Mathematical Analysis and Optimal Control of Giving up the Smoking Model

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. 'emodel under consideration describes the interaction between the potential smokers (P), 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. 'e 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. 'e existence of the optimal control pair is discussed, and by using Pontryagin minimum principle, these two optimal controls are characterized. 'e 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.


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. e active substances of the smoke are rapidly transmitted from the lungs to the bloodstream and therefore to the brain where they operate. e 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][4][5][6][7][8].
Smoking dynamics were modelled by several authors in order to study the interaction between the model compartments.
e 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 (Q P ). e authors studied the local stability of the equilibria and suggest a nonlinear smoking relapse model. Later, in 2008, another compartment of temporarily quit smokers (Q T ) 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. e stochastic study of the PSQ T Q P 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]. e authors studied the following PLSQ P smoking model: 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 c 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 I R (P, L): � ((2PL)/(P + L)). We note that for square-root incidence function I R (P, L): � �� � PL √ , and the PLSQ P smoking problem is tackled by [13]; while with bilinear incidence function I R (P, L): � PL, the PLSQ T Q P smoking problem is studied in [14].
In this paper and inspired by the previous works, we are interested in studying PLSQ T Q P smoking problem by taking into consideration the saturated incidence rate I R (P, L): � ((2PL)/(P + L)); indeed, this saturated rate describes the crowding effect of potential and occasional smokers in smoking areas. e problem under consideration will take the following form: e reversion to the smoking phenomenon is highlighted in this model by the parameter α which represents the ratio of temporarily quit smokers who start to smoke again, while η is the portion of quitters who quit smoking permanently. e flowchart of our suggested PLSQ T Q P 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][19][20][21][22][23].
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.
is paper is organized as follows. In Section 2, we will establish the wellposedness of our suggested PLSQ T Q P 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. e 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.
e numerical results will show the importance of the optimal control in reducing the smokers' addiction. In Section 7, we will give concluding remarks.

e Wellposedness Result.
Let C � C(R + , R 5 ) be the Banach space of continuous functions mapping R + into R 5 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 (P(t), L(t), S(t), Q T (t), Q P (t)) of system (2) with initial data (P(0), L(0), S(0), Q T (0), Q P (0)) ∈ C. e following proposition guarantees that this solution exists globally.

Proposition 1.
e solution of the studied model (2) remains positive and bounded for all t ≥ 0. Moreover, we have Proof. e solution P(t) is nonnegative, for all t ≥ 0. Supposing the contrary, let t 0 > 0 be the first time such that 2 International Journal of Differential Equations P(t 0 ) � 0 and _ P(t 0 ) ≤ 0. From the first equation of system (2), we have _ P(t 0 ) � Λ > 0, which leads a contradiction. Similar proofs show the positivity of L(t), S(t), Q T (t), and Q P (t), for all t ≥ 0. Indeed, according to system (2), we have is completes the proof of positivity.
About boundedness, let the total population From system (2), we have erefore, At t � 0, we have en, Consequently,

e Smokers' Generation Number.
e 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 matrix FV − 1 to calculate the smokers' generation number R 0 [25]. e formula that gives us the smokers' generation number is R 0 � ρ(FV − 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

e Smoking Problem Equilibria.
It is straightforward to check that model (2) has two steady states developed as follows:
Proof. e Jacobian matrix of system (2) at E 0 is given by Figure 1: e flowchart of the PLSQ T Q P smoking model.
International Journal of Differential Equations 3 e characteristic polynomial of J E 0 is erefore, the eigenvalues of J(E 0 ) are given as follows: We conclude that E 0 is locally asymptotically stable when R 0 < 1.

Local Stability of the Smoking-Present Equilibrium.
In order to study the local stability of the smoking-present equilibrium E * (P * , L * , S * , Q * T , Q * P ), let us first give the Jacobian matrix of system (2) at E * : e characteristic polynomial of J E * is with such that p � (1/R 0 ) and q � 1 − (1/R 0 ) (remark that p + q � 1). e first eigenvalue of the above characteristic polynomial is ξ 1 � − (δ + μ) < 0; the second eigenvalue is us, it is easy to verify that a 2 > 0, a 1 a 2 − a 3 > 0, and a 3 > 0. en, by application of Routh-Hurwitz theorem, the other eigenvalues, ξ 3 , ξ 4 , and ξ 5 , have negative real parts.We conclude that E * is locally asymptotically stable in case when R 0 > 1.

Global Stability of the Smoking-Free Equilibrium
Theorem 1. If R 0 ≤ 1, then the smoking-free equilibrium E 0 is globally asymptotically stable.
Proof. First, we denote d � δ + μ, P f � (Λ/(δ + μ)), and we consider the following Lyapunov function defined in R 5 e time derivative is given by If R 0 ≤ 1, then _ V 0 ≤ 0; consequently, according to the Lyapunov theorem, the smoking-free equilibrium E 0 is globally asymptotically stable.

Global Stability of the Smoking-Present Equilibrium
Theorem 2. If R 0 > 1, then the smoking-present equilibrium point E * is globally asymptotically stable.
Proof. We consider the following Lyapunov function defined in R 5 + as follows: e time derivative is given by International Journal of Differential Equations It is easy to verify that So, en, By application of the relation between geometric and arithmetic means, we obtain We conclude that the smoking-present equilibrium point E * is globally asymptotically stable when R 0 > 1.

e Smoking Optimization Problem.
In order to formulate our smoking optimization problem, we will introduce to problem (2) two controls u 1 and u 2 . e smoking problem with controls becomes International Journal of Differential Equations e first control u 1 represents the government prohibition of smoking in public area, while the second control u 2 symbolizes the education campaign and the increase of cigarette cost. e 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 u i ∈ [0, 1]. e smoking optimization problem consists in maximizing the objective functional defined as follows: where t f represents the time needed for the undertaken control measures and the two positive constants C 1 and C 2 are based on the costs of each strategy u 1 (t) and u 2 (t), respectively. e 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 where U is the control set defined by

e Smoking Optimality System.
Pontryagin's minimum principle [26] transforms our optimization problem into a Hamiltonian maximization problem, where the Hamiltonian H is given by H(P, L, S, Q T , Q P , u 1 , u 2 , ϕ 1 , e optimality system is summarized by the following result.

Proposition 3.
For any solution (P * , L * , S * , Q * T , Q * P ) of the smoking optimization problem, there exist the adjoint variables, ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 , and ϕ 5 , satisfying the system of equations under the transversality conditions, In addition, the optimal control is given by (33)

International Journal of Differential Equations
Proof.
e five adjoint equations and the transversality conditions can be obtained via Pontryagin's minimum principle [26] such that Simple calculations lead to e two optimal controls u * 1 and u * 2 can be found from the following optimality conditions: However, we have Since the two controls belong to the interval [0, 1], we will have u * 1 � min 1, max 0, In addition, we have □

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][29][30]. e numerical method will be implemented under Algorithm 1. e parameters values of our numerical simulations are given in Table 1.

8
International Journal of Differential Equations e smoking-free equilibrium stability result is illustrated numerically in Figure 2. 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, R 0 � 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 E f � (3.12, 0, 0, 0, 0) which means that our numerical result supports the theoretical result concerning the stability of the first steady state.
On the contrary, the smoking-present equilibrium stability is represented in Figure 3. 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, R 0 � 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 smokingpresent equilibria are in good agreement with our numerical results.
Our third numerical result concerns the comparison between the model with control strategy and the model without any control. Indeed, Figure 4 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 Step 1 (initial conditions): Step 2 (numerical solving): for i � 0, . . . , n − 1, do: Step 3 (the solution and optimal controls):     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. is 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.
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.

Conclusion
We have studied in this paper mathematically and numerically the dynamics of giving up smoking behavior. e International Journal of Differential Equations 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. e 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. e existence and the characterization of the two optimal controls are discussed by using the Pontryagin minimum principle. e 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.

Data Availability
No data were used to support this study.