# Mathematical Analysis and Optimal Control of Giving up the Smoking Model

Authors: Omar Khyar; Jaouad Danane; Karam Allali
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

In this study, we are going to explore mathematically the dynamics of giving up smoking behavior. For this purpose, we will perform a mathematical analysis of a smoking model and suggest some conditions to control this serious burden on public health. The model under consideration describes the interaction between the potential smokersP, the occasional smokers L, the chain smokers S, the temporarily quit smokers QT, and the permanently quit smokers QP. Existence, positivity, and boundedness of the proposed problem solutions are proved. Local stability of the equilibria is established by using Routh–Hurwitz conditions. Moreover, the global stability of the same equilibria is fulfilled through using suitable Lyapunov functionals. In order to study the optimal control of our problem, we will take into account a two controls’ strategy. The first control will represent the government prohibition of smoking in public areas which reduces the contact between nonsmokers and smokers, while the second will symbolize the educational campaigns and the increase of cigarette cost which prevents occasional smokers from becoming chain smokers. The existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to check the equilibria stability, confirm the theoretical findings, and show the role of optimal strategy in controlling the smoking severity.


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## Body

## 1. Introduction

Smoking is a behavior in which a substance is heated or burned, and the derived smoke is breathed through the mouth or the nose and goes directly to the lungs. The active substances of the smoke are rapidly transmitted from the lungs to the bloodstream and therefore to the brain where they operate. The most common smoking substance is known as tobacco which is consumed by nearly 1.3 billion smokers worldwide [1]. More than 9% of all deaths are related to tobacco consumption [2], making the burden of smoking a global health priority. Smoking is one of the main risk factors of many noncommunicable diseases such as diabetes, cancer, cardiovascular, and chronic respiratory diseases [3–8].Smoking dynamics were modelled by several authors in order to study the interaction between the model compartments. The simplest system that describes such dynamics was introduced in 1997 by Castillo-Garsow et al. [9], in which we find three main classes representing the potential smokers P, the chain smokers S, and the permanently quit smokers QP. The authors studied the local stability of the equilibria and suggest a nonlinear smoking relapse model. Later, in 2008, another compartment of temporarily quit smokers QT was added to the previous one [10]. Both the local and the global stability of the equilibria were performed depending on the smoking generation number. The stochastic study of the PSQTQP smoking model was tackled in [11]; the authors consider Brownian motion as perturbation of their equations and establish the sufficient condition for mean square stability. Recently, the class of occasional smokers L was taken into account in [12]. The authors studied the following PLSQP smoking model:(1)dPdt=Λ−βIRP,L−δ+μP,dLdt=βIRP,L−σ+δ+μL,dSdt=σL−γ+δ+μS,dQPdt=γS−δ+μQP,where Λ is the recruitment rate, β is the transmission rate of potential smokers to occasional smoking class, σ denotes the rate of becoming chain smoker for a occasional smoker, and γ is the rate of becoming quit smoker. Furthermore, the natural death rate of all of the subpopulations is denoted by μ and their death rate related to smoking behavior is denoted by δ. Here, the incidence rate is given by IRP,L:=2PL/P+L. We note that for square-root incidence function IRP,L:=PL, and the PLSQP smoking problem is tackled by [13]; while with bilinear incidence function IRP,L:=PL, the PLSQTQP smoking problem is studied in [14].In this paper and inspired by the previous works, we are interested in studyingPLSQTQP smoking problem by taking into consideration the saturated incidence rate IRP,L:=2PL/P+L; indeed, this saturated rate describes the crowding effect of potential and occasional smokers in smoking areas. The problem under consideration will take the following form:(2)dPdt=Λ−β2PLP+L−δ+μP,dLdt=β2PLP+L−σL−δ+μL,dSdt=σL−γS−δ+μS+αQT,dQTdt=γ1−ηS−αQT−δ+μQT,dQPdt=γηS−δ+μQP.The reversion to the smoking phenomenon is highlighted in this model by the parameterα which represents the ratio of temporarily quit smokers who start to smoke again, while η is the portion of quitters who quit smoking permanently. The flowchart of our suggested PLSQTQP smoking model is illustrated in Figure 1. Our main interest is to study the equilibria stability of model (2) and to study the role of the optimal control strategy in reducing the number of smokers. For this last reason, we will introduce two controls; the first will represent the government prohibition of smoking in public areas, while the second control will symbolize the educational campaigns and the increase of cigarette cost. It will worthy to notice that the optimal control strategy have shown its efficiency in controlling many diseases such as tuberculosis and COVID-19 [15, 16]. Also, the optimal control has been studied in some fractional mathematical models [17]. Finally, as applications of many fractional derivatives on smoking modelling issues, we can cite [18–23].Figure 1
The flowchart of thePLSQTQP smoking model.Different studies on smoking [16] have shown the clear impact of tobacco consumption on the rapid deterioration of health in COVID-19 patients. In fact, smokers are more vulnerable to COVID-19 because they already have respiratory problems, which considerably increase the risk that their condition worsens more quickly than nonsmokers. Hence, our main ambition is to check the role of governement of prohibition of smoking in public area, education campaign, and the increase of cigarette cost in controlling the severity of the smokers’ addiction.This paper is organized as follows. In Section2, we will establish the wellposedness of our suggested PLSQTQP mathematical model by proving the existence, positivity, and boundedness of solutions. We fulfilled the mathematical analysis of the model, by proving the local and global stability of the equilibria in Sections 3 and 4. The optimization analysis of our model is given in Section 5; in Section 6 an appropriate numerical algorithm is constructed and some numerical simulations are given. The numerical results will show the importance of the optimal control in reducing the smokers’ addiction. In Section 7, we will give concluding remarks.


## 2. The Wellposedness Result and Smokers’ Generation Number

### 2.1. The Wellposedness Result

LetC=Cℝ+,ℝ5 be the Banach space of continuous functions mapping ℝ+ into ℝ5 with the topology of uniform convergence. By using the fundamental theory of functional differential equations [24], we prove that there exists a unique local solution Pt,Lt,St,QTt,QPt of system (2) with initial data P0,L0,S0,QT0,QP0∈C. The following proposition guarantees that this solution exists globally.Proposition 1.
The solution of the studied model (2) remains positive and bounded for all t≥0. Moreover, we have(3)Nt≤Λδ+μ+N0,whereNt=Pt+Lt+St+QTt+QPt.Proof. The solutionPt is nonnegative, for all t≥0. Supposing the contrary, let t0>0 be the first time such that Pt0=0 and P˙t0≤0. From the first equation of system (2), we have P˙t0=Λ>0, which leads a contradiction. Similar proofs show the positivity of Lt, St, QTt, and QPt, for all t≥0. Indeed, according to system (2), we have L˙|L=0=0≥0, S˙|S=0=σL+αQT≥0, QT.|QT=0=γ1−ηS≥0, and QP.|QP=0=γηS≥0. This completes the proof of positivity.About boundedness, let the total populationNt=Pt+Lt+St+QTt+QPt.From system (2), we have(4)dNtdt=Λ−δ+μNt.Therefore,(5)Nt=Λδ+μ+Ke−δ+μt.Att=0, we have(6)N0=Λδ+μ+K.Then,(7)Nt=Λδ+μ+N0−Λδ+μe−δ+μt.Consequently,(8)Nt≤Λδ+μ+N0e−δ+μt.Since0<e−δ+μt≤1, for all t≥0, we conclude that Nt≤Λ/δ+μ+N0.


### 2.2. The Smokers’ Generation Number

The smokers’ generation number is defined as the average number of new smokers resulting from one smoker, in a population consisting of potential smokers only. We use the next generation matrixFV−1 to calculate the smokers’ generation number R0 [25].The formula that gives us the smokers’ generation number isR0=ρFV−1, where ρ stands for the spectral radius, F is the nonnegative matrix of new smokers, and V is the matrix of smoking transition associated with model (2). After a simple resolution, we obtain(9)R0=2βσ+δ+μ.


## 2.1. The Wellposedness Result

LetC=Cℝ+,ℝ5 be the Banach space of continuous functions mapping ℝ+ into ℝ5 with the topology of uniform convergence. By using the fundamental theory of functional differential equations [24], we prove that there exists a unique local solution Pt,Lt,St,QTt,QPt of system (2) with initial data P0,L0,S0,QT0,QP0∈C. The following proposition guarantees that this solution exists globally.Proposition 1.
The solution of the studied model (2) remains positive and bounded for all t≥0. Moreover, we have(3)Nt≤Λδ+μ+N0,whereNt=Pt+Lt+St+QTt+QPt.Proof. The solutionPt is nonnegative, for all t≥0. Supposing the contrary, let t0>0 be the first time such that Pt0=0 and P˙t0≤0. From the first equation of system (2), we have P˙t0=Λ>0, which leads a contradiction. Similar proofs show the positivity of Lt, St, QTt, and QPt, for all t≥0. Indeed, according to system (2), we have L˙|L=0=0≥0, S˙|S=0=σL+αQT≥0, QT.|QT=0=γ1−ηS≥0, and QP.|QP=0=γηS≥0. This completes the proof of positivity.About boundedness, let the total populationNt=Pt+Lt+St+QTt+QPt.From system (2), we have(4)dNtdt=Λ−δ+μNt.Therefore,(5)Nt=Λδ+μ+Ke−δ+μt.Att=0, we have(6)N0=Λδ+μ+K.Then,(7)Nt=Λδ+μ+N0−Λδ+μe−δ+μt.Consequently,(8)Nt≤Λδ+μ+N0e−δ+μt.Since0<e−δ+μt≤1, for all t≥0, we conclude that Nt≤Λ/δ+μ+N0.


## 2.2. The Smokers’ Generation Number

The smokers’ generation number is defined as the average number of new smokers resulting from one smoker, in a population consisting of potential smokers only. We use the next generation matrixFV−1 to calculate the smokers’ generation number R0 [25].The formula that gives us the smokers’ generation number isR0=ρFV−1, where ρ stands for the spectral radius, F is the nonnegative matrix of new smokers, and V is the matrix of smoking transition associated with model (2). After a simple resolution, we obtain(9)R0=2βσ+δ+μ.


## 3. The Problem Equilibria and Local Stability

### 3.1. The Smoking Problem Equilibria

It is straightforward to check that model (2) has two steady states developed as follows:(i)
The smoking-free equilibrium state isE0Λ/δ+μ,0,0,0,0.(ii)
The smoking-present equilibrium point isE∗P∗,L∗,S∗,QT∗,QP∗ such that(10)P∗=ΛR0βR0−1+δ+μR0,L∗=R0−1P∗,S∗=σδ+μ+αR0−1γ+δ+μα+δ+μ−αγ1−ηP∗,QT∗=σγ1−ηR0−1γ+δ+μα+δ+μ−αγ1−ηP∗,QP∗=σγηδ+μ+αR0−1δ+μγ+δ+μα+δ+μ−αγ1−ηP∗.


#### 3.1.1. Local Stability of the Smoking-Free Equilibrium

The local stability of the smoking-free equilibriumE0Λ/δ+μ,0,0,0,0 is given by the following result.Proposition 2.
(1)
IfR0<1, then the smoking-free equilibrium, E0, is locally asymptotically stable.(2)
IfR0>1, then E0 is unstable.Proof. The Jacobian matrix of system (2) at E0 is given by(11)JE0=δ+μ−2β00002β−σ+δ+μ0000σ−γ+δ+μα000γ1−η−α+δ+μ000γη0−δ+μ.The characteristic polynomial ofJE0 is(12)PE0λ=σ+δ+μδ+μ+λ2R0−1−λαγ1−η−γ+δ+μα+δ+μ+γ+δ+μα+δ+μλ−λ2.Therefore, the eigenvalues ofJE0 are given as follows:(13)λ1=−δ+μ<0,λ2=2β−σ−δ−μ,λ3=R0−1,λ4=−γ+δ+μα+δ+μ−Δ2<0,λ5=−γ+δ+μα+δ+μ+Δ2<0,with Δ=γ+δ+μα+δ+μ2−4γ+δ+μα+δ+μ−αγ1−η. It is clear that λ2<0 and λ3<0 if R0<1.We conclude thatE0 is locally asymptotically stable when R0<1.


#### 3.1.2. Local Stability of the Smoking-Present Equilibrium

In order to study the local stability of the smoking-present equilibriumE∗P∗,L∗,S∗,QT∗,QP∗, let us first give the Jacobian matrix of system (2) at E∗:(14)JE∗=−2β1−1R02−δ+μ−2βR020002β1−1R022βR02−σ+δ+μ0000σ−γ+δ+μα000γ1−η−α+δ+μ000γη0−δ+μ.The characteristic polynomial ofJE∗ is(15)PE∗ξ=δ+μ+ξ2βq2+δ+μ+ξξ3+a1ξ2+a2ξ+a3,with(16)a1=2βq2+δ+μσ+δ+μ−2βδ+μp2+δ+μ+α+δ+μ,a2=2βq2+δ+μσ+δ+μ−2βδ+μp2δ+μ+α+δ+μ+δ+μα+δ+μ−αγ1−η,a3=2βq2+δ+μσ+δ+μ−2βδ+μp2δ+μα+δ+μ−αγ1−η,such that p=1/R0 and q=1−1/R0 (remark that p+q=1).The first eigenvalue of the above characteristic polynomial isξ1=−δ+μ<0; the second eigenvalue is ξ2=−2βq2+δ+μ<0.IfR0>1, then 2βq2+δ+μσ+δ+μ−2βδ+μp2>0; consequently, a1>0.Thus, it is easy to verify thata2>0, a1a2−a3>0, and a3>0. Then, by application of Routh–Hurwitz theorem, the other eigenvalues, ξ3, ξ4, and ξ5, have negative real parts.We conclude that E∗ is locally asymptotically stable in case when R0>1.


## 3.1. The Smoking Problem Equilibria

It is straightforward to check that model (2) has two steady states developed as follows:(i)
The smoking-free equilibrium state isE0Λ/δ+μ,0,0,0,0.(ii)
The smoking-present equilibrium point isE∗P∗,L∗,S∗,QT∗,QP∗ such that(10)P∗=ΛR0βR0−1+δ+μR0,L∗=R0−1P∗,S∗=σδ+μ+αR0−1γ+δ+μα+δ+μ−αγ1−ηP∗,QT∗=σγ1−ηR0−1γ+δ+μα+δ+μ−αγ1−ηP∗,QP∗=σγηδ+μ+αR0−1δ+μγ+δ+μα+δ+μ−αγ1−ηP∗.


### 3.1.1. Local Stability of the Smoking-Free Equilibrium

The local stability of the smoking-free equilibriumE0Λ/δ+μ,0,0,0,0 is given by the following result.Proposition 2.
(1)
IfR0<1, then the smoking-free equilibrium, E0, is locally asymptotically stable.(2)
IfR0>1, then E0 is unstable.Proof. The Jacobian matrix of system (2) at E0 is given by(11)JE0=δ+μ−2β00002β−σ+δ+μ0000σ−γ+δ+μα000γ1−η−α+δ+μ000γη0−δ+μ.The characteristic polynomial ofJE0 is(12)PE0λ=σ+δ+μδ+μ+λ2R0−1−λαγ1−η−γ+δ+μα+δ+μ+γ+δ+μα+δ+μλ−λ2.Therefore, the eigenvalues ofJE0 are given as follows:(13)λ1=−δ+μ<0,λ2=2β−σ−δ−μ,λ3=R0−1,λ4=−γ+δ+μα+δ+μ−Δ2<0,λ5=−γ+δ+μα+δ+μ+Δ2<0,with Δ=γ+δ+μα+δ+μ2−4γ+δ+μα+δ+μ−αγ1−η. It is clear that λ2<0 and λ3<0 if R0<1.We conclude thatE0 is locally asymptotically stable when R0<1.


### 3.1.2. Local Stability of the Smoking-Present Equilibrium

In order to study the local stability of the smoking-present equilibriumE∗P∗,L∗,S∗,QT∗,QP∗, let us first give the Jacobian matrix of system (2) at E∗:(14)JE∗=−2β1−1R02−δ+μ−2βR020002β1−1R022βR02−σ+δ+μ0000σ−γ+δ+μα000γ1−η−α+δ+μ000γη0−δ+μ.The characteristic polynomial ofJE∗ is(15)PE∗ξ=δ+μ+ξ2βq2+δ+μ+ξξ3+a1ξ2+a2ξ+a3,with(16)a1=2βq2+δ+μσ+δ+μ−2βδ+μp2+δ+μ+α+δ+μ,a2=2βq2+δ+μσ+δ+μ−2βδ+μp2δ+μ+α+δ+μ+δ+μα+δ+μ−αγ1−η,a3=2βq2+δ+μσ+δ+μ−2βδ+μp2δ+μα+δ+μ−αγ1−η,such that p=1/R0 and q=1−1/R0 (remark that p+q=1).The first eigenvalue of the above characteristic polynomial isξ1=−δ+μ<0; the second eigenvalue is ξ2=−2βq2+δ+μ<0.IfR0>1, then 2βq2+δ+μσ+δ+μ−2βδ+μp2>0; consequently, a1>0.Thus, it is easy to verify thata2>0, a1a2−a3>0, and a3>0. Then, by application of Routh–Hurwitz theorem, the other eigenvalues, ξ3, ξ4, and ξ5, have negative real parts.We conclude that E∗ is locally asymptotically stable in case when R0>1.


## 3.1.1. Local Stability of the Smoking-Free Equilibrium

The local stability of the smoking-free equilibriumE0Λ/δ+μ,0,0,0,0 is given by the following result.Proposition 2.
(1)
IfR0<1, then the smoking-free equilibrium, E0, is locally asymptotically stable.(2)
IfR0>1, then E0 is unstable.Proof. The Jacobian matrix of system (2) at E0 is given by(11)JE0=δ+μ−2β00002β−σ+δ+μ0000σ−γ+δ+μα000γ1−η−α+δ+μ000γη0−δ+μ.The characteristic polynomial ofJE0 is(12)PE0λ=σ+δ+μδ+μ+λ2R0−1−λαγ1−η−γ+δ+μα+δ+μ+γ+δ+μα+δ+μλ−λ2.Therefore, the eigenvalues ofJE0 are given as follows:(13)λ1=−δ+μ<0,λ2=2β−σ−δ−μ,λ3=R0−1,λ4=−γ+δ+μα+δ+μ−Δ2<0,λ5=−γ+δ+μα+δ+μ+Δ2<0,with Δ=γ+δ+μα+δ+μ2−4γ+δ+μα+δ+μ−αγ1−η. It is clear that λ2<0 and λ3<0 if R0<1.We conclude thatE0 is locally asymptotically stable when R0<1.


## 3.1.2. Local Stability of the Smoking-Present Equilibrium

In order to study the local stability of the smoking-present equilibriumE∗P∗,L∗,S∗,QT∗,QP∗, let us first give the Jacobian matrix of system (2) at E∗:(14)JE∗=−2β1−1R02−δ+μ−2βR020002β1−1R022βR02−σ+δ+μ0000σ−γ+δ+μα000γ1−η−α+δ+μ000γη0−δ+μ.The characteristic polynomial ofJE∗ is(15)PE∗ξ=δ+μ+ξ2βq2+δ+μ+ξξ3+a1ξ2+a2ξ+a3,with(16)a1=2βq2+δ+μσ+δ+μ−2βδ+μp2+δ+μ+α+δ+μ,a2=2βq2+δ+μσ+δ+μ−2βδ+μp2δ+μ+α+δ+μ+δ+μα+δ+μ−αγ1−η,a3=2βq2+δ+μσ+δ+μ−2βδ+μp2δ+μα+δ+μ−αγ1−η,such that p=1/R0 and q=1−1/R0 (remark that p+q=1).The first eigenvalue of the above characteristic polynomial isξ1=−δ+μ<0; the second eigenvalue is ξ2=−2βq2+δ+μ<0.IfR0>1, then 2βq2+δ+μσ+δ+μ−2βδ+μp2>0; consequently, a1>0.Thus, it is easy to verify thata2>0, a1a2−a3>0, and a3>0. Then, by application of Routh–Hurwitz theorem, the other eigenvalues, ξ3, ξ4, and ξ5, have negative real parts.We conclude that E∗ is locally asymptotically stable in case when R0>1.


## 4. The Smoking Problem Global Stability Analysis

### 4.1. Global Stability of the Smoking-Free Equilibrium

Theorem 1.
IfR0≤1, then the smoking-free equilibrium E0 is globally asymptotically stable.Proof. First, we denoted=δ+μ, Pf=Λ/δ+μ, and we consider the following Lyapunov function defined in ℝ+5 by(17)V0P,L,S,QT,QP=12P+L+S+QT+QP−Pf2+L.The time derivative is given by(18)V˙0P,L,S,QT,QP=P−Pf+L+S+QT+QPΛ−dP−dL−dS−dQT−dQP+2βLPP+L−σ+dL,=P−Pf+L+S+QT+QPdPf−dP−dL−dS−dQT−dQP+2βLPP+L−σ+dL,≤−dP+L+S+QT+QP−Pf2+2βL−σ+dL,≤−dP+L+S+QT+QP−Pf2+σ+dR0−1L.IfR0≤1, then V˙0≤0; consequently, according to the Lyapunov theorem, the smoking-free equilibrium E0 is globally asymptotically stable.


### 4.2. Global Stability of the Smoking-Present Equilibrium

Theorem 2.
IfR0>1, then the smoking-present equilibrium point E∗ is globally asymptotically stable.Proof. We consider the following Lyapunov function defined inℝ+5 as follows:(19)V1P,L,S,QT,QP=12P−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2+L∗P−P∗−P∗lnPP∗+P∗L−L∗−L∗lnLL∗.The time derivative is given by(20)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2+L∗1−P∗PP˙+P∗1−L∗LL˙.It is easy to verify that(21)P∗+L∗=R0P∗,L∗P∗=R0−1.So,(22)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2+L∗P−P∗PΛ−2β2PLP+L−dP+P∗L−L∗2β2PLP+L−2βP∗P∗+L∗.Then,(23)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2−dL∗P∗2P+2dL∗P∗−dL∗P−2βLL∗2P−P∗2PP∗+L∗P+L+2βP∗L−L∗P+LP∗+L∗L∗P−P∗L∗−2PL∗−P∗L.Therefore,(24)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2−dP∗2R0−1PP∗+P∗P−2−2βLL∗P−P∗2R0PP∗+L∗P+LR0−1−2βP∗2P+LP∗+L∗L−L∗2.By application of the relation between geometric and arithmetic means, we obtain(25)PP∗+P∗P−2≥0.SinceR0>1, we have that R0−1>0. Consequently, V˙1<0.We conclude that the smoking-present equilibrium pointE∗ is globally asymptotically stable when R0>1.


## 4.1. Global Stability of the Smoking-Free Equilibrium

Theorem 1.
IfR0≤1, then the smoking-free equilibrium E0 is globally asymptotically stable.Proof. First, we denoted=δ+μ, Pf=Λ/δ+μ, and we consider the following Lyapunov function defined in ℝ+5 by(17)V0P,L,S,QT,QP=12P+L+S+QT+QP−Pf2+L.The time derivative is given by(18)V˙0P,L,S,QT,QP=P−Pf+L+S+QT+QPΛ−dP−dL−dS−dQT−dQP+2βLPP+L−σ+dL,=P−Pf+L+S+QT+QPdPf−dP−dL−dS−dQT−dQP+2βLPP+L−σ+dL,≤−dP+L+S+QT+QP−Pf2+2βL−σ+dL,≤−dP+L+S+QT+QP−Pf2+σ+dR0−1L.IfR0≤1, then V˙0≤0; consequently, according to the Lyapunov theorem, the smoking-free equilibrium E0 is globally asymptotically stable.


## 4.2. Global Stability of the Smoking-Present Equilibrium

Theorem 2.
IfR0>1, then the smoking-present equilibrium point E∗ is globally asymptotically stable.Proof. We consider the following Lyapunov function defined inℝ+5 as follows:(19)V1P,L,S,QT,QP=12P−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2+L∗P−P∗−P∗lnPP∗+P∗L−L∗−L∗lnLL∗.The time derivative is given by(20)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2+L∗1−P∗PP˙+P∗1−L∗LL˙.It is easy to verify that(21)P∗+L∗=R0P∗,L∗P∗=R0−1.So,(22)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2+L∗P−P∗PΛ−2β2PLP+L−dP+P∗L−L∗2β2PLP+L−2βP∗P∗+L∗.Then,(23)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2−dL∗P∗2P+2dL∗P∗−dL∗P−2βLL∗2P−P∗2PP∗+L∗P+L+2βP∗L−L∗P+LP∗+L∗L∗P−P∗L∗−2PL∗−P∗L.Therefore,(24)V˙1P,L,S,QT,QP=−dP−P∗+L−L∗+S−S∗+QT−QT∗+QP−QP∗2−dP∗2R0−1PP∗+P∗P−2−2βLL∗P−P∗2R0PP∗+L∗P+LR0−1−2βP∗2P+LP∗+L∗L−L∗2.By application of the relation between geometric and arithmetic means, we obtain(25)PP∗+P∗P−2≥0.SinceR0>1, we have that R0−1>0. Consequently, V˙1<0.We conclude that the smoking-present equilibrium pointE∗ is globally asymptotically stable when R0>1.


## 5. Optimal Control of Smoking Problem

### 5.1. The Smoking Optimization Problem

In order to formulate our smoking optimization problem, we will introduce to problem (2) two controls u1 and u2. The smoking problem with controls becomes(26)dPdt=Λ−β1−u12PLP+L−δ+μP,dLdt=β2PLP+L−σ+δ+μL,dSdt=σ1−u2L−γ+δ+μS+αQT,dQTdt=γ1−ηS−α+δ+μQT,dQPdt=γηS−δ+μQP,with P0≥0, L0≥0, S0≥0, QT0≥0, and QP0≥0.The first controlu1 represents the government prohibition of smoking in public area, while the second control u2 symbolizes the education campaign and the increase of cigarette cost. The aim of the first control is to reduce the contact between nonsmokers and smokers; the role of the second control is to prevent occasional smokers to become chain smokers. Since the two controls represent the effectiveness of each strategy, we will have the fact that ui∈0,1.The smoking optimization problem consists in maximizing the objective functional defined as follows:(27)Ju1,u2=∫0tfPt−C12u12t+C22u22tdt,where tf represents the time needed for the undertaken control measures and the two positive constants C1 and C2 are based on the costs of each strategy u1t and u2t, respectively. The interest is to maximize the objective functional (27) which would mean that we want to increase the nonsmokers’ number and to decrease the cost of each strategy. More precisely, we are looking for optimal controls u1∗,u2∗ such that(28)Ju1∗,u2∗=maxJu1,u2:u1,u2∈U,where U is the control set defined by(29)U=uitmeasurable,0≤uit≤1,t∈0,tf,withi=1,2.


### 5.2. The Smoking Optimality System

Pontryagin’s minimum principle [26] transforms our optimization problem into a Hamiltonian maximization problem, where the Hamiltonian ℋ is given by ℋP,L,S,QT,QP,u1,u2,ϕ1,ϕ2,ϕ3,ϕ4,ϕ5=C1/2u12+C2/2u22−P+∑i=15ϕifi with(30)f1=Λ−β1−u12PL/P+L−δ+μP,f2=β1−u12PLP+L−σ+δ+μL,f3=σ1−u2L−γ+δ+μS+αQT,f4=γ1−ηS−α+δ+μQT,f5=γηS−δ+μQP.The optimality system is summarized by the following result.Proposition 3.
For any solutionP∗,L∗,S∗,QT∗,QP∗ of the smoking optimization problem, there exist the adjoint variables, ϕ1,ϕ2,ϕ3,ϕ4, and ϕ5, satisfying the system of equations(31)ϕ1′t=1+ϕ1tδ+μ+β1−u12L∗2P∗+L∗2ϕ1t−ϕ2t,ϕ2′t=β1−u12P∗2P∗+L∗2ϕ1t−ϕ2t+σ+δ+μϕ2t−σ1−u2ϕ3t,ϕ3′t=γ+δ+μϕ3t−γ1−ηϕ4t−γηϕ5t,ϕ4′t=−αϕ3t+α+δ+μϕ4t,ϕ5′t=δ+μϕ5t,under the transversality conditions,(32)ϕitf=0,i=1,…,5.
In addition, the optimal control is given by(33)u1∗=min1,max0,1C12βP∗L∗P∗+L∗ϕ2t−ϕ1t,u2∗=min1,max0,1C2σL∗tϕ3t.Proof.
The five adjoint equations and the transversality conditions can be obtained via Pontryagin’s minimum principle [26] such that(34)ϕ1′t=−∂ℋ∂Pt,ϕ1tf=0,ϕ2′t=−∂ℋ∂Lt,ϕ2tf=0,ϕ3′t=−∂ℋ∂St,ϕ3tf=0,ϕ4′t=−∂ℋ∂QTt,ϕ4tf=0,ϕ5′t=−∂ℋ∂QPt,ϕ5tf=0.
Simple calculations lead to(35)ϕ1′t=1+ϕ1tδ+μ+β1−u12L∗2P∗+L∗2ϕ1t−ϕ2t,ϕ2′t=β1−u12P∗2P∗+L∗2ϕ1t−ϕ2t+σ+δ+μ2t−σ1−u2ϕ3t,ϕ3′t=γ+δ+μϕ3t−γ1−ηϕ4t−γηϕ5t,ϕ4′t=−αϕ3t+α+δ+μϕ4t,ϕ5′t=δ+μϕ5t.
The two optimal controlsu1∗ and u2∗ can be found from the following optimality conditions:(36)∂ℋ∂u1t=0,∂ℋ∂u2t=0.
However, we have(37)∂ℋ∂u1t=C1u1t+2βPLP+Lϕ1t−ϕ2t,∂ℋ∂u2t=C2u2t−σϕ3tLt.
Since the two controls belong to the interval0,1, we will have(38)u1∗=min1,max0,1C12βP∗L∗P∗+L∗ϕ2t−ϕ1t,u2∗=min1,max0,1C2σL∗tϕ3t.
In addition, we have(39)ϕitf=0,i=1,…,5.


## 5.1. The Smoking Optimization Problem

In order to formulate our smoking optimization problem, we will introduce to problem (2) two controls u1 and u2. The smoking problem with controls becomes(26)dPdt=Λ−β1−u12PLP+L−δ+μP,dLdt=β2PLP+L−σ+δ+μL,dSdt=σ1−u2L−γ+δ+μS+αQT,dQTdt=γ1−ηS−α+δ+μQT,dQPdt=γηS−δ+μQP,with P0≥0, L0≥0, S0≥0, QT0≥0, and QP0≥0.The first controlu1 represents the government prohibition of smoking in public area, while the second control u2 symbolizes the education campaign and the increase of cigarette cost. The aim of the first control is to reduce the contact between nonsmokers and smokers; the role of the second control is to prevent occasional smokers to become chain smokers. Since the two controls represent the effectiveness of each strategy, we will have the fact that ui∈0,1.The smoking optimization problem consists in maximizing the objective functional defined as follows:(27)Ju1,u2=∫0tfPt−C12u12t+C22u22tdt,where tf represents the time needed for the undertaken control measures and the two positive constants C1 and C2 are based on the costs of each strategy u1t and u2t, respectively. The interest is to maximize the objective functional (27) which would mean that we want to increase the nonsmokers’ number and to decrease the cost of each strategy. More precisely, we are looking for optimal controls u1∗,u2∗ such that(28)Ju1∗,u2∗=maxJu1,u2:u1,u2∈U,where U is the control set defined by(29)U=uitmeasurable,0≤uit≤1,t∈0,tf,withi=1,2.


## 5.2. The Smoking Optimality System

Pontryagin’s minimum principle [26] transforms our optimization problem into a Hamiltonian maximization problem, where the Hamiltonian ℋ is given by ℋP,L,S,QT,QP,u1,u2,ϕ1,ϕ2,ϕ3,ϕ4,ϕ5=C1/2u12+C2/2u22−P+∑i=15ϕifi with(30)f1=Λ−β1−u12PL/P+L−δ+μP,f2=β1−u12PLP+L−σ+δ+μL,f3=σ1−u2L−γ+δ+μS+αQT,f4=γ1−ηS−α+δ+μQT,f5=γηS−δ+μQP.The optimality system is summarized by the following result.Proposition 3.
For any solutionP∗,L∗,S∗,QT∗,QP∗ of the smoking optimization problem, there exist the adjoint variables, ϕ1,ϕ2,ϕ3,ϕ4, and ϕ5, satisfying the system of equations(31)ϕ1′t=1+ϕ1tδ+μ+β1−u12L∗2P∗+L∗2ϕ1t−ϕ2t,ϕ2′t=β1−u12P∗2P∗+L∗2ϕ1t−ϕ2t+σ+δ+μϕ2t−σ1−u2ϕ3t,ϕ3′t=γ+δ+μϕ3t−γ1−ηϕ4t−γηϕ5t,ϕ4′t=−αϕ3t+α+δ+μϕ4t,ϕ5′t=δ+μϕ5t,under the transversality conditions,(32)ϕitf=0,i=1,…,5.
In addition, the optimal control is given by(33)u1∗=min1,max0,1C12βP∗L∗P∗+L∗ϕ2t−ϕ1t,u2∗=min1,max0,1C2σL∗tϕ3t.Proof.
The five adjoint equations and the transversality conditions can be obtained via Pontryagin’s minimum principle [26] such that(34)ϕ1′t=−∂ℋ∂Pt,ϕ1tf=0,ϕ2′t=−∂ℋ∂Lt,ϕ2tf=0,ϕ3′t=−∂ℋ∂St,ϕ3tf=0,ϕ4′t=−∂ℋ∂QTt,ϕ4tf=0,ϕ5′t=−∂ℋ∂QPt,ϕ5tf=0.
Simple calculations lead to(35)ϕ1′t=1+ϕ1tδ+μ+β1−u12L∗2P∗+L∗2ϕ1t−ϕ2t,ϕ2′t=β1−u12P∗2P∗+L∗2ϕ1t−ϕ2t+σ+δ+μ2t−σ1−u2ϕ3t,ϕ3′t=γ+δ+μϕ3t−γ1−ηϕ4t−γηϕ5t,ϕ4′t=−αϕ3t+α+δ+μϕ4t,ϕ5′t=δ+μϕ5t.
The two optimal controlsu1∗ and u2∗ can be found from the following optimality conditions:(36)∂ℋ∂u1t=0,∂ℋ∂u2t=0.
However, we have(37)∂ℋ∂u1t=C1u1t+2βPLP+Lϕ1t−ϕ2t,∂ℋ∂u2t=C2u2t−σϕ3tLt.
Since the two controls belong to the interval0,1, we will have(38)u1∗=min1,max0,1C12βP∗L∗P∗+L∗ϕ2t−ϕ1t,u2∗=min1,max0,1C2σL∗tϕ3t.
In addition, we have(39)ϕitf=0,i=1,…,5.


## 6. Numerical Simulations

In order to solve numerically the optimization system, we will use the classical discretized scheme based on forward and backward finite-difference approximation scheme [28–30]. The numerical method will be implemented under Algorithm 1. The parameters values of our numerical simulations are given in Table 1.Algorithm 1:
Step 1 (initial conditions):P0, L0, S0, QT0, QP0, u10=0, u20=0.ϕ1n=0,ϕ2n=0,ϕ3n=0,ϕ4n=0,ϕ5n=0.Step 2 (numerical solving):fori=0,…,n−1, do:Pi+1=Pi+hΛ−β1−u1i2PiLi/Pi+Li−δ+μPi,Li+1=Li+hβ1−u1i2PiLi/Pi+Li−σ+δ+μLi,Si+1=Si+hσ1−u2iLi−γ+δ+μSi+αQTi,QTi+1=QTi+hγ1−ηSi−α+δ+μQTi,QPi+1=QPi+hγηSi−δ+μQPi,ϕ1n−i−1=ϕ1n−i−h1+ϕ1n−iδ+μ+β1−u12L∗2/P∗+L∗2ϕ1n−i−ϕ2n−iϕ2n−i−1=ϕ2n−i−hβ1−u12P∗2/P∗+L∗2ϕ1n−i−ϕ2n−i+σ+δ+μϕ2n−i−σ1−u2ϕ3n−i,ϕ3n−i−1=ϕ3n−i−hγ+δ+μϕ3n−i−γ1−ηϕ4n−i−γηϕ5n−i,ϕ4n−i−1=ϕ4n−i−h−αϕ3n−i+α+δ+μϕ4n−i,ϕ5n−i−1=ϕ5n−i−hδ+μϕ5n−i,R1i+1=1/C12βPi+1Li+1/Pi+1+Li+1ϕ2n−i−1−ϕ1n−i−1R2i+1=1/C2σLi+1ϕ3n−i−1,u1i+1=min1,maxR1i+1,0,u2i+1=min1,maxR2i+1,0,end forStep 3 (the solution and optimal controls):fori=1,…,n, writeP∗ti=Pi,L∗ti=Li,S∗ti=Si,QT∗ti=QTi,QP∗ti=QPi,u1∗ti=u1i,u2∗ti=u2i.end forTable 1
The estimation of the problem parameters.
ParametersFigure2Figure3Figure4ReferenceΛ0.250.250.25[12, 27]β0.0060.0320.032[12, 27]μ0.080.01110.0111[12, 27]δ4×10−50.00190.0019[12, 27]σ0.020.0210.021[12, 27]γ2.74×10−40.0410.041[12, 27]η0.80.60.6Estimatedα0.00140.0140.014EstimatedThe smoking-free equilibrium stability result is illustrated numerically in Figure2. In fact, we clearly see that the number of potential smokers increase and reach its maximum value. However, the other components of our smoking problem representing occasional smokers, chain smokers, temporarily quit smokers, and permanently quit smokers vanish. Within the used parameters in this figure (see Table 1), we can easily calculate the smokers’ generation number, R0=0.119952, which means that this number is less than unity, and we have confirmation, from our theoretical findings, that the smoking-free equilibrium is stable. Moreover, we observe that all the curves converge toward the smoking-free equilibrium Ef=3.12,0,0,0,0 which means that our numerical result supports the theoretical result concerning the stability of the first steady state.Figure 2
The evolution of the population whenR0=0.059.On the contrary, the smoking-present equilibrium stability is represented in Figure3. Here, all the smoking problem compartments remain at strictly positive levels. From Table 1, we can calculate the smokers’ generation number relative to this figure, R0=1.88. Since it is greater than unity, we have confirmation from our theoretical results that the smoking-present equilibrium is stable. Both the numerical and the theoretical results agree about the stability of the smoking-present equilibrium. From this same figure, we see the convergence of the different curves toward the smoking-present equilibrium E∗=5.81,5.13,2.36,1.43,4.48 which means this second steady state can be found numerically or calculated theoretically. We conclude that the local and global theoretical results concerning the stability of the smoking-free and the smoking-present equilibria are in good agreement with our numerical results.Figure 3
The evolution of the population whenR0=1.88.Our third numerical result concerns the comparison between the model with control strategy and the model without any control. Indeed, Figure4 shows the evolution of potential smokers, occasional smokers, chain smokers, temporarily quit smokers, permanently quit smokers, and the two controls as the function of time. We clearly observe that, with control strategy, we will have more potential smokers than without control measures. Also, it is clearly seen that, with control measures, a significant reduce of occasional and chain smokers is observed. This means the roles of control strategies are to reduce the number of occasional and chain smokers which can lead to save many lives in smoking population. Besides, with control strategy, we observe that, without any control strategy, the curves representing the different smoking compartment converge toward E∗=5.81,5.13,2.36,1.43,4.48. However, with taking significant control measures, those curves converge toward Ec∗=19.04,7.09×10−2,3.1×10−2,1.91×10−2,4.91×10−2 which confirms the role of control in reducing the smoking habit. In addition, we remark from the two controls’ behavior that we should give more importance to the first control than the second. It is also observed that the two controls should be also maintained at a constant level in order to have the desired good result.Figure 4
The effect of the control on the evolution of the population.
(a)(b)(c)(d)(e)(f)


## 7. Conclusion

We have studied in this paper mathematically and numerically the dynamics of giving up smoking behavior. The suggested smoking model consists of a system of five differential equations representing the potential smokers, the occasional smokers, the chain smokers, the temporarily quit smokers, and the permanently quit smokers. We have established the wellposedness result of our smoking problem by proving the existence, positivity, and boundedness of the smoking problem solution. Both the equilibria local and the global stability results are fulfilled. To study the role of control measures on our smoking problem dynamics, two different controls are introduced to the model. The role of the first control is to reduce the contact between nonsmokers and smokers, while the objective of the second is to prevent occasional smokers to become chain smokers. The existence and the characterization of the two optimal controls are discussed by using the Pontryagin minimum principle. The optimality system is solved numerically using the classical forward and backward difference numerical scheme. It was established that the numerical simulations concerning the stability of the smoking-free and the smoking-present equilibria are in good agreement with the local and global theoretical results. Moreover, the numerical results confirm the important role of control measures in reducing the number of occasional and chain smokers which may save significant number of lives in the smoking population.



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