OPTIMAL CONTROL AND DISCRETE-TIME MODELLING OF ALCOHOL MODEL WITH PHYSICAL AND PSYCHOLOGICAL COMPLICATIONS

In this paper, we propose a discrete mathematical model that describes the interaction between the classes of drinkers, namely, potential drinkers (P), moderate drinkers (M), heavy drinkers(H), heavy drinkers with physical complications(C1), the individuals with psychological complications(C2) and recovered and quitters of drinking (Q). We also focus on the importance of treatment of heavy drinkers with and without complications to aiming to find the optimal strategies to minimize the number of drinkers and maximize the number of the individuals who recovered and quitters of drinking. We use three controls which represent awareness programs through media and education for the potential drinkers, efforts treatment for the heavy drinkers and treatment for the heavy drinkers with physical complications. We use Pontryagin’s maximum principle in discrete time to characterize these optimal controls. The resulting optimality system is solved numerically by Matlab. Consequently, the obtained results confirm the performance of the optimization strategy.


INTRODUCTION
Alcoholism is a social and global phenomenon that is harmful to all social groups due to some physiological, genetic or even neurological factors that cause this disease, which need to drink isolated and uncontrollable alcohol. This is why an alcoholic will continue to drink despite serious family, social, professional, and legal problems arising from his addiction.
Harmful use of alcohol can harm others such as family members and bankruptcy may occur, bankruptcy in the relationship (divorce, separation, isolation), economic (loss of employment, financial difficulties), legal bankruptcy (loss of driver's license, arrest, imprisonment) Psychological (loss of self-esteem, dignity, freedom, autonomy, quality of life and well-being, physical and neurological complications, anxiety, depression, etc) [15]. According to the World Health Organization's 2016 report, alcohol drinking increases the risk of heart disease, high blood pressure and stroke. Risk increases in younger people due to party drinking, which can lead to accidents or violence, and there are 3 million deaths a year, which accounts for 6% of all deaths due to drinking alcohol [15]Alcoholism is a serious problem that causes many physical and psychological complications. Concerning physical complications, people who drink too much alcohol are at risk of harmful, potentially life-threatening liver problems. When you drink a lot, your liver breaks down alcohol and removes it from your blood. Drinking too much alcohol in a short period of time can lead to fatty liver and my father is considered a chronic condition that involves accumulation of bad fat in the liver. Which leads to the problem of obesity, liver failure and type 2 diabetes. Other serious liver complications are associated with prolonged and excessive alcohol consumption. There are other problems which are caused by drinking a lot of alcohol which are alcoholic hepatitis, brosis and cirrhosis. In order too treat these health problems, They require a proper medical diagnosis and intensive treatment plan. According to the World Health Organization's 2016 report, deaths caused by liver cirrhosis disease, age- deaths globally consuming too much alcohol in 2016 is linked to high blood pressure, irregular heartbeat, trouble pumping blood through the body, blood clots, stroke, cardiomyopathy, or heart attack Excessive alcohol use, both directly and through malnutrition, can also lead to anemia. For example, it affects the circulatory system, veins and arteries negatively, leading to high blood pressure, as well as atherosclerosis and heart attacks, which may lead to death in some cases. It also causes ulcers and gastrointestinal problems, where pancreatitis, central nervous system, respiratory infections, immune system dysfunction, brain damage, malnutrition and vitamin deficiency, osteoporosis, Accidents and injuries.
In addition to the physical complications caused by alcoholism, the frequent use of alcohol leads to serious psychological complications. It may lead to personality changes: alcohol use can cause major changes to your personality. Attributes of the natural personality disappear during intoxication and are replaced by angry, selfish and aggressive behavior, mood swings are very common, and a general deterioration of morals also occurs. Alcohol slows the tangles in the brain and changes your body chemically by affecting levels of serotonin, the chemical responsible for transmitting mood signals to your mind. These physical changes make your emotions out of control and cause you to say and do things that you wouldn't normally do. Concomitant disorders: often there are psychological and mood disorders common among alcoholics. It can be caused by mental effects of light drinking, or it can be caused by heavy drinking in the first place. Often, a person with a common disorder such as depression or anxiety becomes addicted to alcohol after trying to treat his symptoms on his own. For example: depression, partial depression disorder, anxiety, compulsive behavior, deprivation, alcohol-induced depression disorder, alcohol-induced bipolar disorder, alcohol-induced sleep disorder, severe depression and some anxiety disorders, schizophrenia, bipolar disorder, etc., And mental disorder caused by alcohol. It changes the sleep pattern and personality, as it has cognitive effects such as short attention and reduced coordination problems. Several mathematicians did a lot of work in order to understand the dynamics of drinking and reduce its harm on the drinker and society as well as minimizing the number of addicted drinkers. For example [1,4,6,7,14]: H. F. Huo and N. Song [6] divided heavy drinkers in thier study into two types: those who confess drinking and those who do not and they proposed a two-stage model for binge drinking problem taking into consideration the transition of drinkers from the class of susceptible individuals towards the class of admitting drinkers. H. F. Huo, and Q. Wang [7] developed a non-linear mathematical model with the effect of awareness programs on the binge drinking where they show that awareness programs are an effective measure in reducing alcohol problems. H. F. Huo et al [6] proposed a new social epidemic model to depict alcoholism with media coverage which was proven to be an effective way in pushing people to quit drinking.
S. H. Ma et al [14] modeled alcoholism as a contagious disease and used an optimal control to study their mathematical model with awareness programs and time delay. Wang et al [4] proposed and analyzed a non-linear alcoholism model and used optimal control for the purpose of hindering interaction between susceptible individuals and infected individuals. I. k. Adu et al [1] used a non-linear SHT R mathematical model to study the dynamics of drinking epidemic, they divided their population into four classes: non-drinkers (S), heavy drinkers (H), drinkers receiving treatment (T ) and recovered drinkers (R). They discussed the existence and stability of drinking-free and endemic equilibrium. Other mathematical models has also been widely used to study this phenomenon (For example, [10,12,13.....]). In addition, most of these previous researches have focused on continuous-time modeling. In this research, we will adopt the discrete-time modeling as the statistical data are collected at discrete time (day, week, month and year) as well as the treatment and vaccination of some patients are done in discrete-time.
So, it is more direct, more convenient, and more accurate to describe the phenomena by using the discrete-time modeling than the continuous-time modeling and the use of discrete time models avoids some mathematical complications such as choosing a space, function, and regularity of the solution. Hence, difference equations appear as a more natural way to describe the epidemic models and discrete problems. Moreover, numerical solutions of differential equations use discretization and this encourages us to employ difference equations directly. The numerical exploration of discrete-time models is rather straightforward and therefore can be easily implemented by non mathematicians. Besides these works, we will study the dynamics of a mathematical alcohol model PMHC 1 Q C 2 which contains the following additions: • A discrete-time mathematical modeling.
• A compartment C 1 that represents the number of the heavy drinkers with physical complications.
• A compartment C 2 that represents the number of individuals with psychological complications.
• The death rate induced by the heavy drinkers δ 1 .
• The death rate induced by the heavy drinkers with physical complications δ 2 .
The drinkers classes of this model are divided into six compartments: Potential drinkers In order to achieve this purpose, we use optimal control strategies associated with three types of controls: the first represents awareness programs for potential drinkers, the second is the treatment for heavy drinkers and the third is the treatment for heavy drinkers with physical complicatons (C 1 ).
The paper is organized as follows. In Section 2, we present our PMHC 1 QC 2 discrete mathematical model that describes the dynamics of the classes of drinkers. In Section 3 and 4, we present the optimal control problem for the proposed model where we give some results concerning the existence of the optimal controls and we characterize these optimal controls using Pontryagin's Maximum Principle in discrete time. Numerical simulations are given in Section 5. Finally, we conclude the paper in Section 6.

MODEL FORMULATION
We propose a discrete model PMHC 1 QC 2 to describe the dynamics of population and the transmission of drinking. The population is divided into six compartments denoted by P, M, H,C 1 , Q and C 2 .
The graphical representation of the proposed model is shown in Figure1. (1) The compartment P: contains the potential drinkers who represent individuals whose age is over adolescence and adulthood and may become drinkers. This compartment is increased by the recruitment rate denoted by b and decreased by an effective contact with the moderate drinkers at a rate β 1 and natural death µ. It is assumed that potential drinkers can acquire drinking behavior and can become moderate drinkers through effective contact with moderate drinkers in some social occasions like weddings, celebrating graduation ceremonies, week-end parties and end of the year celebration. In other words, it is assumed that the acquisition of a drinking behavior is analogous to acquiring disease infection.
The compartment M: is composed of the moderate drinkers who drink alcohol in a controlled manner during some events and occasions or in a way that is unapparent to their social environment. It is increased by potential drinkers who turn to be moderate drinkers at a rate β 1 .
This compartment is decreased when moderate drinkers become heavy drinkers at a rate β 2 , and those who quit drinking with psychological complication (C 2 ) at a rate α 4 and also by natural death at rate µ.
The compartment H: comprises the heavy drinkers. This compartment becomes larger as the number of heavy drinkers increases by the rate β 2 and decreases when some of them give up drinking who quit drinking at a rate α 2 and outher becomes heavy drinkers with physical complications (C 1 ) at rate α 1 (α 1 is a rate of heavy drinkers becomes heavy drinkers with physical complications) and some becomes heavy drinkers with psychological complications (C 2 ) at rate α 3 (α 3 is a rate of heavy drinkers becomes individuals with psychological complications). In addition, this compartment decreases by natural death µ and due to deaths caused by diseases resulted from excessive alcohol intake at a rate δ 1 .
The compartment C 1 : represents the number of heavy drinkers with physical complications (C 1 ). This compartment is increased by the rate α 1 and decreased by the rates γ 1 and (µ + δ 2 ) , where γ 1 is the individiuals heavy drinkers with physical complications who quit drinking and treated and δ 2 is the death rate induced by the heavy drinkers with physical complications.
The compartment Q: contains the number of individuals who recovered from physical complications and quitters of drinking.This compartment is increased by the rates α 2 and γ 1 and decreased by the rates γ 2 (γ 2 is a rate of the individuals who recovered from physical complications and quitters of drinking becomes individuals with psychological complications) and µ.
The compartment C 2 : encompasses the individuals who recovered from physical complications and quitters of drinking and becomes individuals with psychological complications for alcohol-induced depressive disorder, alcohol-induced bipolar disorder, alcohol-induced sleep disorder, alcohol-induced psychotic disorder, and more. These disorders are temporary and can occur after significant intoxication and/or withdrawal. mental illnesses which can develop separately from the disorder and may even predate it, such as major depression, some anxiety disorders, schizophrenia, bipolar disorder, and others. It is increased with the recruitment of individuals who have been treated at rate γ 2 and α 4 ( α 4 is a rate of the moderate drinkers who becomes to individuals with psychological complications). It also increases at the rate α 3 of those who quit and recovered alcohol and decreases at the rate µ due to natural deaths.
The total population size at time k is denoted by N k with N k = P k +M k +H k +C 1,k +Q k +C 2,k and it is supposed as constant.

THE OPTIMAL CONTROL PROBLEM
The objective of the proposed control strategy is to minimize the number of heavy drinkers with and without complications and maximize the number of the recovered and quitters of drinking (Q) in the community during the time step k = 0 to k = T . The cost spent in the awareness programs and treatment are also to be minimized.
In order to achieve these objectives, we introduce three control variables. The first control u 1 represents the effort of the awareness programs to protect the potential drinkers not to be drinkers. The second control u 2 measures the treatment of the heavy drinkers . We note that the control function εu 2 represents the fraction of the heavy drinkers who will be treated and become moderate drinkers and the fraction (1 − ε)u 2 of those leaving the heavy drinkers class who will receive treatment and quit drinking and u 3 measures the treatment of the heavy drinkers with physical complications.
So, the controlled mathematical system is given by the following system of difference equations: The optimal control problem to minimize the objective functional is given by: Where the parameters A 1,k > 0, A 2,k > 0 and A 3,k > 0 are selected to weigh the relative importance of the cost of awareness programs and treatment respectively.
The aim is to find an optimal control u * 1 , u * 2 and u * 3 such that : Where U ad is the set of admissible controls defined by (5) U ad = (u 1,k , u 2,k , u 3,k )/ 0 ≤ u j min ≤ u j,k ≤ u j max ≤ 1; j = 1, 2, 3 and k = 0, 1, 2, ..., T − 1 The sufficient condition for the existence of an optimal control (u * 1 , u * 2 , u * 3 ) for problem (2) and (3) comes from the following theorem. Theorem 1. There exists the optimal control (u * 1 , u * 2 , u * 3 ) such that subject to the control system (2) with initial conditions. Since there is a finite number of uniformly bounded sequences, there exist (u * 1 , u * 2 , u * 3 ) ∈ U 3 ad and P * , M * , H * ,C * 1 , Q * and C * 2 ∈ IR T +1 such that, on a subsequence, Lim Finally, due to the finite dimensional structure of system (2) and the objective function J(u 1 , is an optimal control with corresponding states P * , M * , H * , C * 1 , Q * and C * 2 . Therefore inf

CHARACTERIZATION OF THE OPTIMAL CONTROLS
We apply the discrete version of Pontryagin's Maximum Principle [2,3,8,11,16]. The key idea is introducing the adjoint function to attach the system of difference equations to the objective functional resulting in the formation of a function called the Hamiltonian. This principle converts the problem of finding the control to optimize the objective functional subject to the state difference equation with initial condition to find the control to optimize Hamiltonian pointwise (with respect to the control).
Now we have the HamiltonianĤ at time step k, defined by: where f i;k+1 the right-hand side of the system of difference equations (2) of the i th state variable at time step k + 1.

NUMERICAL SIMULATION
In this section, we shall solve numerically the optimal control problem for our PMHC 1 QC 2 model. Here, we obtain the optimality system from the state and adjoint equations. The proposed optimal control strategy is obtained by solving the optimal system which consists of six difference equations and boundary conditions. The optimality system can be solved by using an iterative method. Using an initial guess for the control variables, u 1,k ,u 2,k and u 3,k , the state vari- The proposed control strategy in this work helps to achieve several objectives:  To achieve this objective, we only use the control u 1 and u 2 i.e. treatment of the heavy drinkers. In Figure3(a), it is observed that there is a significant decrease in the number of moderate drinkers with control compared to a situation when there is no control where the decrease reaches 41.20% at the end of the proposed control strategy. Figure 3(b) shows that the number of the heavy drinkers decreased from 636.88 (without control) to 338.88 ( with control) at the end of the proposed control. Figure 3(c) shows that the number of the heavy drinkers with physical complications C 1 decreased from 4310 ( without control) to 2200 ( with control) at the end of the proposed control. Figure 3(d) shows that the number of the individuals with psychological complications decreased from 71.25 ( without control) to 52.72 ( with control) at the end of the proposed control. Figure 3(e) shows that the number of people who recovered and quitters of drinking without control increases and approaches a value of 97.39. It's increase appears with control u 1 and u 2 . To meet this objective, we use the controls u 1 , u 2 and u 3 i.e. awareness programs for the potential drinkers, treatment for heavy drinkers and treatment for heavy drinkers with physical complications (C 1 ). Figure 4(a) shows that the number of the heavy drinkers increases starting from the early days a value 636.88 (without controls) to 425.96 (with controls). Also, Figure   4 to 5169 (with controls). As a result, the objective set before has been achieved.

CONCLUSION
In this research paper, we introduced a discrete modeling of drinking with and without complications for the purpose of minimizing the number of drinkers with and without complications and maximizing the number of recovered and quitters of drinking. Unlike some other previous models, we have taken into account the impact the treatment for physical complications on alcoholics. The results showed that treatment have substantial influence on the dynamics of alcoholism and can greatly impact the spread of the drinking, thus, it is crucial to urge people to know complications to recovered and quitters of drinking. We also presented three controls which represent awareness programs and treatment respectively. 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.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.