ3Q,It=I0+CFI0δδ1S+δ2T+ε2Q−ε1−α−γ4I.We denoteS by H1, T by H2, Q by H3, and I by H4.Let(13)K1t,H1=π1−δ1H4+α+β1H1,K2t,H2=π2−δ2H4+α+β2H2,K3t,H3=ε1H4−ε2H4+α+β3+γ3H3,K4t,H4=δ1H1+δ2H2+ε2H3−ε1−α+γ4H4.Then, (12) becomes
(14)Hit=Hi0+1−δKit,Hi+δ∫0tKiy,Hidy,i=1,2,3,4.We now suppose(15)Hit≤θi,for i=1,2,3,4. We have the following theorem.Theorem 4.
Fori=1, 2, 3, and 4, Ki satisfy the Lipschitz condition in the second variable, that is,
(16)Kit,H2i−Kit,H1i≤MiH2i−H1i,where
(17)M1=α+β1+η1+δ1θ4,M2=δ1θ1+ε2θ3+ε1+α+β4,M3=α+β3+η3+ε2θ4,M4=α+β2+η2+δ2θ1.
In addition,Ki is contraction if 0≤Mi<1 for i=1,2,3,4.Proof.
We consider onlyK1. The other can be done analogously. Let H11 and H21 be two functions. Then,
(18)K1t,H21−K1t,H11=δ1H4+α+β1H21−δ1H4+α+β1H11≤δ1H4+α+β1+γ1H21−H11,by triangle inequality.We now define the following recurrence formula:(19)Hnit=1−δKit,Hn−1i+δ∫0tKiy,Hn−1idy,where H0it=Hi0,i=1,2,3,4.Letϕnit=Hnit−Hn−1it. Then, Hnit=∑j=0nϕjit, where ϕ0it=Hi0.Lemma 5.
LetMi be as defined in (17). We have
(20)ϕnit≤Hi01−δ+δtnMin.Proof.
SinceKi satisfies the Lipschitz condition with Lipschitz condition Mi, we have
(21)Hni−Hn−1i=1−δKit,Hn−1i+δ∫0tKiy,Hn−1idy−1−δKit,Hn−2i+δ∫0tKiy,Hn−2idy≤1+δ+δtMiHn−1i−Hn−1i.
It follows that(22)ϕnt≤1+δ+δtMiϕn−1t.
Applying (22) recursively, we get
(23)ϕnt≤Hi01+δ+δtnMin.
We can easily show that(24)Hit−Hi0=Hnit+Bnit,where
(25)Bnit=1−δKit,Hi−Kit,Hn−1i+δ∫0tKiy,Hi−Kiy,Hn−1idy.For the purpose of the next theorem, we state and prove the following lemma.Lemma 6.
LetBint, Mi, and θi be defined as in (25), (17), and (15), respectively. Then,
(26)Bnit≤1−δ+δtn+1Min+1θi.Proof.
(27)Bnit=1−δKit,Hi−Kit,Hn−1i+δ∫0tKiy,Hi−Kiy,Hn−1idy≤1−δMiHi−Hn−1i+δMi∫0tHi−Hn−1idy≤1−δMiHi−Hn−1i+δMitHi−Hn−1i.
Applying (27) recursively, we get
(28)Bnit≤1−δ+δtn+1Min+1θi.Theorem 7.
If1−δ+δtMi<1, then limn⟶∞Bnit=0. Consequently, the solution of the system (7)–(10) exists.Proof.
Using Lemma6, limn⟶∞Bnit=0 when 1−δ+δtMi<1. It follows that limn⟶∞Bn=0. Hence, using (24), limn⟶∞Hnit=Hit−Hi0.Theorem 8.
If1−δ+δtMi≤1, then the solution of the system (7)–(10) is unique.Proof.
LetHit and Rit be solutions of the system (7)–(10). We notice that
(29)Hit−Rit=1−δKit,Hi−Kit,Ri+δ∫0tKiy,Hi−Kiy,Ridy≤1−δM1Hit−Rit+δMi∫0tHit−Ritdy≤1−δ+δtMiHit−Rit.
It follows thatHit−Rit1−Mi1−δ+δt≤0. So if 1−δ+δtMi≤1, then Hit−Rit=0. Hence, Hit=Rit.


## 5. Invariant Region, Positivity, and Boundedness

The dynamics of the Caputo–Fabrizio fractional model (7)–(10) is explored in a feasible region Ω⊂ℝ+4 such that
(30)Ω=xt∈ℝ+4:Nt≤11+ν1−δ1−δπ1+π2+N0,where xt=St,Tt,Qt,It and Nt=St+Tt+Qt+It.Lemma 9.
The regionΩ⊂ℝ+4 is positively invariant with nonnegative initial conditions for model (7)–(10) in ℝ+4.Proof.
After adding the components of human population in model (7)–(10), we get
(31)CFD0δNt=CFD0δSt+CFD0δTt+CFD0δQt+CFD0δIt.
Then, we have(32)CFD0δNt+νNt≤π1+π2.
By applying Laplace transform and then its inverse, we obtain(33)Nt≤11+ν1−δ1−δπ1+π2+N0.
Thus, the solution of the model (7)–(10) with the nonnegative conditions in Ω remains in Ω. So, the region Ω is positively invariant and attracts all the solutions in ℝ+4. Now, for the positivity of the system solution, let
(34)ℝ+4=xt∈∣xt≥0,xt=St,Tt,Qt,Itt.Corollary 10 (see [30]).
Supposegt∈Ca,b and CFD0δgt∈Ca,b, where δ∈0,1. Then,
(1)
ifCFD0δgt≥0,∀y∈a,b, then gt is nondecreasing(2)
ifCFD0δgt≤0,∀y∈a,b, then gt is nonincreasingTheorem 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(35)CFD0δSts=0=π1>0,CFD0δTtT=0=π2>0,CFD0δQtQ=0=ε1I≥0,CFD0δItI=0=0.
By Corollary10, we have
(36)St>0,Tt>0,Qt>0,It>0,fort>0.
SinceNt≤1/1+ν1−δ1−δπ1+π2+N0, the solution is bounded.


## 6. Equilibrium Points and Reproduction Number

Let us denoteA1=α+β1,A2=α+β2,A3=α+β3+γ3, A4=α+γ4, and B=ε1+A4. The smoking-free equilibrium point is M0=S0,T0,Q0,I0, where
(37)S0=π1α+β1,T0=π2α+β2,Q0=0,I0=0.We need to distinguish new infections from all other changes in population to compute reproduction number [31]. Let F be the rate of appearance of new infections in each compartment and V 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
(38)CFD0δxt=F−V,where xt=StTtQtTtt,
(39)F=000δ1IS+δ2IT+ε2IQ,V=δ1I+A1S−π1δ2I+A2T−π2ε2I+A3Q−ε1IBI.The Jacobian matrices ofF and V at the smoking-free equilibrium point are, respectively,
(40)DF=000000000000000δ1π1A1+δ2π2A2,DV=A100δ1π1A10A20δ2π2A200A3−ε1000B.We see that(41)DFDV−1=1A1A2A3B000000000000000A1A2A3δ1π1A1+δ2π2A2.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 DFDV−1 [33], that is,
(42)R0=ρDFDV−1=1Bδ1π1A1+δ2π2A2.IfI≠0, then
(43)δ1S+δ2T+ε2Q=ε1+α+γ4.Substituting(44)S=π1δ1I+A1,T=π2δ2I+A2,Q=ε1Iε2I+A3,in (43), we get
(45)δ1π1δ1I+A1+δ2π2δ2I+A2+ε1ε2Iε2I+A3=B.LetGx=δ1π1/δ1x+A1+δ2π2/δ2x+A2+ε1ε2x/ε2x+A3−B. Then, the following holds true.
(1)
G is continuous on 0.∞(2)
limx⟶0+Gx>0ifR0>1(3)
limx⟶∞Gx<0Consequently, if(46)ε1ε2A3ε2x+A32<δ12π1δ1x+A12+δ22π2δ2x+A22,then Gx has a unique zero in 0,∞. Let I∗ be the unique solution of (45). The endemic equilibrium point is M∗=S∗,T∗,Q∗,I∗, where
(47)S∗=π1δ1I∗+A1,T∗=π2δ2I∗+A2,Q∗=ε1I∗ε2I∗+A3.


## 7. Stability

Consider the following fractional-order linear system described by the Caputo–Fabrizio derivative:(48)CFD0δxt=Axt,where xt∈ℝn,A∈ℝn×ℝn, and 0<δ<1.Definition 12 (see [34]).
The characteristic equation of system (48) is
(49)sI−1−δA−δA=0.Theorem 13 (see [34]).
IfI−1−δA 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 pointM0 of the system (7)–(10) with R0<1 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 M0 is
(50)sI−1−δJM0−δJM0=0,where JM0 is the Jacobian matrix at M0, that is,
(51)JM0=−A100−δ1π1A10−A20−δ2π2A200−A3ε1000BR0−1.
The roots of the characteristic equation (50) are
(52)s1=−δA11+1−δA1<0,s2=−δA21+1−δA2<0,s3=−δA31+1−δA3<0,s4=δBR0−11−B1−δR0−1<0.


## 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 Oh3, and we will show in what follows the truncation error of Adams-Moulton methods is of Oh4. Let tj=t0+jh, where h=tend−t0/n and j=0,1,2,⋯,n, be the discretization of the interval t0=0,tend. We will define the recursive formula as follows.
(53)Hitj+1−Hitj=1−δKitj,Hi−Kitj−1,Hi+δ∫tjtj+1Kiy,Hidy,i=1,2,3,4.We approximateKit,Hi by a Lagrange polynomial of degree 3
(54)Pi,3τ=∑k=j−2j+1LjKitj,Hi,where
(55)Liτ=∏k=j−2,k≠ij+1τ−tkti−tk,i=j−2,j−1,j,j+1.Letu=τ−tj/h. Then,
(56)∫tjtj+1Kiy,Hidy=∫tjtj+1∑k=j−2j+1LjKitj,Hi=h24Ki,j−2−5Ki,j−1+19Ki,j+9Ki,j+1.Substituting (56) into (53), we get
(57)Hitj+1=Hitj+1−δKitj,Hi−Kitj−1,Hi+δh24Ki,j−2−5Ki,j−1+19Ki,j+9Ki,j+1,i=1,2,3,4.Equation (57) is a type of three-step Adams-Moulton method. It is an implicit method because its right-hand side contains Hitj+1. So the left-hand side of (57) can be calculated using Adams-Bashforth methods
(58)Hitj+1=Hitj+1−δKitj,Hi−Kitj−1,Hi+δh125Ki,j−2−16Ki,j−1+23Ki,j,i=1,2,3,4.We now write the methods as follows:(59)Hitj+1=Hitj+1−δKitj,Hi−Kitj−1,Hi+δh24Ki,j−2−5Ki,j−1+19Ki,j+9K^i,j+1,i=1,2,3,4,where K^i,j+1=Kitj+1,H∧itj+1 and
(60)H∧itj+1=Hitj+1−δKitj,Hi−Kitj−1,Hi+δh125Ki,j−2−16Ki,j−1+23Ki,j,i=1,2,3,4.In (60), Hit1 and Hit2 can be computed using the method
(61)Hitj+1=Hitj+1−δ+δhKitj,Hi.The truncation error for the three-step Adams-Moulton methods can be estimated by using the error estimate for the Lagrange interpolating polynomial, namely,(62)Kit,Hi=Pi,3t+Ei,3t,where
(63)Ei,3t=f4ξj,Hi24t−tj−2t−tj−1t−tjt−tj+1,ξj∈tj−2,tj+1.We notice that(64)∫tjtj+1Ei,3tdt=∫tjtj+1f4ξj,Hi24t−tj−2t−tj−1t−tjt−tj+1dt=h5f4ξj,Hi24∫01u+2u+1uu−1dubytakingu=t−tj−2/h=−h51930f4ξj,Hi24.We denote the right-hand side of (59) by H~i. The total truncation error of the use of formula (59) is
(65)Hitj+1−H~ih=−δ19h4720f4ξj,Hi.


## 9. Numerical Results and Discussions

For the purpose of numerical simulations, we utilize the values of the initial conditionsS0=0.15,T0=0.05,Q0=0.1,I0=0.2 and parameters π1=0.02,π2=0.01,ε1=0.15,ε2=0.068,δ1=0.02,δ2=0.01,α=0.005,β1=0.02,β2=0.05,β3=0.04,γ3=0.0015,γ4=0.003. Thus, we have
(66)M0=S0,T0,Q0,I0=0.7692,0.1815,0,0,R0=0.1124<1.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 Qt and smokers It goes to zero when t⟶∞. Hence, the smoking-free equilibrium point is asymptotically stable for δ=0.85,0.90,0.95,1, and S and T eventually disappear from the system.Remark 15.
δ=1 represents the standard derivative.Figure 2
Time series plot of all variables for distinct values ofδ.Figure 3
Time series plot of each ofSt,Tt,Qt, and It for distinct values of δ.In Figure3, St,Tt,Qt, and It are plotted for different values of δ. We can see that as δ increases, St and Tt increase and converge to smoking-free equilibrium point S0=0.7692 and T0=0.1815, respectively, whereas quitters Qt and smokers It decrease and converge to smoking-free equilibrium point Q0=0 and I0=0, 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 whenR0<1. Furthermore, the three-step Adams-Moulton method was applied to compute the numerical solutions. The solution depends on the fractional order δ and as δ⟶1− 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.



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