Optimal Control Strategy for a Discrete Time Smoking Model with Specific Saturated Incidence Rate

The aim of this paper is to study and investigate the optimal control strategy of a discrete mathematical model of smoking with specific saturated incidence rate. The population that we are going to study is divided into five compartments: potential smokers, light smokers, heavy smokers, temporary quitters of smoking, and permanent quitters of smoking. Our objective is to find the best strategy to reduce the number of light smokers, heavy smokers, and temporary quitters of smoking. We use three control strategies which are awareness programs through media and education, treatment, and psychological support with follow-up. Pontryagins maximum principle in discrete time is used to characterize the optimal controls. The numerical simulation is carried out using MATLAB. Consequently, the obtained results confirm the performance of the optimization strategy.


Introduction
Following the WHO report on the global tobacco epidemic, which was published on 19 July 2017 in the United Nations High-Level Political Forum for Sustainable Development in New York, tobacco consumption is the world's leading cause of death with a rate of more than 7 million deaths a year [1]. Tobacco cosumption is known as the main cause of death of lethal diseases such as lung cancer, oral cavity, stomach ulcer, and a probable cause of death for cancers of the larynx, bladder, pancreas, and renal pelvis. Comparative data on smoking show that the risk of heart attack among smokers is 70% higher than that of nonsmokers. In addition to that, the economic costs are also enormous totaling more than 1400 billion dollars (US $) in health expenditure and loss of productivity [2].
Lung cancer in smokers is ten times higher than in nonsmokers, and one in ten smokers will die of lung cancer. In Spain, it is estimated that about 55,000 deaths a year are attributable to smoking [3]. However, smokingrelated illnesses cause more than 440,000 deaths each year in the United States and more than 105,000 deaths in the United Kingdom each year. Moreover, about 4 million people die from smoking-related diseases worldwide and half of all smokers die from smoking-related diseases, while the number of new smokers continues to increase [4].
As far as Morocco is concerned, a new report by the World Health Organization revealed that the number of smokers has notably risen, expecting it to reach over 7 million by 2025 [5]. The WHO has urged the Moroccan government to increase taxes on cigarettes and other tobacco products to discourage use and decrease the rising number of smokers across the country. In its Global Report on Trends in Prevalence of Tobacco Smoking in 2015, WHO estimated that up to 21% of Morocco's population (approximately 4,820,500 persons) smoked in 2010.
More specifically, the report noted that about 42% of men and up to 2% of women smoked in Morocco in 2010. It goes on to add that the highest rate of smoking among men was seen in the 25-39 age groups and 15-24 age groups among women. WHO recommends that at least one adult survey 2 Discrete Dynamics in Nature and Society and one youth survey be completed every five years. In the event that the Moroccan government does not adopt new measures to discourage the use of tobacco, the UN estimated that the number of smokers could be over 7 million by 2025. Member states, including Morocco, adopted a voluntary global target to reduce tobacco use 30% (smokers and smokeless) by 2025. However, based on the current smoking trend, Morocco will not achieve the target, WHO concluded [6].
Mathematical modeling of smoking has been studied by many researchers [3,[7][8][9]. We observe that most of those researchers focused on the continuous-time models described by the differential equations. It is noted that, in recent years, more and more attention has been given to discrete time models (see [10][11][12][13] and the references cited therein). The reasons for adopting discrete modeling are as follows: Firstly, the statistical data are collected at discrete moments (day, week, month, or year). So, it is more direct and more accurate and timely to describe the disease using discrete time models than continuous time models. Secondly, the use of discrete time models can avoid some mathematical complexities such as choosing a function space and regularity of the solution. Thirdly, the numerical simulations of continuous time models are obtained by the way of discretization.
Based on the aforementioned reasons, we will develop in this paper a discrete time model studying the dynamics of smokers and introduce a saturated incidence rate to be analysed in detail in the next section. Also, we add to our model two elements which were not taken into consideration in the most previous researches. Those two elements are a group of light smokers who quit smoking permanently and a group of heavy smokers who died due to diseases generated by the excess of smoking.
In addition, in order to find the best strategy to reduce the number of light smokers, heavy smokers, and temporary quitters of smoking we will use three control srategies which are awareness programs through media and education, treatment, and psychological support with follow-up.
In this paper, we construct a discrete PLSQ Q Mathematical Smoking Model with Specific Saturated Incidence Rate and introduce the control of awareness measures. In Section 2, the mathematical model is proposed. In Section 3, we investigate the optimal control problem for the proposed discrete mathematical model. Section 4 consists of numerical simulation through MATLAB. The conclusion is given in Section 5.

Formulation of the Mathematical Model
In this section, we present a discrete PLSQ Q Mathematical Smoking Model. The population under invistegation is divided into five compartments: potential smokers (nonsmokers) P , light smokers L , heavy smokers S , smokers who temporarily quit smoking Q , and smokers who permanently quit smoking Q , respectively. Following what has been done in many works [2], we introduce a saturated incidence rate ( /(1 + )) in discete time (where ∈ { , , }, −for = 1, 2, − is the contact rate between and , −for = 1, 2, 3− is a positive constant) to describe the crowding effect among potential smokers, light smokers, and heavy smokers. measures the infection force of the smoking and 1/(1 + ) describes the crowding effect and the "psychological" effect from the behavioral change of the individuals when their number increases.

2.1.
Description of the Model. The compartment P: the potential smokers (nonsmokers), poeple who have not smoked yet but might become smokers in the future. This compartment is increased by the recruitment of individials at rate Λ and P is decreased with the rates 1 (1 − )( /(1 + 1 )), 1 ( /(1 + 2 )) and some of the poeple vacate at a constant death rate of due to the total natural death rate . The compartment L: the occasional smokers whose number increases when the potential smokers start to smoke with a saturated incidence rate 1 (1 − )( /(1 + 1 )). Some other individials will leave the compartment with the saturated incidence rate 2 ( /(1 + 3 )), the rate 3 , and . Here, 3 is the rate of light smokers who permanently quit smoking.
The compartment S: the people who are heavy smokers and whose number increases by the saturated incidence rate ) and the rate of temporary quitters who revert back to smoking. Some others will leave at the rates , , and . Here, is the rate of death due to heavy somking and is the rate of quitting smoking.
The compartment : the individuals who temporarily quit smoking, whose number increases at the rate (1 − ) and decreases at the rates and , where (1 − ) is the fraction of heavy smokers who temporarily quit smoking (at a rate ).
The compartment : the individuals who permamently quit smoking, whose number increases with the rates and 3 . Some people of this compartment will die with the rate , where is the remaining fraction of heavy smokers who permanently quit smoking (at a rate ).
The following diagram will demonstrate the flow directions of individuals among the compartments. These directions are going to be represented by directed arrows in Figure 1.

Model Equations.
Through the addition of the rates at which individuals enter the compartment and also by subtructing the rates at which people vacate the compartment, we obtain an equation of difference for the rate at which the individuals of each compartment change over discrete time. Hence, we present the smoking Figure 1: The flow between the five compartments .
infection model by the following system of difference equations:

The Optimal Control Problem
The strategies of control that we adopt consist of an awareness program through media and education, treatment, and psychological support with follow-up. Our main goal in adopting those strategies is to minimize the number of occasional smokers, heavy smokers, and the temporarily quitters of smoking during the time steps = 0 to and also minimizing the cost spent in apllying the three strategies. In this model, we include the three controls , V , and that represent consecutively the awareness program through media and education, treatment, and psychological support with follow-up as measures at time . So, the controlled mathematical system is given by the following system of difference equations:  There are three controls = ( 0, 1, . . . ), V = (V 0, V 1, . . . V ), and = ( 0, 1, . . . ). The first control can be interpreted as the proportion to be adopted for the awareness program through media and education. So, we note that is the proportion of the light smoker individuals who moved to the individuals who permanently quit smoking at time step . The second control can be interpreted as the proportion to be subjected to treatment. So, we note that V is the proportion of the individuals who will move from the class of heavy smokers towards the class of the individuals who permanently quit smoking at time step . The third control can also be interpreted as the proportion to get psychological support with follow-up. So, we note that is the proportion of the individuals who temporarily quit smoking and who will transform into the individuals who permanently quit smoking at time step . Indeed, the system above (2) presents eight different models as Table 1 explains.
The problem that we face here is how to minimize the objective functional: where the parameters > 0, > 0, > 0, > 0, > 0, and > 0 are the cost coefficients; they are selected to weigh the relative importance of , , , , V , and at time . is the final time.
In other words, we seek the optimal controls , V , and such that where is the set of admissible controls defined by The sufficient condition for the existence of optimal controls ( , V, ) for problem (2) and (4) comes from the following theorem.
In order to derive the necessary condition for optimal control, the pontryagins maximum principle in discrete time given in [10,11,[14][15][16] was used. This principle converts into a problem of minimizing a Hamiltonian at time step defined by Discrete Dynamics in Nature and Society 5 where , +1 is the right side of the system of difference equations (2) of the ℎ state variable at time step + 1.
By the bounds in of the controls, it is easy to obtain * , V * , and * in the form of (10).

Algorithm.
In this section, we present the results obtained by solving numerically the optimality system. This system consists of the state system, adjoint system, initial and final time conditions, and the controls characterization. So, the optimality system is given by the following.
Step 1. 0 = 0 , 0 = 0 , 0 = 0 , 0 = 0 , 0 = 0 , 1, = 5, = 0, 2, = , 3, = , 4, = , and given * ,0 , V * ,0 , and * ,0 Step 2. For = 0; 1; . . . ; − 1 do: end for Step 3. For = 0; 1; . . . ; ; write: * = , * = , Discrete Dynamics in Nature and Society 7 end for In this formulation, there were initial conditions for the state variables and terminal conditions for the adjoints. That is, the optimality system is a two-point boundary value problem with separated boundary conditions at time steps = 0 and = . We solve the optimality system by an iterative method with forward solving of the state system followed by backward solving of the adjoint system. We start with an initial guess for the controls at the first iteration and then before the next iteration we update the controls by using the characterization. We continue until convergence of successive iterates is achieved.

Discussion.
In this section, we study and analyse numerically the effects of optimal control strategies such as awarness program through media and education, treatment, and psychological support with follow-up for the infected smokers (Table 2 ).

Strategy A: Control with Awareness Program.
Given the importance of the awareness programs in restricting the spreading of smoking, we propose an optimal strategy for this purpose. Hence, we activate the optimal control variable which represents the awareness program for light smokers. Figure 2  smokers with and without control in which the effect of the proposed awareness program through media and education is proven to be positive in decreasing the number of light smokers.

Strategy B: Control with Treatment and Psychological
Support with Follow-Up. When the number of smokers is so high, it is obligatory to resort to some strategies such as treatment in order to reduce the number of smokers. Therefore, we propose an optimal strategy by using the optimal control V in the beginning. In spite of using the optimal control V, we observe a temporary decrease of the heavy smokers number which is increased again (Figure 3(a)). The reason of this increase is justified by the fact that heavy smokers revert back to smoking after giving up. For improving the effectiveness of this strategy, we add the elements of follow-up and psychocological support which are represented in the proposed strategy by the optimal control variable ( Figure 3(b)). Combining follow-up and psychocological support with treatment results in an obvious decrease in the number of heavy smokers. Also, the proposed strategy has an additional effect in decreasing clearly the number of temporary quitters of smoking.

Strategy C: Control with Awareness Program, Treatment, and Psychological Support with Follow-Up.
In this strategy, we combine the two previous strategies to achieve better results. We notice that the numbers of light smokers (Figure 4(a)), heavy smokers (Figure 4(b)), and temporary quitters of smoking (Figure 4(c)) are decreased markedly which leads to satisfactory results.

Conclusion
In this paper, we introduced a discrete modeling of smokers in order to minimize the number of light smokers, heavy smokers, and temporary quitters of smoking. We also introduced three controls which, respectively, represent awareness program through education and media, treatment, and psychological support with followup. We applied the results of the control theory and we managed to obtain the characterizations of the optimal controls. The numerical simulation of the obtained results showed the effectiveness of the proposed control strategies.

Data Availability
The disciplinary data used to support the findings of this study have been deposited in the Network Repository (http://www.networkrepository.com).