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Mathematical assessment of the dynamics of the tobacco smoking model: An application of fractional theory

  • In this paper we consider fractional-order mathematical model describing the spread of the smoking model in the sense of Caputo operator with tobacco in the form of snuffing. The threshold quantity R0 and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of of the solution of the considered model. The new version of numerical approximation's framework for the approximation of Caputo operator is used. Finally, the numerical results are presented to justify the significance of the arbitrary fractional order derivative. The analysis shows fractional-order model of tobacco smoking in Caputo sense gives useful information as compared to the classical integer order tobacco smoking model.

    Citation: Peijiang Liu, Taj Munir, Ting Cui, Anwarud Din, Peng Wu. Mathematical assessment of the dynamics of the tobacco smoking model: An application of fractional theory[J]. AIMS Mathematics, 2022, 7(4): 7143-7165. doi: 10.3934/math.2022398

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  • In this paper we consider fractional-order mathematical model describing the spread of the smoking model in the sense of Caputo operator with tobacco in the form of snuffing. The threshold quantity R0 and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of of the solution of the considered model. The new version of numerical approximation's framework for the approximation of Caputo operator is used. Finally, the numerical results are presented to justify the significance of the arbitrary fractional order derivative. The analysis shows fractional-order model of tobacco smoking in Caputo sense gives useful information as compared to the classical integer order tobacco smoking model.



    Mathematical biology is a field of great concern both for mathematicians and biologists in the current century. This field has many applications. The main focus of a researcher is to describe the dynamics of infectious diseases and their control aspects in terms of mathematical language. It was Brownlee [1] who developed the initial framework for mathematical biology and provide a concrete base for the subject. He used a probabilistic approach and within three years and proposed a law about the spread of infection [2]. A detailed mathematical touch to the subject was given in the work of Kermark and McKendrik [3]. Following the approach of Kermark-McKendrik, different types of infections were modelled and analyzed in a more sophisticated way (see [4,5,6,7,8,9,10,11,12,13,14]). Similar to infectious diseases, we have a very dangerous social habit common in all part of the world which not causes the severe diseases but kills. This social habit is nothing but smoking. It may be defined as the process in which an individual inhales the smoke of tobacco or the process in which tobacco smoke is taken by the mouth and then released through pipes or cigars. The habit of smoking in Europe was initially introduced with the entrance of Columbus in the 16th-century [15]. However, before this habit, other species of strange nature had adversely affected the human habitat and the whole ecosystem. It was Nicot who widely spread the use of tobacco in England as money yield and promoted it as a business. Due to this connection, the word "Nicotine'' was derived from his name. Late in the nineteenth century, cigarette manufacturing equipment was invented whose production speed was 200 units per minute and currently it is 9000. The habit of smoking causes many deadly diseases which include mouth, lung and throat cancer [16,17,18,19].

    In the 18th century, Fourier and Reimann-Liouville did remarkable work in ordinary calculus. Then many other researchers started to work in fractional calculus (FC) [20,21,22,23]. Fractional calculus is a generalized form and we can obtain the equation of ordinary calculus by substituting specific values for the chosen parameters. Fractional calculus has many applications in-memory processes, mathematical modeling and in different hereditary process. The related application can be found in [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. This field attracts more researchers due to their vast applications and use of the fractional order of the differential as well as the integral calculus.

    The Caputo derivative is very useful when dealing with real-world problem because, it allows traditional initial and boundary conditions to be included in the formulation of the problem and in addition the derivative of a constant is zero; however, functions that are not differentiable do not have fractional derivative. In this paper we study the fractional tobacco smoking model. We calculate the stability and equilibrium for the disease free point of the proposed model. We also give the theoretical results to validate our results obtained via fractional order method with numerical results. The main focus of this work is to construct fractional operator for the tobacco smoking system under Caputo derivative with fractional operator ζ(0,10). The ordinary differential equations which contains the integer order derivative contains the integer order derivative can be globalize to the fractional differential equations by fractional order (FDEs). The FDEs with parameter 0<α<1 may be found in [20,21,22,23,27].

    The outline of this paper is organized as follows. In Section 2, we give the essential definitions. Section 3, presents the formulation of the smoking model in classical integer and fractional case. The threshold quantity and equilibria of the proposed model are determined in Section 4. The existence theory of the proposed model given in Section 5. Also, the numerical methods to solve the considered problems are given in Section 6. Indeed, numerical results of the proposed models under different values of fractional orders are given in Section 7. Finally, the conclusion regarding the present finding is given in Section 8.

    In this section we provide some basic definitions, theorems and results that will be used in this article.

    Definition 1. [23] For a integrable function g, the Caputo derivative of fractional order ζ(0,1) is given by

    cDζg(t)=1Γ(mζ)t0g(m)(v)(tv)ζm+1dv,m=[ζ]+1.

    Also, the corresponding fractional integral of order ζ with Re(ζ)>0 is given by

    cIζg(t)=1Γ(ζ)t0(tv)ζ1g(v)dv.

    Lemma 1. [26] Let S(θ)C(0,T), then the solution of fractional differential equation

    {CDp0(θ)=S(θ),θ[0,T],(0)=S0

    is given by

    I(θ)=pj=0Njθj+1Γ(μ)θ0(θξ)μ1S(ξ)dξ.

    For NjR,j=0,1,2,3,,p.

    Lemma 2. [22,23,27] "The following result holds for fractional differential equations

    Iζ[cDζθ(t)]=θ(t)+α0+α1t+α2t2+...+αm1tm1,

    for arbitrary αiR, i=0,1,2,3,...,m1, where m=[ζ]+1 and [ζ] symbolizes the integer part of ζ".

    Lemma 3. [27] Let θACn[0,T],ζ>0 and n=[ζ], then the following result holds

    Iζ[cDζθ(ϑ)]=θ(ϑ)n1j=0Djθ(a)j!(ta)j.

    Lemma 4. [23,27]In view of Lemma (3), the solution of Dζθ(t)=y(t), n1<ζ<n is given by

    θ(t)=Iζy(t)+c0+c1(t)+c2t2++cn1tn1

    where cjR.

    Definition 2. [23] Suppose we have Caputo's fractional differential equation of order ζ,

    cDζθ(t)=f(t,θ(t))

    Then the solution is given as

    θ(tn+)=θ(tn)+f(tn,θn)hΓ(ζ){2hζtζn+1tζ+1n+1ζ+1+hζtζntζ+1ζ}f(tn1,θn1)hΓ(ζ{hζtζn+1tζ+1n+1ζ+1+tζnζ+1}+Rζn(t).

    Where Rζn(t) represent the remainder term. For the study of convergence and uniqueness of the solution of the scheme, we refer to [23].

    Theorem 1. [22,23] "Let G be a Banach space and P:XG is compact and continuous, if the set

    E={θG:θ=mPθ,m(0,1)},

    is bounded, then P has a unique fixed point''.

    According to the report of the center for disease control and prevention (CDC) smoking is the leading cause of premature and preventable deaths worldwide. By the above report approximately 440000 deaths occurs in the United State annually and the United Kingdom 105000 [37]. Further half of the smokers die due to smoking related diseases. Also smoking reduce the life expectancy by ten to twelve years. Moreover the ratio of heart attack is more than 70% as compare to non-smokers. Also this cause the lung cancer and it is also ten time higher than the non-smokers. The other cancer diseases are also linked with smoking. This includes pancreas, breast, cervix, stomach, mouth and throat cancers. These all diseases are occurring due to cigarette smoking, because one cigarette include aroung 4000 toxins and chemical compounds. Recently, Din et al. [39] proposed a stochastic smoking model, in which the total population is divided into five classes; susceptible individuals V(t), snuffing individuals Y(t), casual smokers X(t), chain W(t) and quit smokers Z(t) at any time t, so total population N(t)=V(t)+Y(t)+X(t)+W(t)+Z(t), and the model is expressed as follows:

    dV(t)=[ΠβV(t)Y(t)NdV(t)+λZ(t)]dt+α1V(t)dB1(t),dY(t)=[βV(t)Y(t)NδY(t)X(t)N(γ+d)Y(t)]dt+α2Y(t)dB2(t),dX(t)=[δY(t)X(t)N(μ+ω+d)X(t)]dt+α3X(t)dB3(t),dW(t)=[ωX(t)(κ+d)W(t)]dt+α4W(t)dB4(t),dZ(t)=[κW(t)(λ+d)Z(t)]dt+α5Z(t)dB5(t). (3.1)

    The biological interpretation of parameters used in the model are presented in Table 1.

    Table 1.  Parameters description.
    Symbols Description
    Π Inflow rate (either by birth or through migration)
    β Rate at which susceptible starts snuffing
    λ Relapse rate
    γ Tobacco related death rate in snuffing compartment
    d Natural death rate
    ω Rate at which occasional smokers tend to be chain smokers
    μ Death due to tobacco related diseases
    δ Rate through which snuffing population become casual smokers
    κ Quitting rate

     | Show Table
    DownLoad: CSV

    Where Bi(t) for i=15 stand for the independent Brownian motions and α1, α2,α3,α4, and α5 are the intensities of the white noises. In this work, we put Brownian motions intensities zero in model (3.1), then we obtain the following ordinary differential equations system:

    ´V(t)=ΠβV(t)Y(t)NdV(t)+λZ(t),´Y(t)=βV(t)Y(t)NδY(t)X(t)N(γ+d)Y(t),´X(t)=δY(t)X(t)N(μ+ω+d)X(t),´W(t)=ωX(t)(κ+d)W(t),´Z(t)=κW(t)(λ+d)Z(t), (3.2)

    with initial conditions

    V(0)>0,Y(0)>0,X(0)>0,W(0)>0,Z(0)>0. (3.3)

    For the better understanding the dynamics of proposed model, the author's generalized the aforementioned model to fractional order, due to its great degree of freedom, glob in nature and more reliable. We generalize the model (3.2) under the corresponding Caputo fractional order derivative as

    cDζV(t)=ΠβV(t)Y(t)NdV(t)+λZ(t),cDζY(t)=βV(t)Y(t)NδY(t)X(t)N(γ+d)Y(t),cDζX(t)=δY(t)X(t)N(μ+ω+d)X(t),cDζW(t)=ωX(t)(κ+d)W(t),cDζZ(t)=κW(t)(λ+d)Z(t). (3.4)

    With given initial conditions,

    V(0)>0,Y(0)>0,X(0)>0,W(0)>0,Z(0)>0. (3.5)

    In this section, we will interrogate the stability analysis of system (3.2) [18,24,25].

    First, we interrogate the disease-free steady-state of the proposed system (3.2) which is indicated by E0. We use Y(t)=X(t)=W(t)=Z(t)=0, so the smoking free equilibrium point E0 is

    E0=(V0,Y0,X0.W0,Z0)=(Πd,0,0,0,0)andN0=V0.

    Next-generation technique [12,16,18] is utilized to determined the reproduction value of the system (3.2). We calculated the reproduction parameter of the system (3.2) in the following next-generation method

    F=[βΠdN000000000],
    V=[(γ+d)000(μ+ω+d)00ω(κ+d)].

    The dominant eigenvalue of FV1=βΠdN0(γ+d), so

    R0=βΠdN0(γ+d).

    The system (3.2) is also have equilibrium point (EE) which denoted by E1=(V1,Y1,X1.W1,Z1), using the left side of system (3.2) is equal to zero, as follows:

    V1=N1(δW1+(γ+d))β,Y1=(μ+ω+d)δ,X1=(κ+d)(γ+d)[δd(R01)β(μ+ω+d)](κ+d)βδω+(μ+ω+d)(βδ((κ+d)+δ2d)),W1=ωX1(κ+d),Z1=κW1λ+d,N1=V1+Y1+X1+W1+Z1.

    Theorem 2. If R0<1, the infection-free steady-state (E0) of system (3.2) is

    (i) Locally asymptotically stable.

    (ii) Globally asymptotically stable.

    Otherwise unstable.

    Proof (i). For the local stability at E0, the Jacobian of system (3.2) is

    J(E0)=(dβΠdN000λ0βΠdN0(γ+d)00000(μ+ω+d)0000ω(κ+d)0000κ(d+λ)),

    the above matrix has eigenvalues λ1,λ2,λ3, λ4 and λ5,

    λ1=d<0,λ2=(γ+d)(R01),λ3=(μ+ω+d)<0,λ4=(κ+d)<0,λ4=(d+λ)<0,

    implying that λ2<0 for R0<1,λ2=0 for R0=1 and λ2>0 for R0>1. Thus, the eigenvalues are negative which proves the conclusion.

    Proof (ii). For the proof of condition (ii), first we construct the Lyapunov function L as

    L=lnVV0+lnYY0+X+W. (4.1)

    Differentiating Eq (4.1) with respect to time

    L=ΠVβYN+κWVd+βVNδXN(κ+d)(μ+ω+d)X,

    using the values of E0 in the above equation,

    L=dd+βΠdN0(γ+d),=R0(γ+d)(γ+d),=(γ+d)(R01), (4.2)

    therefore, if R0<1, then L<0, which implies that the system (3.2) is globally stable around E0.

    Theorem 3. The Smoking present equilibrium point E1 of system (3.2) is locally asymptotically stable if R0<1 and unstable if R0>1.

    Proof. For the local stability at E1 the Jacobian of system (3.2) is

    J(E1)=(βY1dN1βV1N100λβY1N1βV1N1δX1N1(γ+d)δY1N1000δX1N1δY1N1(d+ω+d)0000ω(κ+d)0000κ(λ+d)),

    easily we can get one eigenvalue of the above matrix is

    λ5=(λ+d)<0, (4.3)

    and

    J(E1)=(βY1dN1βV1N100βY1N1βV1NδX1N1(γ+d)δX1N100δX1N10000ω(κ+d)),

    using the basic matrix properties then we can get the following matrix

    J(E1)=(βY1N1dβV1N1ω(μ+ω+d)00βVN1+dδXβY1δY1N1dδY1βY1+ω(μ+ω+d)0βY1N1δX1N1δW1N1000ω(κ+d)).

    For simplification, this matrix can also be written as

    J(E1)=(ABCD).

    Here,

    A=(βY1N1dβV1N10βV1N1+dδXβY1),B=(ω(μ+ω+d)0δY1N1dδY1βY1+ω(μ+ω+d)0),C=(0βV1N1+dδX1βY100),D=(δW1N10ω(κ+d)).

    Since the eigenvalues of J(E) depend on the eigenvalues of A and D, the eigenvalues of A are given as follows:

    λ1=βY1N1d<0,λ2=(γ+d)+dδX1βY1Π(ΠY1(γ+d)R0),

    if R0>δΠ(μ+ω+d)(γ+d), then λ2<0.

    Now, the eigenvalues of D are

    λ3=δY1N1<0,λ4=(κ+d)<0,

    which is the required proof.

    In this section we construct the conditions for the existence and uniqueness of the solution, to get the desired results, we construct the following function:

    {ϑ1(t,V,Y,X,W,Z)=ΠβV(t)Y(t)NdV(t)+λZ(t),ϑ2(t,V,Y,X,W,Z)=βV(t)Y(t)NδY(t)X(t)N(γ+d)Y(t),ϑ3(t,V,Y,X,W,Z)=δY(t)X(t)N(μ+ω+d)X(t),ϑ4(t,V,Y,X,W,Z)=ωX(t)(κ+d)W(t),ϑ5(t,V,Y,X,W,Z)=κW(t)(λ+d)Z(t). (5.1)

    Suppose that the considered space C[0,T]=B be a Banach space with norm

    ||θ(t)||=supt[0,T][|V(t)|+|Y(t)|+|X(t)|+|W(t)|+|Z(t)|],

    where

    θ(t)={V(t)Y(t)X(t)W(t)Z(t),θ0(t)={V0Y0X0W0Z0,T(t,θ(t))={ϑ1(t,V,Y,X,W,Z)ϑ2(t,V,Y,X,W,Z)ϑ3(t,V,Y,X,W,Z)ϑ4(t,V,Y,X,W,Z)ϑ5(t,V,Y,X,W,Z). (5.2)

    With the help of (5.2), the system (3.4) can be written in as

    cDζθ(t)=T(t,θ(t)),t[0,T],θ(0)=θ0. (5.3)

    By Lemma (1), Eq (5.3) converts into the following form

    θ(t)=θ0+t0(ts)ζ1Γ(ζ)T(s,θ(s))ds,tJ=[0,T]. (5.4)

    To prove the existence of the solution we make the following assumptions

    (P1) constants K1,M1

    |T(t,θ(t))|K1|θ|q+M1.

    (P2) L>0, for each θ,ˉθ

    |T(t,θ)T(t,ˉθ)|L||θˉθ||.

    And let an operator P:BB be defined as:

    Pθ(t)=θ0+t0(ts)ζ1Γ(ζ)T(s,θ(s))ds. (5.5)

    Theorem 4. When the assumptions (P1) and (P2) are true it verifies that the problem (5.3), has at least one fixed point which implies that the problem of our study has also at least one solution.

    Proof. Furthermore we proceed as:

    Step I. First we have to show that P is continuous. To acquire the results we suppose that Tj is continuous for j=1,2,3,4,5,6. Which implies that T(s,θ(s)) is also continuous. Assume θn.θGθnθ, we must have PθnPθ.

    For this we consider

    ||PθnPθ||=maxtJ=[0,T]|t0(ts)ζ1Γ(ζ)Tn(s,θn(s))ds1Γ(ζ)t0(ts)ζ1T(s,θ(s))ds|,maxtJ=[0,T]t0|(ts)ζ1Γ(ζ)||Tn(s,θn(s))T(s,θ(s))|ds,TtΓ(ζ+1)||TnT||0asn. (5.6)

    As T is continuous, therefore PθnPθ, yields that P is continuous.

    Step II. Now to prove that P is bounded for any θG, we make of the supposition that P satisfies the growth condition:

    ||Pθ||=maxt[0,T]|θ0+1Γ(ζ)t0(ts)ζ1T(s,θ(s))ds|,|θ0|+maxt[0,T]1Γ(ζ)t0|(ts)ζ1||T(s,θ(s))|ds,|θ0|+TζΓ(ζ+1)[K1||θ||q+M1]. (5.7)

    Here we assume a S, the subset of G with the property of boundedness and we need to prove that P(S) is also bounded. To reach our destination, we assume that for any θS, now as S is bounded so Kq0

    ||θ||Kq,θS. (5.8)

    Further for any θS by using the growth condition, we have

    ||Pθ|||θ0|+TζΓ(ζ+1)[K1||θq||+M1]|θ0|+TζΓ(ζ+1)[K1Kq+M1]. (5.9)

    Therefore, P(S) is bounded.

    Step III. Here we attempt to prove that operator we defined is equi continuous, for this we assume that t2t1J=[0,T], then

    |Pθ(t1)Pθ(t2)|=|1Γ(ζ)t0(t1s)ζ1T(s,θ(s))ds1Γ(ζ)t0(t2s)ζ1T(s,θ(s))ds|,|1Γ(ζ)t0(t1s)ζ11Γ(ζ)t0(t2s)ζ1||T(s,θ(s))|ds,TζΓ(ζ+1)[K1||θ||q+M1][t1t2]. (5.10)

    By taking advantage of Arzela-Ascoli theorem, we can say that P(S) is relative compact.

    Step IV. In this step we need to prove that the set defined below is bounded

    E={θG:θ=mPθ,m(0,1)}. (5.11)

    To prove this we suppose that θE, for each tJ where J=[0,T] we have

    ||θ||=m||Pθ||m[|θ0|+TζΓ(ζ+1)K1||θ||q+M1]. (5.12)

    From here we can claim that the set defined above is bounded. By using Schaefer's FPT the operator we defined i.e., P has atleast one fixed point, and hence the model we studied in this paper has atleast one solution.

    Theorem 5. The problem (5.3) is unique solution, if TζK1Γ(ζ+1)<1.

    Proof. Let θ,ˉθG, then

    ||PθPˉθ||maxtJ=[0,T]t0|(ts)ζ1Γ(ζ)||T(s,θ(s))T(s,ˉθ(s))|ds,TζLTΓ(ζ+1)||θˉθ||. (5.13)

    Hence we can say that the fixed point is unique and therefor our solution is unique.

    This section is devoted to the numerical solution of the proposed model. For this we will use the well-known two steps fractional order Adam's Bashforth method. The considered model is given as

    cDζV(t)=ΠβV(t)Y(t)NdV(t)+λZ(t),cDζY(t)=βV(t)Y(t)NδY(t)X(t)N(γ+d)Y(t),cDζX(t)=δY(t)X(t)N(μ+ω+d)X(t),cDζW(t)=ωX(t)(κ+d)W(t),cDζZ(t)=κW(t)(λ+d)Z(t). (6.1)

    To obtain the desired results, we apply the fundamental theorem of fractional calculus to system (3.4) gives

    V(t)=V(0)+1Γ(ζ)t0G1(β,V(β))(tβ)ζ1dβ,Y(t)=Y(0)+1Γ(ζ)t0G2(β,Y(β))(tβ)ζ1dβ,X(t)=X(0)+1Γ(ζ)t0G3(β,X(β))(tβ)ζ1dβ,W(t)=W(0)+1Γ(ζ)t0G4(β,W(β))(tβ)ζ1dβ,Z(t)=Z(0)+1Γ(ζ)t0G5(β,Z(β))(tβ)ζ1dβ.

    The unknown terms G1,G2,G3,G4,G5 are given below. Now for t=tn+1, we get

    V(tn+1)=V(0)+1Γ(ζ)tn+10G1(t,V(t))(tn+1t)ζ1dt,Y(tn+1)=Y(0)+1Γ(ζ)tn+10G2(t,Y(t))(tn+1t)ζ1dt,X(tn+1)=X(0)+1Γ(ζ)tn+10G3(t,X(t))(tn+1t)ζ1dt,W(tn+1)=W(0)+1Γ(ζ)tn+10G4(t,W(t))(tn+1t)ζ1dt,Z(tn+1)=Z(0)+1Γ(ζ)tn+10G5(t,Z(t))(tn+1t)ζ1dt. (6.2)

    For t=tn we get the following

    V(tn)=V(0)+1Γ(ζ)tn0G1(t,V(t))(tnt)ζ1dt,Y(tn)=Y(0)+1Γ(ζ)tn0G2(t,Y(t))(tnt)ζ1dt,X(tn)=X(0)+1Γ(ζ)tn0G3(t,X(t))(tnt)ζ1dt,W(tn)=W(0)+1Γ(ζ)tn0G4(t,W(t))(tnt)ζ1dt,Z(tn)=Z(0)+1Γ(ζ)tn0G5(t,Z(t))(tnt)ζ1dt. (6.3)

    By V(tn+1)V(tn), Y(tn+1)Y(tn), X(tn+1)X(tn), W(tn+1)W(tn), Z(tn+1)Z(tn) in (6.2) and (6.3), we obtain

    V(tn+1)=V(tn)+G1ζ,1+G1η,2,Y(tn+1)=Y(tn)+G2ζ,1+G2η,2,X(tn+1)=X(tn)+G3ζ,1+G3η,2,W(tn+1)=W(tn)+G4ζ,1+G4η,2,Z(tn+1)=Z(tn)+G5ζ,1+G5η,2. (6.4)

    Where

    G1ζ,1=1Γ(ζ)tn+10G1(t,V(t))(tn+1t)ζ1dt,A2ζ,1=1Γ(ζ)tn+10G2(t,Y(t))(tn+1t)ζ1dt,G3ζ,1=1Γ(ζ)tn+10G3(t,X(t))(tn+1t)ζ1dt,G4ζ,1=1Γ(ζ)tn+10G4(t,W(t))(tn+1t)ζ1dt,G5ζ,1=1Γ(ζ)tn+10G5(t,Z(t))(tn+1t)ζ1dt.

    and

    G1ζ,2=1Γ(ζ)tn0G1(t,V(t))(tnt)ζ1dt,G2ζ,2=1Γ(ζ)tn0G2(t,Y(t))(tnt)ζ1dt,G3ζ,2=1Γ(ζ)tn0G3(t,X(t))(tnt)ζ1dt,G4ζ,2=1Γ(ζ)tn0G4(t,W(t))(tnt)ζ1dt,G5ζ,2=1Γ(ζ)tn0G5(t,Z(t))(tnt)ζ1dt.

    By approximating A1ζ,1, A1ζ,2, A2ζ,1, A2ζ,2, A3ζ,1, A3ζ,2, A4ζ,1, A4ζ,2, A5ζ,1, A5ζ,2 with the help of Lagrange's polynomials and the plugging back in (6.4) we get the following solution

    V(tn+1)=V(tn)+G1(tn,V(tn))Γ(ζ){2ζtζn+1tζ+1n+1ζ+1+ζtζntζ+1ζ}+G1(tn1,Vn1)Γ(ζ){ζtζn+1tζ+1n+1ζ+1+tζnζ+1}+Rζ1,n(t),Y(tn+1)=Y(tn)+G2(tn,Y(tn))Γ(ζ){2ζtζn+1tζ+1n+1ζ+1+ζtζntζ+1ζ}+G2(tn1,Yn1)Γ(ζ){ζtζn+1tζ+1n+1ζ+1+tζnζ+1}+Rζ2,n(t),X(tn+1)=X(tn)+G3(tn,X(tn))Γ(ζ){2ζtζn+1tζ+1n+1ζ+1+ζtζntζ+1ζ}+G3(tn1,Xn1)Γ(ζ){ζtζn+1tζ+1n+1ζ+1+tζnζ+1}+Rζ3,n(t),W(tn+1)=W(tn)+G4(tn,W(tn))Γ(ζ){2ζtζn+1tζ+1n+1ζ+1+ζtζntζ+1ζ}+G4(tn1,Wn1)Γ(ζ){ζtζn+1tζ+1n+1ζ+1+tζnζ+1}+Rζ4,n(t),Z(tn+1)=Z(tn)+G5(tn,Z(tn))Γ(ζ){2ζtζn+1tζ+1n+1ζ+1+ζtζntζ+1ζ}+G5(tn1,Vn1)Γ(ζ){ζtζn+1tζ+1n+1ζ+1+tζnζ+1}+Rζ5,n(t). (6.5)

    Where

    G1=ΠβV(t)Y(t)NdV(t)+λZ(t),G2=βV(t)Y(t)NδY(t)X(t)N(γ+d)Y(t),G3=δY(t)X(t)N(μ+ω+d)X(t),G4=ωX(t)(κ+d)W(t),G5=κW(t)(λ+d)Z(t).

    And Rζ1,n(t), Rζ2,n(t), Rζ3,n(t), Rζ4,n(t), Rζ5,n(t) are the remainder's terms.

    This section aims to present the numerical simulations of the fractional-order model (3.4) with Caputo fractional derivative. The time level is taken up to 40 units. The parameters and initial values are given in Table 2 for a biologically feasible way.

    Table 2.  The parameter values used in the simulations.
    Parameters value Source
    Π 0.80 [39]
    d 0.01 [39]
    β 0.70 assumed
    γ 0.07 [39]
    μ 0.01 [39]
    κ 0.50 assumed
    λ 0.002 [39]
    δ 0.010 [39]
    ω 0.001 [39]
    V(0) 60 [39]
    Y(0) 50 [39]
    X(0) 40 [39]
    W(0) 35 [39]
    Z(0) 15 [39]

     | Show Table
    DownLoad: CSV

    We use the numerical solution introduced for the approximation of Caputo fractional derivative to model (3.4). For this we used the values given in Table 2. We took the interval for time between 0 and 40, i.e., [040]. The initial data for the population has been taken from the Table 2 for all the classes considers in this article. These includes five different classes namely susceptible class V(t), snuffing population Y(t), casual smokers X(t), chain W(t) and quit smokers Z(t). The numerical results obtained for the above classes are plotted in Figures 13. The Figures 1(a)1(e) representing the comparison of model (3.4) when ζ=0.90, and Figures 2(a)2(e) representing the comparison of the system (3.4) and its deterministic version, when ζ=1.0. We approximate the numerical solution of the governing model (3.4) under Caputo non-integer order operator by iterative scheme. All the compartment, i.e., (V(t),Y(t),X(t),W(t),Z(t)) of the governing model against the parameters values given in Table 2 for different values of ζ=0.95,0.85,0.75,0.65,0.55. We show the dynamical behavior of the different compartments for the system (3.4) in Figures 3(a)3(e).

    Figure 1.  Simulations of susceptible, snuffing population, casual smokers, chain and quit smokers individuals of model (3.4), when ζ=90.0.
    Figure 2.  Simulations of susceptible, snuffing population, casual smokers, chain and quit smokers individuals of model (3.4), when ζ=1.0.
    Figure 3.  Simulations of susceptible, snuffing population, casual smokers, chain and quit smokers individuals of model (3.4) when ζ=0.95,0.85,0.75,0.65,0.55.

    First of all it is observed that the decay in susceptible class is very rapid and then become stable with the passage of time. Analogously the infection cases is also decreasing at various fractional order of ζ. In this case of chain and quit smokers has been achieved for their maximum peak. From these results it is clearly seen that the the governing model depends on the fractional order ζ, This gives more flexible information about the behavior of the model that cannot be obtained with the classical integer-order model.

    In this work, the formulation of a model containing a snuffing class is presented; then the equilibrium points that are smoking free and smoking positive are discussed. Further, we formulated a Caputo-fractional derivative to describe the transmission dynamics of a mathematical model describing the dynamics of tobacco smoking disease. We have discussed various fundamental properties of the model. The existence and uniqueness property of proposed model of the disease discussed. With the help of fractional adam's bashforth scheme, an approximate solution of the model has been determined. For the numerical solution of the proposed model are performed and we assumed the initial populations greater than zero for t>0. This can be clearly seen from the numerical results that our solution satisfy the initial data if the right hand side of the proposed system vanishes only at some specific conditions. The obtained results for various values of the fractional-order have been discussed. Our simulations show that the results obtained for the fractional model under the Caputo-derivatives provide a more realistic analysis than the classical integer-order tobacco smoking disease. The proposed method we can also study for different epidemics.

    This work was supported by the National Natural Science Foundation of China (Nos. 11901114, 62002068), and Guangzhou Science and technology innovation general project (No. 201904010010), Young innovative talents project of Guangdong Provincial Department of Education (No. 2017KQNCX081), Natural Science Foundation of Guangdong Province (Grant No. 2017A030310598), and Finance and Accounting Innovation Research Team under Guangdong, Hong Kong and Macau Greater Bay Area Capital Market (No. 2020WCXTD009).

    The authors declare that they have no conflict of interest.



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