# A Mathematical Analysis on the New Fractal-Fractional Model of Second-Hand Smokers via the Power Law Type Kernel: Numerical Solutions, Equilibrium Points, and Sensitivity Analysis

Authors: S. Rezapour; S. Etemad; M. Sinan; J. Alzabut; A. Vinodkumar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The second-hand smoke is a phenomenon that needs to be investigated, and its effects on the health of the people are to be examined. To analyze such an issue, the mathematical models are the best tools that help us to study the dynamical behaviors of this phenomenon. For this purpose, in the present paper, we consider a three-compartmental fractal-fractional mathematical model of a specific population of smokers or people that are exposed to second-hand smoke. By assuming some conditions onϕ-ψ-contractions and compact operators, we prove some theorems in relation to the existence of solutions. The Banach principle for the usual contractions is used for proving the uniqueness of solutions. Next, by some notions of functional analysis, two types of Ulam-Hyers stability for the fractal-fractional second-hand smoker model are established. Moreover, we have a steady-state analysis and obtain equilibrium points and basic reproduction number R0. Then, we investigate the sensitivity of the fractal-fractional system with respect to each parameter. For numerical simulation, the Adams-Bashforth (AB) method is used to derive numerical schemes for plotting and simulating the approximate solutions. Finally, the obtained solutions are tested with real data and different values of fractal dimensions and fractional orders.


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## Body

## 1. Introduction

The notion of the second-hand smoke is considered as the combination of smoke caused by the burning end of a cigarette, the smoke that is exhaled by a smoker, or smoke caused due to other tobacco products [1]. Therefore, being in the exposure of second-hand smoke, it involves an unintentional inhalation of smoke that happens near the people smoking or inhalation in an indoor environment where tobacco has been recently used. The people may be exposed to the second-hand smoke in different places that includes public places, home, private or public transport, workplace, home of relatives, and buses [2, 3]. Nowadays, the exposure to second-hand smoke is considered one of the main risk factor for a class of diseases and harmful health-related results at a vast scale and is evaluated to cause more than 600,000 deaths in each year.There exists an apparent relation between the factor of the second-hand smoke and an increased danger of stroke. Being in exposure to the second-hand smoke regularly in some public environment increases the chance of stroke by fifty percent [4]. At the same time, the second-hand smoke is as damaging to a fetus as if the mothers were inhaling the smoke directly from a cigarette [5]. Only 30 minutes of exposure to second-hand smoke can result in heart diseases similar to that of habitual smokers [6]. In view of these items, being in exposure to second-hand smoke is regarded a subject of great concern for all of the people due to its familiar harmful effects on the human health.Some of researchers have discussed about the adverse effects of second-hand smoke on the health problem as well. In 2011, Lubick et al. [7] conducted a research on the global health burden of the second-hand smoke, and in the same year, Burton [8] focused on an alarming consequence of smoking in indoor places. All the aforementioned studies indicate that the second-hand smoke is a significant problem worldwide.The importance of this issue led to a number of researches being done in precise mathematical formats, because we have to model these phenomena to study their exact behaviors. Even during the recent years, researchers have modeled many phenomena and diseases by applying new mathematical operators and analyzed such systems numerically and analytically. Their findings help us to take steps about that specific disease and to control the speed of its spread. Instances of such a modeling can be found in [9, 10] by studying electrical circuits and some processes in engineering, in [11–14] for analyzing COVID-19, in [15, 16] for cancer treatment, in [17–19] for investigating different diseases, in [20–22] for controlling some viruses, in [23, 24] for studying some social problems, in [25, 26] for investigating some phenomena in relation to animals and agriculture, etc. More specifically, the operators with Mittag-Leffler-type kernels play an important role in recent mathematical modelings. To see these important phenomena, we can even enumerate some new works in this regard. Khan et al. [27] studied a fractional COVID-19 epidemic model with a convex incidence rate with the help of the Atangana-Baleanu operators in the Caputo sense and analyzed the optimal control on the amount of the infection. In [28], Akgul solved an Atangana-Baleanu fractional differential equation with the reproducing kernel Hilbert space method.In this direction, the second-hand smoke is also a phenomenon which is important to investigate its effects based on mathematical models. To know some previous works about smoking models, Alkhudhari et al. [29] conducted a research on the global dynamics in relation to smoking on temporary quitters in 2014. One year later, Verma et al. [30] investigated a new model for the smoking cessation and the effects of media campaigns on this issue. Recently in 2019, Adhana et al. [31] studied a model of smoking tobacco in the form of a case study in Ethiopia and analyzed the smoking generation number (SGN) in relation to the given mathematical system. Pulecio-Montoya et al. [32] designed a model of the growth of tobacco consumers and simulated it by the fourth-order Runge-Kutta techniques.In last decades, Atangana [33] introduced a new advanced kind of derivatives entitled fractal-fractional derivative that connects the two topics of fractional calculus and fractal calculus. Further, he extended the relevant fractal-fractional integral. The construction of such operators is in the form of the convolution of the power-law, exponential-law, and generalized Mittag-Leffler-law type kernels with fractal derivatives. Fractal-fractional operators have two components: one is the fractional order and the second is the fractal dimension (order). Actually, differential equations furnished with the fractal-fractional derivative transfer the order and dimension of the supposed system into a rational order system.According to this property, we can extend the usual differential equations to generalized systems with arbitrary order of derivatives and dimensions. In other words, the basic aim for defining these derivatives is to study nonlocal BVPs/IVPs in nature that contain fractal behaviors. In this direction, some mathematicians established several results and designed some fractal-fractional models that show better simulations for describing mathematical structures. For instance, Gomez-Aguilar et al. [34] analyzed the transmission of malaria with the help of these fractal-fractional operators. The situation and spread of coronavirus in Pakistan were studied in the form of a fractal-fractional model by Shah et al. in 2020 [35]. Also, one year later, Ali et al. developed another fractal-fractional model of COVID-19 based on the data extracted from Wuhan [36]. In [37], Farman et al. analyzed the solutions of a fractal fractional Atangana-Baleanu model of COVID-19 via the Atangana-Toufik scheme. In other paper, published by Amin et al. [38], the authors used the same fractal-fractional operators for investigating the effect of vaccination to control COVID-19. More recently, Alqhtani and Saad [39] used three types of fractal-fractional operators via the power-law, exponential decay, and Mittag-Leffler kernels for modeling Michaelis-Menten Enzymatic Reaction and compared their numerical results with the classical results. Also, Saad et al. [40] used the Caputo-Fabrizio fractal-fractional derivatives to model the hepatitis C virus infection and analyzed numerical solutions and their chaos with respect to different values of parameters.In the present study, we will consider a three-compartmental mathematical model of a specific population of smokers or people that are exposured to second-hand smoke, and we will analyze and interpret our findings and model graphically, numerically, and analytically. Also, note that due to the importance of fixed point theory in proving the existence results, most of researchers use the main theorems of this field for confirming the existence of solutions for vide range of mathematical models. For instance, there are different theorems such as the Leray-Schauder fixed point theorem, Krasnoselskii’s fixed point theorem [41], some fixed point theorems in partial metric spaces [42], or some special contractions such as F-contractions [43] and ϕ-ψ-contractions [44]. Here, we will use these new contractions for proving the existence of solutions.We emphasize that the basic contribution and also the novelty of this work is that we compute and obtain our results based on a new model of second-hand smokers designed by the fractal-fractional derivatives for the first time. In the mentioned structure, we use power-law type kernel for this fractal-fractional derivative. Also, for the first time, to prove the existence of solutions,ϕ-admissible maps and ϕ-ψ-contractions play an important role in this study. Moreover, we try to investigate different stability results for the given model and at the end of the paper, we see that our simulative graphs show the accuracy and applicability of the fractal-fractional operators in comparison to other usual fractional operators. Also, we again emphasize that this is the first work on the application of the fractal-fractional derivatives for modeling dynamics of second-hand smokers. The numerical and graphical results obtained in this work show that we can analyze different qualitative behaviors of mathematical models under the effect of the fractal dimension and fractional order of these new operators and obtain better and more accurate results with the help of real data. This confirms the considerable advantages of the fractal-fractional operators.This model will be studied from several aspects. In the first place, after describing the suggested fractal-fractional model, we investigate existence theory based on two criteria in relation toϕ-ψ-contractions and compact operators. The Banach principle for usual contractions is utilized for proving the uniqueness result. Next, by notions of functional analysis, two types of Ulam-Hyers stability of the fractal-fractional system are established. In the sequel, we have a steady-state analysis and obtain equillibria and reproduction number R0, and then we investigate the sensitivity of the fractal-fractional system with respect to each parameter. For numerical simulation, a fractional type of two-step Lagrange polynomial known as the fractional Adams-Bashforth (AB) method is utilized to derive numerical schemes for plotting and simulating the results. Finally, the obtained solutions are tested with real data and different values of fractal and fractional orders.


## 2. Preliminaries

In this section, we recall some definitions and properties on the fractal-fractional operators and some self-maps includingϕ-ψ-contractions.We consider the familyΨ of all increasing functions ψ:0,∞⟶0,∞ such that
(1)∑j=1∞ψjt<∞,ψt<t,∀t>0.Definition 1 (see [44]).
LetF:X⟶X and ϕ:X2⟶ℝ≥0, where X is a normed space. Then,
(1)
Forϰ1,ϰ2∈X, F is ϕ-ψ-contraction if(2)ϕϰ1,ϰ2dFϰ1,Fϰ2≤ψdϰ1,ϰ2.(2)
F is ϕ-admissible if(3)ϕϰ1,ϰ2≥1⟹ϕFϰ1,Fϰ2≥1.Definition 2 (see [33]).
Let a continuous functionF:a,b⟶0,∞ be fractal differentiable of fractal order ν. Then, the fractal-fractional derivative of F equipped with the power-law-type kernel of order ω in the sense of Riemann-Liouville is defined by
(4)FFPDa,tω,νFt=1Γn−ωddtν∫att−wn−ω−1Fwdw,where dFw/dwν=limt⟶wFt−Fw/tν−wν is the fractal derivative and n−1<ω,ν≤n∈ℕ.One can simply observed that by lettingν=1, the fractal-fractional derivative FFPDa,tω,ν is the same standard Riemann-Liouville derivative RLDa,tω of order ω.Definition 3 (see [33]).
A continuous functionF defined on a,b is fractal-fractional integrable of the fractional and fractal orders ω and ν, respectively, via the power-law-type kernel if the integral
(5)IFFPa,tω,νFt=νΓω∫atwν−1t−wω−1Fwdwexists, where ν,ω>0.


## 3. Description of the Model for Second-Hand Smokers

This model of second-hand smoker tobacco involves a system of three differential equations [45]. The compartments are Pt, St, and Qt. It is notable that Pt represents the second-hand smokers or those (at risk of) exposure to others smoking, Qt denotes a group of persons who have cessated smoking but are at risk because of their previous smoking habit, and St is a group of persons who are addicted to tobacco and smoke it yet. The variable as well as parameters used in the model is all nonnegative. Meaning of variables and parameters is given in the sequel. With the above assumptions, and based on [45], the second-hand smoker (SHS) model is provided by a system of ODEs in the form:
(6)P′t=θ−s+q1+rPt−bStPt,S′t=bPtSt+r1QtSt−q1+q2+r2St,Q′t=r2St−r1StQt−q3+q1+γQt,where θ is the number of healthful individuals who are also at risk of smoker people, q1 stands for the natural mortality rate per total population, s is the death rate of second-hand smoker persons because of exposure to second-hand smoke, q2 is the death rate of individuals by smoking tobacco, q3 is the death rate of quit because of smoking habit before transferring to the phase Q, r is the exit rate of second-hand smoker to the healthful individuals, γ is the exit rate of persons who have cessation smoking to the healthful people, b is the infection rate from P to S, r2 is the exit rate from S to Q, and r1 is the infection rate from Q to S. The limitations for these parameters are θ>0, 0<s≤1, 0<q1<1, and 0≤r,b,r1,q2,r2,q3,γ≤1. The initial conditions are P0=P0≥0, S0=S0≥0, and Q0=Q0≥0.Motivated by the above standard model, we here consider the fractal-fractional model of the second-hand smoker in the following structure:(7)DFFP0,tω,νPt=θ−s+q1+rPt−bStPt,DFFP0,tω,νSt=bPtSt+r1QtSt−q1+q2+r2St,DFFP0,tω,νQt=r2St−r1StQt−q3+q1+γQt,subject to
(8)P0=P0≥0,S0=S0≥0,Q0=Q0≥0,where FFPD0,tω,ν is the fractal-fractional derivative with the fractional order ω∈0,1 and the fractal order ν∈0,1 via the power-law-type kernel. We impose several required assumptions on the model: the parameters of (7) are nonnegative and
(9)Nt=Pt+St+Qt,where Nt stands for the total population at the time t∈I≔0,T,T>0.


## 4. Existence of Solutions

In this section, the existence criterion is ensured by fixed point theory. Here, for the qualitative analysis, we define the Banach spaceX=M3, where M=CI,ℝ under the norm
(10)AX=P,S,QX=maxKt:t∈I,for which K≔P+S+Q. We rewrite the right-hand side of the fractal-fractional SHS-model (7) as
(11)W1t,Pt,St,Qt=θ−s+q1+rPt−bStPt,W2t,Pt,St,Qt=bPtSt+r1QtSt−q1+q2+r2St,W3t,Pt,St,Qt=r2St−r1StQt−q3+q1+γQt.Since, the integral is differentiable, we write the fractal-fractional SHS-model (7) in the following form:
(12)DRL0,tω,νPt=vtv−1W1t,Pt,St,Qt,DRL0,tω,νSt=vtv−1W2t,Pt,St,Qt,DRL0,tω,νQt=vtv−1W3t,Pt,St,Qt.By (12), the developed system is illustrated by the following IVP
(13)RLD0,tω,νAt=vtv−1Wt,At,ω,ν∈0,1,where
(14)At=Pt,St,QtT,A0=P0,S0,Q0T,Wt,At=W1t,Pt,St,Qt,W2t,Pt,St,Qt,W3t,Pt,St,Qt,t∈I.Now, we operate on both sides of the Equation (13) by the fractal-fractional integral which is given by Definition 3, and we get
(15)At=A0+νΓω∫0twν−1t−wω−1Ww,Awdw.In other words, the extended form of the above fractal-fractional integral is represented as(16)Pt=P0+νΓω∫0twν−1t−wω−1W1w,Pw,Sw,Qwdw,St=S0+νΓω∫0twν−1t−wω−1W2w,Pw,Sw,Qwdw,Qt=Q0+νΓω∫0twν−1t−wω−1W3w,Pw,Sw,Qwdw.To transform into a fixed point problem, we defineG:X⟶X by
(17)GAt=A0+νΓω∫0twν−1t−wω−1Ww,Awdw.In the preceding, we recall the required fixed point theorem in connection with our aim for proving the existence results.Theorem 4 (see [44]).
Assume thatX,d is a complete metric space, ϕ:X×X⟶ℝ, ψ∈Ψ, and W:X⟶X are an ϕ-ψ-contractive map such that
(1)
W is ϕ-admissible self map on X(2)
For someù0∈X,ϕù0,Wù0≥1(3)
For any sequenceùn in X with ùn⟶ù and ϕùn,ùn+1≥1 for all n≥1, we have ϕùn,ù≥1 for all n≥1
Then, there is a fixed point forW.Now, the first existence result is proved here under some special operators.Theorem 5.
Suppose that there are a mapℏ:ℝ×ℝ⟶ℝ, a continuous function W:I×X⟶X, and a nondecreasing function ψ∈Ψ. Assume that
℘1 For any A1,A2∈X and t∈I,
(18)Wt,A1t−Wt,A2t≤ℓ~ψA1t−A2t,with ℏA1t,A2t≥0, where ℓ~=Γν+ω/νTν+ω−1Γν.
℘2 There is some A0∈X such that for each t∈I,
(19)ℏA0t,GA0t≥0,and also the inequality
(20)ℏA1t,A2t≥0gives
(21)ℏGA1t,GA2t≥0,for each A1,A2∈X and t∈I.
℘3 For each convergent sequence Ann≥1 that belongs to X with An⟶A and
(22)ℏAnt,An+1t≥0,for each n and t∈I, we get
(23)ℏAnt,At≥0.
Then, there is a solution for the fractal-fractional IVP (13), and so there is a solution to the given fractal-fractional SHS-model (7).Proof.
LetA1 and A2 be two members belonging to X with
(24)ℏA1t,A2t≥0,for each t∈I. Then, by definition of the Beta function, we may write
(25)GA1t−GA2t≤νΓω∫0twν−1t−wω−1Ww,A1w−Ww,A2wdw≤νΓω∫0twν−1t−wω−1ℓ~ψA1w−A2wdw≤νℓ~Tν+ω−1Bν,ωΓωψA1−A2X=νTν+ω−1ΓνΓν+ωℓ~ψA1−A2X.
Consequently, we have(26)GA1−GA2X≤νTν+ω−1ΓνΓν+ωℓ~ψA1−A2X=ψA1−A2X.
Now, a functionϕ:X×X⟶0,∞ is introduced by the rule
(27)ϕA1,A2=1ifℏA1t,A2t≥0,0otherwise,for each A1,A2∈X. Then, for every A1,A2∈X, we will get
(28)ϕA1,A2dGA1,GA2≤ψdA1,A2.
Thus,G is found as an ϕ-ψ-contraction. To verify that G is ϕ-admissible, let A1,A2∈X be arbitrary and ϕA1,A2≥1. By definition of ϕ, we have
(29)ℏA1t,A2t≥0.
Then, byP2, ℏGA1t,GA2t≥0 is satisfied. Again, the definition of ϕ gives ϕGA1,GA2≥1. Thus, G is ϕ-admissible.
On the other hand, the conditionP2 guarantees the existence of A0∈X. In this case, for each t∈I, ℏA0t,GA0t≥0 holds. Clearly, we get ϕA0,GA0≥1. These show that the conditions (1) and (2) of Theorem 4 are fulfilled.
Now, we assume thatAnn≥1⊆X such that An⟶A and for all n, ϕAn,An+1≥1. By virtue of the definition of the nonnegative function ϕ,
(30)ℏAnt,An+1t≥0.
Therefore, in the light of hypothesisP3, we obtain
(31)ℏAnt,At≥0.
This indicates thatϕAn,A≥1 for every n. This guarantees the condition (3) of Theorem 4. Ultimately, by using Theorem 4, we conclude that there is a fixed point for G like A∗∈X. This implies thatA∗=P∗,S∗,Q∗T is interpreted as a solution of the fractal-fractional model of second-hand smoker (7), and the proof is completed.In the sequel, we use the Leray-Schauder fixed point theorem to prove the existence result.Theorem 6 (see [41]).
LetX be a Banach space, E a bounded convex closed set in X, and O⊂E an open set with 0∈O. Then, for the continuous and compact mapping G:O¯⟶E, either
(P1) There isù∈O¯ such that ù=Gù or
(P2) There isù∈∂O and 0<μ<1 such that ù=μGù.Theorem 7.
Suppose thatW∈CI×X,X and
(C1) There areφ∈L1I,ℝ+ and increasing function A∈C0,∞,0,∞ such that for each t∈I and A∈X,
(32)Wt,At≤φtAAt.
(C2) There isα>0 such that
(33)αA0+νTν+ω−1Γν/Γν+ωφ0∗Aα>1,where φ0∗=supt∈Iφt.
Then, there is a solution for the fractal-fractional problem (13), and so there is a solution for the given fractal-fractional model of second-hand smokers (7) on I.Proof.
To begin the proof, considerG:X⟶X formulated by (17) and the ball
(34)Nε=A∈X:AX≤ε,for some ε>0. In the first place, the continuity of W yields that of the operator G. Now, by (C1), we have
(35)GAt≤A0+νΓω∫0twν−1t−wω−1Ww,Awdw≤A0+νΓω∫0twν−1t−wω−1φwAAwdw≤A0+νTν+ω−1Bν,ωΓωφ0∗AAX≤A0+νTν+ω−1ΓνΓν+ωφ0∗Aε,for each A∈Nε. In consequence, we obtain
(36)GAX≤A0+νTν+ω−1ΓνΓν+ωφ0∗Aε<∞.
Hence,G is uniformly bounded on X. In the sequel, the equicontinuity of G is investigated. To prove such a claim, for every t,t∗∈0,T such that t<t∗ and for each A∈Nε, by letting
(37)supt,A∈I×NεWt,At=W∗<∞,we have
(38)GAt∗−GAt≤νΓω∫0t∗wν−1t∗−wω−1Ww,Awdw−νΓω∫0twν−1t−wω−1Ww,Awdw≤νW∗Γω∫0t∗wν−1t∗−wω−1dw−∫0twν−1t−wω−1dw≤νW∗Bν,ωΓωt∗ν+ω−1−tν+ω−1=νW∗ΓνΓν+ωt∗ν+ω−1−tν+ω−1,which is independent of A, and the right-hand side of (38) converges to 0 as t∗⟶t. Therefore, this implies that
(39)GAt∗−GAtX⟶0,as t∗⟶t. Thus, G is equicontinuous and is compact on Nε by referring to the Arzelá–Ascoli thoerem. We found that the conditions of Theorem 6 are valid on G. So, one of (P1) or (P2) will be fulfilled. By (C2), set
(40)O≔A∈X:AX<α,for some α>0 via A0+νTν+ω−1Γν/Γν+ωφ0∗Aα<α. With the help of (C1) and by (36), we write
(41)GAX≤A0+νTν+ω−1ΓνΓν+ωφ0∗AA.
Now, we assume the existence ofA∈∂O and 0<μ<1 with A=μGA. For these selections of A and μ, and by (41), one may write
(42)α=AX=μGAX<A0+νTν+ω−1ΓνΓν+ωφ0∗AAX<A0+νTν+ω−1ΓνΓν+ωφ0∗Aα<α,and this cannot occur. Therefore, the case (P2) does not hold and G has a fixed–point in O¯ by Theorem 6 which is interpreted as a solution of the fractal-fractional model of second-hand smoker (SHS) (7), and the proof is completed.


## 5. Uniqueness result

To prove the uniqueness of solution of the given fractal-fractional model of second-hand smoker (7), we use the Lipschitz property of functions Wi, i=1,2,3 given by (11).Lemma 8.
Consider the functionsP,S,Q,P∗,S∗,Q∗∈M≔CI,ℝ. Let
(H1)P≤λ1, S≤λ2, and Q≤λ3 for some constants λ1,λ2,λ3>0.
Then, the functionsW1,W2,W3 introduced by (11) are satisfied the Lipschitz property with respect to the corresponding components if w1,w2,w3>0, where
(43)w1=s+q1+r+bλ2,w2=bλ1+r1λ3+q1+q2+r2,w3=r1λ2+q3+q1+γ.Proof.
We begin with the functionW1. For each P,P∗∈M≔CI,ℝ, we have
(44)W1t,Pt,St,Qt−W1t,P∗t,St,Qt=θ−s+q1+rPt−bStPt−θ−s+q1+rP∗t−bStP∗t≤q1+s+r+bStPt−P∗t≤q1+s+r+bλ2Pt−P∗t=w1Pt−P∗t.
This shows thatW1 is Lipschitz with respect to P with the Lipschitz constant w1>0. For the function W2, for each S,S∗∈M≔CI,ℝ, we have
(45)W2t,Pt,St,Qt−W2t,Pt,S∗t,Qt=bPtSt+r1QtSt−q1+r2+q2St−bPtS∗t+r1QtS∗t−q1+r2+q2S∗t≤b∥Pt∥+r1∥Qt∥+q1+r2+q2∥St−S∗t∥≤bλ1+r1λ3+q1+r2+q2∥St−S∗t∥=w2∥St−S∗t∥.
This shows thatW2 is Lipschitz with respect to S with the Lipschitz constant w2>0. Now, for each Q,Q∗∈M≔CI,ℝ, we have
(46)W3t,Pt,St,Qt−W3t,Pt,St,Q∗t=r2St−r1StQt−q3+q1+γQt−r2St−r1StQ∗t−q3+q1+γQ∗t≤r1St+q3+q1+γQt−Q∗t≤r1λ2+q3+q1+γQt−Q∗t=w3Qt−Q∗t.
Accordingly, this shows thatW3 is Lipschitz with respect to Q with the Lipschitz constant w3>0. Above results show that three functions W1,W2,W3 are Lipschitzian with respect to the corresponding component with the Lipschitz constants w1,w2,w3>0, respectively.According to the obtained results in Lemma8, we investigate the uniqueness property for solution to the supposed fractal-fractional system (7).Theorem 9.
Let (H1) holds. Then, the given fractal-fractional model of second-hand smoker (7) has a unique solution if
(47)νTν+ω−1ΓνΓν+ωwi<1,i∈1,2,3.Proof.
We assume that the conclusion of theorem is not valid. In other words, there is another solution for the given fractal-fractional model of second-hand smoker (7). Assume that P∗t,S∗t,Q∗t is another solution with initial conditions P0,S0,Q0 such that by (16), we have
(48)P∗t=P0+νΓω∫0twν−1t−wω−1W1w,P∗w,S∗w,Q∗wdw,S∗t=S0+νΓω∫0twν−1t−wω−1W2w,P∗w,S∗w,Q∗wdw,Q∗t=Q0+νΓω∫0twν−1t−wω−1W3w,P∗w,S∗w,Q∗wdw.
Now, we can estimate(49)Pt−P∗t≤νΓω∫0twν−1t−wω−1×W1w,Pw,Sw,Qw−W1w,P∗w,S∗w,Q∗wdw≤νΓω∫0twν−1t−wω−1w1P−P∗dw≤νTν+ω−1ΓνΓν+ωw1P−P∗,and so
(50)1−νTν+ω−1ΓνΓν+ωw1P−P∗≤0.
The latter inequality is true ifP−P∗=0, and accordingly, P=P∗. Similarly, from
(51)S−S∗≤νTν+ω−1ΓνΓν+ωw2S−S∗,we get
(52)1−νTν+ω−1ΓνΓν+ωw2S−S∗≤0.
This implies thatS−S∗=0 and so S=S∗. Also,
(53)Q−Q∗≤νTν+ω−1ΓνΓν+ωw3Q−Q∗.
This gives(54)1−νTν+ω−1ΓνΓν+ωw3Q−Q∗≤0.
Hence,Q=Q∗. Consequently, we get
(55)Pt,St,Qt=P∗t,S∗t,Q∗t.
This shows that the fractal-fractional model of second-hand smoker (7) has a unique solution, and this completes our proof.


## 6. Stability

Here, the stability notion in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized versions is established for the system of fractal-fractional SHS-model (7). For more details on the stability analysis, we refer to [46, 47].Definition 10.
The fractal-fractional model of SHS (7) is Ulam–Hyers stable if there are 0<MWi∈ℝ,i∈1,2,3 such that for each εi>0, and for each P∗,S∗,Q∗∈X satisfying
(56)FFPD0,tω,νP∗t−W1t,P∗t,S∗t,Q∗t<ε1,FFPD0,tω,νS∗t−W2t,P∗t,S∗t,Q∗t<ε2,FFPD0,tω,νQ∗t−W1t,Q∗t,S∗t,Q∗t<ε3,there is P,S,Q∈X satisfying the given fractal-fractional model (7) such that
(57)P∗t−Pt≤MW1ε1,S∗t−St≤MW2ε2,Q∗t−Qt≤MW3ε3.Definition 11.
The given fractal-fractional model of second-hand smoker (7) is generalized Ulam–Hyers stable if there are MWi∈Cℝ+,ℝ+,i∈1,2,3 with MWi0=0 such that for each εi>0 and for each P∗,S∗,Q∗∈X satisfying the inequalities (56), there is P,S,Q∈X as a solution of the given fractal-fractional model of second-hand smoker (7) such that
(58)P∗t−Pt≤MW1ε1,S∗t−St≤MW2ε2,Q∗t−Qt≤MW3ε3.Note that Definition11 is derived from Definition 10.Remark 12.
Notice thatP∗,S∗,Q∗∈X is a solution for (56) if and only if there are ℏ1,ℏ2,ℏ3∈C0,T,ℝ (depending on P∗,S∗,Q∗, respectively) such that for each t∈I,
(59)ℏit<εi.(1)
We have(60)DFFP0,tω,νP∗t−W1t,P∗t,S∗t,Q∗t+ℏ1t,DFFP0,tω,νS∗t−W2t,P∗t,S∗t,Q∗t+ℏ2t,DFFP0,tω,νQ∗t−W3t,P∗t,S∗t,Q∗t+ℏ3t,Definition 13.
The given fractal-fractional model of second-hand smoker (7) is Ulam–Hyers–Rassias stable with respect to functions Φi,i∈1,2,3 if there are 0<MWi,Φi∈ℝ such that for each εi>0 and for each P∗,S∗,Q∗∈X satisfying
(61)DFFP0,tω,νP∗t−W1t,P∗t,S∗t,Q∗t<ε1Φ1t,DFFP0,tω,νS∗t−W2t,P∗t,S∗t,Q∗t<ε2Φ2t,DFFP0,tω,νQ∗t−W3t,P∗t,S∗t,Q∗t<ε3Φ3t,there is P,S,Q∈X as a solution of the given fractal-fractional model of second-hand smoker (7) such that
(62)P∗t−Pt≤ε1MW1,Φ1Φ1t,∀t∈I,S∗t−St≤ε2MW2,Φ2Φ2t,∀t∈I,Q∗t−Qt≤ε3MW3,Φ3Φ3t,∀t∈I.Definition 14.
The given fractal-fractional model of second-hand smoker (7) is generalized Ulam–Hyers–Rassias stable with respect to functions Φi if there are 0<MWi,Φi∈ℝ such that for each P∗,S∗,Q∗∈X satisfying
(63)DFFP0,tω,νP∗t−W1t,P∗t,S∗t,Q∗t<Φ1t,DFFP0,tω,νS∗t−W2t,P∗t,S∗t,Q∗t<Φ2t,DFFP0,tω,νQ∗t−W3t,P∗t,S∗t,Q∗t<Φ3t,there is P,S,Q∈X as a solution of the given fractal-fractional model of second-hand smoker (7) such that
(64)P∗t−Pt≤MW1,Φ1Φ1t,S∗t−St≤MW2,Φ2Φ2t,Q∗t−Qt≤MW3,Φ3Φ3t.Note that Definition14 is derived from Definition 13. Also, if we take Φit=1, then Definition 13 gives the Ulam-Hyers property for the stability of solutions.Remark 16.
Notice thatP∗,S∗,Q∗∈X is a solution for (61) if and only if there are ℏ1,ℏ2,ℏ3∈C0,T,ℝ (depending on P∗,S∗,Q∗, respectively) such that ∀t∈I,
(65)ℏit<εiΦiT.(i)
We have(66)FFPD0,tω,νP∗t=W1t,P∗t,S∗t,Q∗t+ℏ1t,FFPD0,tω,νS∗t=W2t,P∗t,S∗t,Q∗t+ℏ2t,FFPD0,tω,νQ∗t=W3t,P∗t,S∗t,Q∗t+ℏ3t.The Ulam–Hyers stability is discussed here to the given fractal-fractional model of second-hand smoker (7).Theorem 17.
If the assumption (H1) is fulfilled, then the given fractal-fractional model of second-hand smoker (7) is Ulam–Hyers stable on I≔0,T and also is generalized Ulam–Hyers stable such that
(67)νTν+ω−1ΓνΓν+ωwi<1,i∈1,2,3,where wi is given by (43).Proof.
Letε1>0 and P∗∈M be arbitrary such that
(68)DFFP0,tω,νP∗t−W1t,P∗t,S∗t,Q∗t<ε1.
Then, from Remark1, we can find a function ℏ1t satisfying
(69)DFFP0,tω,νP∗t=W1t,P∗t,S∗t,Q∗t+ℏ1t,with ℏ1t≤ε1. It follows that
(70)P∗t=P0+νΓω∫0twν−1t−wω−1W1w,P∗w,S∗w,Q∗wdw+νΓω∫0twν−1t−wω−1ℏ1wdw.
By Theorem9, let P∈M be the unique solution of the given fractal-fractional model of second-hand smoker (7). Then, Pt is defined as
(71)Pt=P0+νΓω∫0twν−1t−wω−1W1w,Pw,Sw,Qwdw.
Therefore,(72)P∗t−Pt≤νΓω∫0twν−1t−wω−1ℏ1wdw+νΓω∫0twν−1t−wω−1×W1w,P∗w,S∗w,Q∗w−W1w,Pw,Sw,Qwdw≤νTν+ω−1ΓνΓν+ωε1+νTν+ω−1ΓνΓν+ωw1P∗−P.
Hence, we get(73)P∗−P≤νTν+ω−1Γν/Γν+ωε11−νTν+ω−1Γν/Γν+ωw1.
If we letMW1=νTν+ω−1Γν/Γν+ω/1−νTν+ω−1Γν/Γν+ωw1, then ∥P∗−P∥≤MW1ε1. Similarly, we have
(74)S∗−S≤MW2ε2,Q∗−Q≤MW3ε3,where
(75)MWi=νTν+ω−1Γν/Γν+ω1−νTν+ω−1Γν/Γν+ωwi,i∈2,3.
Hence, the Ulam–Hyers stability of the given SHS-model (7) is fulfilled. Next, by assuming
(76)MWiεi=νTν+ω−1Γν/Γν+ωεi1−νTν+ω−1Γν/Γν+ωwi,i∈2,3,with MWi0=0, clearly, the generalized Ulam–Hyers stability is proved.Theorem 18.
The condition (H1) is assumed to be held, and
H′ There are increasing functions Φi∈C0,T,ℝ+,i∈1,2,3, and ΛΦi>0 such that
(77)DFFP0,tω,νΦit<ΛΦiΦit,i∈1,2,3,∀t∈I.
Then, the given fractal-fractional model of second-hand smoker (7) is the Ulam–Hyers–Rassias and generalized Ulam–Hyers–Rassias stable.Proof.
For eachε1>0 and for each P∗∈M satisfying
(78)DFFP0,tω,νP∗t−W1t,P∗t,S∗t,Q∗t<ε1Φ1t,we can find a function ℏ1t satisfying
(79)DFFP0,tω,νP∗t=W1t,P∗t,S∗t,Q∗t+ℏ1t,with ℏ1t≤ε1Φ1t. It gives
(80)P∗t=P0+νΓω∫0twν−1t−wω−1W1w,P∗w,S∗w,Q∗wdw+νΓω∫0twν−1t−wω−1ℏ1wdw.
By Theorem9, let P∈M be the unique solution of the given fractal-fractional model of second-hand smoker (7). Then, Pt is given by
(81)Pt=P0+νΓω∫0twν−1t−wω−1W1w,Pw,Sw,Qwdw.
Then, by (77),
(82)P∗t−Pt≤νΓω∫0twν−1t−wω−1ℏ1wdw+νΓω∫0twν−1t−wω−1×W1w,P∗w,S∗w,Q∗w−W1w,Pw,Sw,Qwdw≤ε1νΓω∫0twν−1t−wω−1Φ1wdw+νTν+ω−1ΓνΓν+ωw1P∗−P≤ε1ΛΦ1Φ1t+νTν+ω−1ΓνΓν+ωw1P∗−P.
Accordingly, it gives(83)P∗−P≤ε1ΛΦ1Φ1t1−νTν+ω−1Γν/Γν+ωw1.
If we let(84)MW1,Φ1=ΛΦ11−νTν+ω−1Γν/Γν+ωw1,then P∗−P≤ε1MW1,Φ1Φ1t. Similarly, we have
(85)S∗−S≤ε2MW2,Φ2Φ2t,Q∗−Q≤ε3MW3,Φ3Φ3t,where
(86)MWi,Φi=ΛΦi1−νTν+ω−1Γν/Γν+ωwi,i∈2,3.
Hence, the given fractal-fractional model of second-hand smoker (7) is stable in the sense of Ulam–Hyers–Rassias. Along with this, by setting εi=1,i∈1,2,3, the mentioned fractal-fractional model of second-hand smoker (7) is generalized Ulam–Hyers–Rassias stable.


## 7. Steady-State Analysis and Local Stability

Here, we follow our investigation for obtaining equilibrium points of the supposed fractal-fractional system (7).


### 7.1. Disease-free equilibrium point

Define the following homogeneous system of equations:(87)DFFP0,tω,νPt=DFFP0,tω,νSt=DFFP0,tω,νQt=0,or equivalently
(88)θ−s+q1+rPt−bStPt=0,bPtSt+r1QtSt−q1+q2+r2St=0,r2St−r1StQt−q3+q1+γQt=0.In this case, the (disease) smoke-free equilibrium pointE0 of the model of second-hand smoker (7) under no infection when S0=0 and Q0=0 is presented by
(89)E0=θs+q1+r,0,0.


### 7.2. Basic Reproduction Number

Here, we calculate the basic reproduction number with the help of the next generation matrix approach [48]. As we know, this quantity denoted by R0 is considered as the expected value of the rate of infection in each time unit. The infection happens in the susceptible individuals due to the infected individuals, and also, the existence of the endemic equilibrium point to the fractal-fractional second-hand smoker model (7) depends on the value of R0. To do this, we consider the infected compartments St and Qt. By assuming Y=S,QT and from the infected compartments, we have two vectors say f and v in which we have nonlinear terms in f and the negative of linear terms in v, satisfying
(90)dYdtE0=f−v,where
(91)f=bSP00,v=q1+q2+r2S−θ+s+q1+rP+bSP−r2S+r1SQ+q3+q1+γQ.The Jacobian matrices of both matricesf and v are given by
(92)Jf=bSbP0000000,Jv=0q1+q2+r20s+q1+rP+bSbP00−r2+r1Qr1S+q3+q1+γ.Therefore, the Jacobian matrices of both matricesf and v at disease-free equilibrium point E0 obtained as (89) are given by
(93)JfE0=0bθs+q1+r0000000,JvE0=0q1+q2+r20θbθs+q1+r00−r2q3+q1+γ.On the other side,(94)J−1vE0=−bs+q1+rq1+q2+r21θ01q1+q2+r200r2q1+q2+r2q3+q1+γ0−1q3+q1+γ.By some simple calculations, we get(95)JfE0·J−1vE0=bθs+q1+rq1+q2+r200000000.In the final step, the spectral radius of the next generation matrixJfE0·J−1vE0 is the basic reproduction number R0 which is given by
(96)R0=ρJfE0·J−1vE0=bθs+q1+rq1+q2+r2.In Figures1–4, we plot the dynamics of R0 by 3D plots and the contours of the basic production number versus different parameters.Figure 1
The dynamics ofR0 by 3D plot and the contour of R0 vs. b and r2, respectively.
(a)(b)Figure 2
The dynamics ofR0 by 3D plot and the contour of R0 vs. r and r1, respectively.
(a)(b)Figure 3
The dynamics ofR0 by 3D plot and the contour of R0 vs. q1 and q2, respectively.
(a)(b)Figure 4
The dynamics ofR0 by 3D plot and the contour of R0 vs. θ and q1, respectively.
(a)(b)


### 7.3. Endemic Equilibrium Point

As we know, the quantityR0 is a criterion to measure the transmission potential of a infectious disease during a specific time. Whenever R0>1, then the fractal-fractional second-hand smoker model (7) involves an endemic equilibrium point E∗=P∗,S∗,Q∗. To find this point, we must solve the homogeneous system of equation (88) by considering this claim that all state functions Pt,St,Qt are nonzero. Therefore, from the second equation of (88) and by assuming St≠0, we get
(97)bP+r1Q=q1+q2+r2.Now, we rewrite the first and third equations of (88) as
(98)θ−s+q1+r+bSP=0,r2S−r1S+q3+q1+γQ=0,and we derive the following relations
(99)P=θs+q1+r+bS,Q=r2Sr1S+q3+q1+γ.We substitute above relations into (97) and by assuming the constants K1=s+q1+r>0, K2=q1+q2+r2>0 and K3=q3+q1+γ>0, we obtain
(100)bθK1+bS+r1r2Sr1S+K3−K2=0.We have(101)bθr1S+K3+r1r2SK1+bS−K2K1+bSr1S+K3K1+bSr1S+K3=0.Since the denominator of the above equation is nonzero, hence(102)bθr1S+K3+r1r2SK1+bS−K2K1+bSr1S+K3=0.On the other hand, by the new notation, we know thatR0=bθ/K1K2 and so K1K2R0=bθ. Thus, from (102) and by some simple calculations, we derive a quadratic equation with respect to S as
(103)r1r2b−r1bK2S2+r1r2K1−bK2K3+r1K1K2R0−1S+K1K2K3R0−1=0.SetY1=r1br2−r1bK2, and
(104)Y2=r1r2K1−bK2K3+r1K1K2R0−1,and Y3=K1K2K3R0−1. Consequently, we get
(105)S=−Y2±Y22−4Y1Y32Y1.Sincer1br2<r1bK2, thus Y1<0 and if R0>1, then
(106)S∗=−Y2−Y22−4Y1Y32Y1.Simply,P∗ and Q∗ can be obtained by inserting the equation (106) into equation (99), and therefore,
(107)P∗=θs+q1+r+b−Y2−Y22−4Y1Y3/2Y1,(108)Q∗=r2−Y2−Y22−4Y1Y3/2Y1r1−Y2−Y22−4Y1Y3/2Y1+q3+q1+γ.Theorem 19.
IfR0>1 or R0=1 and r1r2K1>bK2K3, then there exists E∗=P∗,S∗,Q∗ as the endemic equilibrium point, where P∗, S∗, and Q∗ are as (106)–(108). Also, If R0<1, then this point does not exist.Proof.
IfR0>1, then by the above discussions, E∗=P∗,S∗,Q∗ exists. If R0=1, then Y2=r1r2K1−bK2K3 and Y3=0. In this case, we have
(109)S=−Y2−Y22Y1.
On the other side, we know thatY1≠0. Hence, if Y2≤0, then S=0 and if Y2>0, then S<0. These show the existence of the endemic equilibrium point E∗.
On the other side,R0<1 gives −Y2>0 and Y3<0. Thus, either S is a complex number or a negative real number, and E∗ does not exist.In Figures5 and 6, we show the stability curves of P, S, and Q at endemic equilibrium point E∗ for some values of the fractal and fractional orders.Figure 5
The stability curves ofPt and St at E∗ for ω=0.97 and ν=0.5.
(a)(b)Figure 6
The stability curves ofQt at E∗ for ω=0.97 and ν=0.5.


### 7.4. Local Asymptotic Stability Analysis

In this place, we aim to investigate the local asymptotic stability of the smoke-free pointE0 found in (89) in relation to the fractal-fractional model (7) of second-hand smokers. First, we obtain the Jacobian matrix of the fractal-fractional system (7) as
(110)J=−s+q1+r−bS−bP0bSbP+r1Q−q1+q2+r2r1S0r2−r1Q−r1S−q3+q1+γ.Theorem 20.
The fractal-fractional model (7) of second-hand smokers is locally asymptotically stable at smoke-free equilibrium point E0 if all roots of the characteristic polynomial detJE0−λI=0 are negative or R0<1; otherwise, it is unstable.Proof.
In view of (110), the Jacobian matrix JE0 of the model (7) at smoke-free equilibrium point E0 is given as
(111)JE0=−s+q1+r−bθs+q1+r00bθs+q1+r−q1+q2+r200r2−q3+q1+γ.
By considering the Jacobian matrix (111) and by detJE0−λI=0, the characteristic polynomial becomes
(112)λ+s+q1+rλ+q3+q1+γλ−bθs+q1+r−q1+q2+r2=0.
The roots of the above polynomial are(113)λ1=−s+q1+r,λ2=−q3+q1+γ,λ3=bθs+q1+r−q1+q2+r2.
Since all parameters are assumed to be positive, we clearly haveλ1<0 and λ2<0. On the other hand, if R0<1, then R0−1<0. So, we can write
(114)bθs+q1+rq1+q2+r2<1.
Hence,(115)bθs+q1+r<q1+q2+r2.
Consequently,λ3=bθ/s+q1+r−q1+q2+r2<0. Thus, all of roots are negative, and so the fractal-fractional second-hand smoker model (7) is locally asymptotically stable around the smoke-free equilibrium point E0. This completes the proof.


### 7.5. Sensitivity Analysis

To find out how sensitive the parameters in each model are to the transmission of infections or diseases, the authors use a criterion introduced by Chitnis et al. [49]. Here, to obtain the sensitivity of R0 in terms of each of its parameters, we compute the partial derivative with respect to every parameter in the reproduction number. In other words, if p is an arbitrary parameter, then the sensitivity index of R0 versus p is calculated by
(116)SpR0=pR0∂R0∂p.Now, according to the above relation, we have(117)SbR0=bR0θq1+r2+q2q1+r+s>0,(118)SθR0=θR0bq1+r2+q2q1+r+s>0,(119)SsR0=sR0−bθq1+r2+q2q1+r+s2<0,(120)Sq1R0=q1R0−bθq1+q2+r2q1+r+s2−bθq1+r2+q22q1+r+s<0,(121)SrR0=rR0−bθq1+r2+q2q1+r+s2<0,(122)Sq2R0=q2R0−bθq1+r2+q22q1+r+s<0,(123)Sr2R0=r2R0−bθq1+r2+q22q1+r+s<0.The numeric values for sensitivity indices (120) are given in Table 1.Table 1
Sensitivity of theR0 versus proposed parameters.
ParameterSValueParameterSValuebSbR0+1.0000θSθR0+1.0000sSsR0-9.1621e-05q1Sq1R0-0.0061rSrR0-0.0366q2Sq2R0-0.0018r2Sr2R0-4.0813e-04In Figure7, when the sign of sensitivity index is positive, then each increase (decrease) in the value of parameters (assuming the remaining parameters to be constant) increases (decreases) the value of R0. The negative sign in this index gives the inverse result for R0. In the sensitivity analysis of our second-hand smoker model (7), it is observed that R0 increases by increasing the values of b,θ and decreases by increasing the values of s,q1,r,q2,r2. For example, Sq1R0=−0.0061 means that an increase in q1 by 10% decreases R0 by 0.061%. Therefore, to reduce the spread of infections, those parameters having negative sensitivity indices must be minimized in the environment.Figure 7
Global sensitivity analysis of model (7) based on R0 (96) and sensitivity analysis (120).


## 7.1. Disease-free equilibrium point

Define the following homogeneous system of equations:(87)DFFP0,tω,νPt=DFFP0,tω,νSt=DFFP0,tω,νQt=0,or equivalently
(88)θ−s+q1+rPt−bStPt=0,bPtSt+r1QtSt−q1+q2+r2St=0,r2St−r1StQt−q3+q1+γQt=0.In this case, the (disease) smoke-free equilibrium pointE0 of the model of second-hand smoker (7) under no infection when S0=0 and Q0=0 is presented by
(89)E0=θs+q1+r,0,0.


## 7.2. Basic Reproduction Number

Here, we calculate the basic reproduction number with the help of the next generation matrix approach [48]. As we know, this quantity denoted by R0 is considered as the expected value of the rate of infection in each time unit. The infection happens in the susceptible individuals due to the infected individuals, and also, the existence of the endemic equilibrium point to the fractal-fractional second-hand smoker model (7) depends on the value of R0. To do this, we consider the infected compartments St and Qt. By assuming Y=S,QT and from the infected compartments, we have two vectors say f and v in which we have nonlinear terms in f and the negative of linear terms in v, satisfying
(90)dYdtE0=f−v,where
(91)f=bSP00,v=q1+q2+r2S−θ+s+q1+rP+bSP−r2S+r1SQ+q3+q1+γQ.The Jacobian matrices of both matricesf and v are given by
(92)Jf=bSbP0000000,Jv=0q1+q2+r20s+q1+rP+bSbP00−r2+r1Qr1S+q3+q1+γ.Therefore, the Jacobian matrices of both matricesf and v at disease-free equilibrium point E0 obtained as (89) are given by
(93)JfE0=0bθs+q1+r0000000,JvE0=0q1+q2+r20θbθs+q1+r00−r2q3+q1+γ.On the other side,(94)J−1vE0=−bs+q1+rq1+q2+r21θ01q1+q2+r200r2q1+q2+r2q3+q1+γ0−1q3+q1+γ.By some simple calculations, we get(95)JfE0·J−1vE0=bθs+q1+rq1+q2+r200000000.In the final step, the spectral radius of the next generation matrixJfE0·J−1vE0 is the basic reproduction number R0 which is given by
(96)R0=ρJfE0·J−1vE0=bθs+q1+rq1+q2+r2.In Figures1–4, we plot the dynamics of R0 by 3D plots and the contours of the basic production number versus different parameters.Figure 1
The dynamics ofR0 by 3D plot and the contour of R0 vs. b and r2, respectively.
(a)(b)Figure 2
The dynamics ofR0 by 3D plot and the contour of R0 vs. r and r1, respectively.
(a)(b)Figure 3
The dynamics ofR0 by 3D plot and the contour of R0 vs. q1 and q2, respectively.
(a)(b)Figure 4
The dynamics ofR0 by 3D plot and the contour of R0 vs. θ and q1, respectively.
(a)(b)


## 7.3. Endemic Equilibrium Point

As we know, the quantityR0 is a criterion to measure the transmission potential of a infectious disease during a specific time. Whenever R0>1, then the fractal-fractional second-hand smoker model (7) involves an endemic equilibrium point E∗=P∗,S∗,Q∗. To find this point, we must solve the homogeneous system of equation (88) by considering this claim that all state functions Pt,St,Qt are nonzero. Therefore, from the second equation of (88) and by assuming St≠0, we get
(97)bP+r1Q=q1+q2+r2.Now, we rewrite the first and third equations of (88) as
(98)θ−s+q1+r+bSP=0,r2S−r1S+q3+q1+γQ=0,and we derive the following relations
(99)P=θs+q1+r+bS,Q=r2Sr1S+q3+q1+γ.We substitute above relations into (97) and by assuming the constants K1=s+q1+r>0, K2=q1+q2+r2>0 and K3=q3+q1+γ>0, we obtain
(100)bθK1+bS+r1r2Sr1S+K3−K2=0.We have(101)bθr1S+K3+r1r2SK1+bS−K2K1+bSr1S+K3K1+bSr1S+K3=0.Since the denominator of the above equation is nonzero, hence(102)bθr1S+K3+r1r2SK1+bS−K2K1+bSr1S+K3=0.On the other hand, by the new notation, we know thatR0=bθ/K1K2 and so K1K2R0=bθ. Thus, from (102) and by some simple calculations, we derive a quadratic equation with respect to S as
(103)r1r2b−r1bK2S2+r1r2K1−bK2K3+r1K1K2R0−1S+K1K2K3R0−1=0.SetY1=r1br2−r1bK2, and
(104)Y2=r1r2K1−bK2K3+r1K1K2R0−1,and Y3=K1K2K3R0−1. Consequently, we get
(105)S=−Y2±Y22−4Y1Y32Y1.Sincer1br2<r1bK2, thus Y1<0 and if R0>1, then
(106)S∗=−Y2−Y22−4Y1Y32Y1.Simply,P∗ and Q∗ can be obtained by inserting the equation (106) into equation (99), and therefore,
(107)P∗=θs+q1+r+b−Y2−Y22−4Y1Y3/2Y1,(108)Q∗=r2−Y2−Y22−4Y1Y3/2Y1r1−Y2−Y22−4Y1Y3/2Y1+q3+q1+γ.Theorem 19.
IfR0>1 or R0=1 and r1r2K1>bK2K3, then there exists E∗=P∗,S∗,Q∗ as the endemic equilibrium point, where P∗, S∗, and Q∗ are as (106)–(108). Also, If R0<1, then this point does not exist.Proof.
IfR0>1, then by the above discussions, E∗=P∗,S∗,Q∗ exists. If R0=1, then Y2=r1r2K1−bK2K3 and Y3=0. In this case, we have
(109)S=−Y2−Y22Y1.
On the other side, we know thatY1≠0. Hence, if Y2≤0, then S=0 and if Y2>0, then S<0. These show the existence of the endemic equilibrium point E∗.
On the other side,R0<1 gives −Y2>0 and Y3<0. Thus, either S is a complex number or a negative real number, and E∗ does not exist.In Figures5 and 6, we show the stability curves of P, S, and Q at endemic equilibrium point E∗ for some values of the fractal and fractional orders.Figure 5
The stability curves ofPt and St at E∗ for ω=0.97 and ν=0.5.
(a)(b)Figure 6
The stability curves ofQt at E∗ for ω=0.97 and ν=0.5.


## 7.4. Local Asymptotic Stability Analysis

In this place, we aim to investigate the local asymptotic stability of the smoke-free pointE0 found in (89) in relation to the fractal-fractional model (7) of second-hand smokers. First, we obtain the Jacobian matrix of the fractal-fractional system (7) as
(110)J=−s+q1+r−bS−bP0bSbP+r1Q−q1+q2+r2r1S0r2−r1Q−r1S−q3+q1+γ.Theorem 20.
The fractal-fractional model (7) of second-hand smokers is locally asymptotically stable at smoke-free equilibrium point E0 if all roots of the characteristic polynomial detJE0−λI=0 are negative or R0<1; otherwise, it is unstable.Proof.
In view of (110), the Jacobian matrix JE0 of the model (7) at smoke-free equilibrium point E0 is given as
(111)JE0=−s+q1+r−bθs+q1+r00bθs+q1+r−q1+q2+r200r2−q3+q1+γ.
By considering the Jacobian matrix (111) and by detJE0−λI=0, the characteristic polynomial becomes
(112)λ+s+q1+rλ+q3+q1+γλ−bθs+q1+r−q1+q2+r2=0.
The roots of the above polynomial are(113)λ1=−s+q1+r,λ2=−q3+q1+γ,λ3=bθs+q1+r−q1+q2+r2.
Since all parameters are assumed to be positive, we clearly haveλ1<0 and λ2<0. On the other hand, if R0<1, then R0−1<0. So, we can write
(114)bθs+q1+rq1+q2+r2<1.
Hence,(115)bθs+q1+r<q1+q2+r2.
Consequently,λ3=bθ/s+q1+r−q1+q2+r2<0. Thus, all of roots are negative, and so the fractal-fractional second-hand smoker model (7) is locally asymptotically stable around the smoke-free equilibrium point E0. This completes the proof.


## 7.5. Sensitivity Analysis

To find out how sensitive the parameters in each model are to the transmission of infections or diseases, the authors use a criterion introduced by Chitnis et al. [49]. Here, to obtain the sensitivity of R0 in terms of each of its parameters, we compute the partial derivative with respect to every parameter in the reproduction number. In other words, if p is an arbitrary parameter, then the sensitivity index of R0 versus p is calculated by
(116)SpR0=pR0∂R0∂p.Now, according to the above relation, we have(117)SbR0=bR0θq1+r2+q2q1+r+s>0,(118)SθR0=θR0bq1+r2+q2q1+r+s>0,(119)SsR0=sR0−bθq1+r2+q2q1+r+s2<0,(120)Sq1R0=q1R0−bθq1+q2+r2q1+r+s2−bθq1+r2+q22q1+r+s<0,(121)SrR0=rR0−bθq1+r2+q2q1+r+s2<0,(122)Sq2R0=q2R0−bθq1+r2+q22q1+r+s<0,(123)Sr2R0=r2R0−bθq1+r2+q22q1+r+s<0.The numeric values for sensitivity indices (120) are given in Table 1.Table 1
Sensitivity of theR0 versus proposed parameters.
ParameterSValueParameterSValuebSbR0+1.0000θSθR0+1.0000sSsR0-9.1621e-05q1Sq1R0-0.0061rSrR0-0.0366q2Sq2R0-0.0018r2Sr2R0-4.0813e-04In Figure7, when the sign of sensitivity index is positive, then each increase (decrease) in the value of parameters (assuming the remaining parameters to be constant) increases (decreases) the value of R0. The negative sign in this index gives the inverse result for R0. In the sensitivity analysis of our second-hand smoker model (7), it is observed that R0 increases by increasing the values of b,θ and decreases by increasing the values of s,q1,r,q2,r2. For example, Sq1R0=−0.0061 means that an increase in q1 by 10% decreases R0 by 0.061%. Therefore, to reduce the spread of infections, those parameters having negative sensitivity indices must be minimized in the environment.Figure 7
Global sensitivity analysis of model (7) based on R0 (96) and sensitivity analysis (120).


## 8. Numerical scheme

In this section, we describe the numerical scheme in relation to the fractal-fractional model of second-hand smoker (7). For this, we have taken help from the technique regarding two-step Lagrange polynomials known as the fractional Adams-Bashforth (AB) method [50]. To begin this process, we present the numerical method of fractal-fractional integral equation (16) using a new approach at tn+1. In other words, we discretize the mentioned equation (16) for t=tn+1, and we have
(124)Ptn+1=P0+νΓω∫0tn+1tn+1−wω−1H1wdw,Stn+1=S0+νΓω∫0tn+1tn+1−wω−1H2wdw,Qtn+1=Q0+νΓω∫0tn+1tn+1−wω−1H3wdw,where
(125)H1w=wν−1W1w,Pw,Sw,Qw,H2w=wν−1W2w,Pw,Sw,Qw,H3w=wν−1W3w,Pw,Sw,Qw.By approximating above integrals, we get(126)Ptn+1=P0+νΓω∑l=0n∫tltl+1tn+1−wω−1H1wdw,Stn+1=S0+νΓω∑l=0n∫tltl+1tn+1−wω−1H2wdw,Qtn+1=Q0+νΓω∑l=0n∫tltl+1tn+1−wω−1H3wdw.In the sequel, we approximate the functionsH1w,H2w,H3w introduced by (125), on the interval tl,tl+1 via two-step Lagrange interpolation polynomials with the step size h=tl−tl−1 as
(127)H1,l∗w≃w−tl−1htlν−1W1wl,Pl,Sl,Ql−w−tlhtl−1ν−1W1wl−1,Pl−1,Sl−1,Ql−1,H2,l∗w≃w−tl−1htlν−1W2wl,Pl,Sl,Ql−w−tlhtl−1ν−1W2wl−1,Pl−1,Sl−1,Ql−1,H3,l∗w≃w−tl−1htlν−1W3wl,Pl,Sl,Ql−w−tlhtl−1ν−1W3wl−1,Pl−1,Sl−1,Ql−1.Then, we have(128)Ptn+1=P0+νΓω∑l=0n∫tltl+1tn+1−wω−1H1,l∗wdw,Stn+1=S0+νΓω∑l=0n∫tltl+1tn+1−wω−1H2,l∗wdw,Qtn+1=Q0+νΓω∑l=0n∫tltl+1tn+1−wω−1H3,l∗wdw.By evaluating above integrals directly, the numerical solutions of the given fractal-fractional model of second-hand smoker (7) are given by
(129)Pn+1=P0+νhωΓω+2∑l=0ntlν−1W1tl,Pl,Sl,QlYn,l−tl−1ν−1W1tl−1,Pl−1,Sl−1,Ql−1Y^n,l,Sn+1=S0+νhωΓω+2∑l=0ntlν−1W2tl,Pl,Sl,QlYn,l−tl−1ν−1W2tl−1,Pl−1,Sl−1,Ql−1Y^n,l,Qn+1=Q0+νhωΓω+2∑l=0ntlν−1W3tl,Pl,Sl,QlYn,l−tl−1ν−1W3tl−1,Pl−1,Sl−1,Ql−1Y^n,l,where
(130)Yn,l=n+1−lωn−l+2+ω−n−lωn−l+2+2ω,Y^n,l=n+1−lω+1−n−lωn−l+1+ω,where ω is the fractional order of the given fractal-fractional system (7).


## 9. Simulations

In this section, we simulate and discuss the behavior of the model based on some parameters provided by [45]. According to this source, we take θ=0.3, s=0.01, q1=0.035, r=0.2, b=0.9001, r1=0.009, q2=0.01, r2=0.01, q3=0.1, and γ=0.1. Finally, the initial values for state functions are assumed as P0=11,S0=9,Q0=30. In different figures, we will show the behaviors of three state functions P,S,Q by assuming different values for the fractal order ν=1,0.9,0.8,0.7,0.6,0.5 and fractional orders ω=1.00,0.99,0.97,0.95,0.93,0.91.More precisely, in Figure8(a), we see the simulation for different fractional orders ω=1.00,0.99,0.97,0.95,0.93,0.91 and fractal order ν=0.99 of the function Pt in which the rate of population who addicts to tobacco increases with time t, while in Figure 8(b), the simulation of the function St for the same values of fractal dimension ν=0.99 and fractional orders ω=1.00,0.99,0.97,0.95,0.93,0.91 shows an increase of addiction to tobacco in the population St by increasing the time t.Figure 8
The simulation of the functionsPt and St during the time t for different fractional orders ω=1,0.99,0.97,0.95,0.93,0.91 and fractal dimension ν=0.99.
(a)(b)In Figure9(a), we see the simulation for fractional orders ω=1.00,0.99,0.97,0.95,0.93,0.91 and fractal order ν=0.99 of the function Qt in which the rate of population who quitting tobacco decreases, while in Figure 9(b), the simulation of the function Pt for different fractal orders ν=1,0.9,0.8,0.7,0.6,0.5 and fractional order ω=0.97 shows an increase in addiction to tobacco in the population at risk.Figure 9
(a) The simulation of the functionQt during the time t for different fractional orders ω=1,0.99,0.97,0.95,0.93,0.91 and fractal dimension ν=0.99. (b) The simulation of the function Pt during the time t for different fractal dimensions ν=1,0.9,0.8,0.7,0.6,0.5 and fractional order ω=0.97.
(a)(b)In Figure10, we see the simulation for different fractal orders ν=1,0.9,0.8,0.7,0.6,0.5 of the functions St and Qt for fractional order ω=0.97 in which the addiction to tobacco increases but the population quitting tobacco decreases.Figure 10
The simulation of the functionsSt and Qt during the time t for different fractal dimensions ν=1,0.9,0.8,0.7,0.6,0.5 and fractional order ω=0.97.
(a)(b)On the other hand, to see the effect of fractional order and fractal dimension simultaneously, we choose different values for these two parameters asω=ν=1,0.98,0.95,0.92. In Figure 11(a), the simulation of the function Pt and (Figure 11(b)) the simulation of the function St during the time t are plotted. We see that the change in the behavior of curves when the fractal dimension ν and fractional order ω decreases, then the slope of curves increases and finally, the curves reach to a stable status. We see similar behavior for the function Pt in Figure 12.Figure 11
The simulation of the functionsPt and St during the time t for fractal dimensions and fractional orders ω=ν=1,0.98,0.95,0.92.
(a)(b)Figure 12
The simulation of the functionQt during the time t for fractal dimensions and fractional orders ω=ν=1,0.98,0.95,0.92.


## 10. Conclusions

Due to the destructive effects of smoking and its resulting smoke on the health of the people in the community, in the present manuscript, we designed a mathematical model of secondhand smokers (SHS) based on three different compartments of the smoker and nonsmoker population and analyzed it numerically and analytically. To model this system, we applied new derivatives entitled the fractal-fractional derivatives with power-law-type kernel. The existence section was proved byϕ-ψ-contractions and compact operators, and the Banach principle for usual contractions was used for proving the uniqueness result. The stability notion was investigated for solutions of the fractal-fractional SHS-model (7). The steady-state analysis including calculation of equilibrium points and basic reproduction number R0 was done, and then we compared the sensitivity of the fractal-fractional SHS-system with each parameter. For numerical simulation, the Adams-Bashforth (AB) method was used and simulated the graphs with the help of real data. The effects of each parameter on the overall result of the calculations showed the increase or decrease in the harmful effect of cigarette smoke on people’s health. Along with these results, we see accurate and better simulations via the fractal-fractional operators. Our graphs and data showed that our analysis on real data for different fractal dimensions and fractional orders yield similar results in comparison to classical operators. In the next works, we can conduct comparative research on different fractal-fractional with exponential decay and power-law type kernels.



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*Source: 3553021-2022-05-10.xml*