MODELING MATHEMATICAL AND ANALYSIS OF AN ALCOHOL DRINKING WITH n COMPLICATIONS

. In this article, we propose a new continuous mathematical approach to model and analyze the dynamics of the population of heavy drinkers and their health complications. The model has several compartments, including a new one that represents the number of heavy drinkers with different complications associated with excessive alcohol consumption. We study the stability of the model using mathematical theories such as the Routh-Hurwitz criteria for local stability, in order to study the equilibrium without and with excessive alcohol consumption, and the construction of Lyapunov functions makes it possible to study global stability. A sensitivity analysis is performed to determine which parameters have the most signiﬁcant impact on the number of reproductions R0. The results were validated using numerical simulations carried out under MATLAB. This model may be useful in guiding public health policies aimed at reducing the number of drinkers and the complications associated with alcohol consumption.


INTRODUCTION
Addiction is a dangerous social scourge that must be combated. But the term "addiction" is defined as a dependence on a substance or behavior, with serious consequences for health and behavior. An addiction is characterized by an often strong, even compulsive desire to consume or engage in a behavior. Thus, so that the person finds himself unable to abstain from taking a substance or engaging in a behaviour; he loses control over substance use or behavior.
This consumption or behavior has negative repercussions on the addict and those around him.
Specialists distinguish between physical addiction and behavioral addiction. Physical addiction specifically means addiction to drugs, alcohol, and sedatives, while behavioral addiction refers to activities that obsess a person and waste significant time in the practice of this activity, that makes his behavior unacceptable. On the contrary, it is a source of evil for him and those around him. In this context, we can talk about addiction to the Internet, shopping, work, food, sports, food, sex, phone ...

Addiction to psychotropic substances.
It defines dependence on tobacco, drugs, and alcohol "if the use of drugs or alcohol continues, it impacts physical or psychological dependence, or both. At the beginningThe drug gives the addict at first a feeling of happiness and calm, then he rushes to gradually increase the dose to enter the cycle of addiction and become a prisoner of drugs and alcohol his only obsession is to get it at any cost. The drug and alcohol addict tries to escape from social events, becomes emotional, and may resort to theft or even murder.
The causes of addiction are multiple: some of these reasons are related to social and cultural factors, the problems and pressures experienced by the individual, also it's related to the accompaniment of bad friends who push young people, especially to the abuse of drugs or alcohol. Moreover, the causes of addiction may be psychological, with some people suffering from mental illness or chronic depression and anxiety, or neurological diseases, or who have a pathological, unstable or antisocial personality, to drugs or alcohol.

Behavioral addiction.
Behavioral addiction can be defined as the control of an idea over a person to turn into an obsession, which takes a lot of his time and interests, and seeks to reach it by all means, despite his awareness of its consequences and risks. This type of addiction is very common, but it is not as visible and obvious as in the case of physical addiction.
We are interested in this work on alcohol dependence. In recent years, with the improvement of our standard of living, lifestyles have diversified, and alcohol consumption has become increasingly an important part of people's daily lives. Yet, the current situation of alcoholism in the world is truly worrying. Alcohol addiction has become one of the public health and social problems facing the world. Alcoholism has very serious consequences such as marital harm, child abuse, crime, social violence, and other serious consequences of criminal acts. It also contributes to traffic accidents. In general, the higher the volume of alcohol consumption, the more alcohol causes about 200 different types of diseases and conditions, including injuries and mental and behavioral disorders.
According to the WHO in 2016, the harmful use of alcohol resulted in some 3 million deaths Alcohol caused approximately 0.4 million of the 11 million deaths worldwide in 2016 from communicable, maternal, perinatal and nutritional diseases, accounting for 3.5% of these deaths.
Harmful use of alcohol caused some 1.7 million deaths from non-communicable diseases in 2016, including some 1.2 million deaths from digestive and cardiovascular diseases (0.6 million for each condition ) and 0.4 million deaths from cancer. Globally, an estimated 0.9 million injury deaths are attributable to alcohol, including around 370,000 deaths from road traffic accidents, 150,000 from self-harm and around 90,000 from interpersonal violence . Among road accidents, 187,000 alcohol-attributable deaths involved people other than drivers. [1,2] Certain diseases caused by excessive alcohol consumption.
Excessive use of alcohol often leads to many harmful consequences for the drinkers themselves and society. And it causes serious health problems, including an increased risk of several diseases, like certain types of cancer such as cancer of the mouth, throat, liver, esophagus, colon and breast, liver diseases (fatty liver, alcoholic liver and cirrhosis), digestive system diseases (gastritis and inflammation of the liver Pancreatic), heart problems lead to high blood pressure and increase the risk of an enlarged heart, heart failure or stroke, neurological complications, weak immune system, sexual function problems and menstrual problems. Alcohol use also contributes to death and disability through road accidents, injuries, violence, crime and suicide, especially among young people. In its Global Status Report on Alcohol and Health, published in 2018, the World Health Organization reported that in 2016, deaths from alcohol consumption were higher than from diseases such as tuberculosis, HIV/AIDS and diabetes. Of the 3 million deaths caused by the harmful use of alcohol (5.3% of all deaths worldwide), 28% of deaths have been attributed to road traffic accidents, violence and suicide, 21% have been attributed to diseases affecting the digestive system, and 19% were attributed to cardiovascular diseases, and 32% were attributed to infectious diseases, cancers, mental disorders, or other conditions. Mathematical epidemiological models have become important tools that predict the dynamics of infectious diseases and provide effective measures to analyze and study and control their spread. Many studies use epidemiological mathematical models to study the dynamics of alcohol consumption, analyze consumer behavior and propose solutions to reduce the risks to consumers and society as well as to minimize the number of excessive alcohol consumers. For example, S. H. Ma et al. [15] modeled alcoholism as a contagious disease and used optimal control to study their mathematical model with sensitization and delay programs. Wang et al. [19] proposed and analyzed a nonlinear model of alcoholism and used optimal control to prevent the interaction between susceptible individuals and infected individuals. Sharma et al. [12] developed a mathematical model of alcohol abuse and discussed the existence, local and global stability of the endemic equilibrium without alcohol consumption and sensitivity analysis of a number of basic reproduction R 0 . Huo et al. [17] focused on the global ownership of a consumption model with public health education campaigns. They conclude that educational campaigns have a positive effect on controlling consumption dynamics. Giacobbe et al. [17] considered a mathematical model that describes the dynamics of a population divided into three categories and used an additional variable that represents an external influence. They studied the existence of an endemic equilibrium and analyzed the stability of the equilibrium. Agrawal et al. [22] developed a nonlinear SHTR mathematical model of alcohol abuse with a nonlinear incidence rate. The stability analysis of the model they proposed shows that the system is locally asymptotically stable at equilibrium without alcohol E 0 when R 0 ≤1.
Motivated by the fact that mathematical models have proven to be useful in understanding the dynamics of several social phenomena, in this study we propose a new model concerned with the study of excessive alcohol consumption and its complications on diseases. We examined the local stability of this model using the Routh-Hurwitz criteria and discussed its global stability using the Lyapunov function So, we will study the dynamics and the analysis of a mathematical model of excessive alcohol consumption and their complications PMHC 1 C 2 C 3 .....C n Q which contains the following additions: • A compartment C i that represents the number of the heavy drinkers with i complications associated with prolonged and excessive alcohol consumption where, i = {1, 2, ......., n} n complications.
• The death rate induced by the heavy drinkers δ 0 .
• The death rate induced by the heavy drinkers with i complications δ i .
The drinkers classes of this model are divided into n+4 compartments: Potential drinkers The paper is organized as follows. In Section 2, we present our PMHC 1 C 2 C 3 .....C n Q mathematical model that illustrates the dynamics of excessive alcohol consumption and their complications.In Section 3; we discuss basic properties and positivity of solutions. In section 4 ; we analyse the local and global stability and the problem of parameters sensitivity. Numerical simulations are given in Section 5. Finally, we conclude the paper in Section 6.

A MATHEMATICAL MODEL
We propose a continuous model PMHC 1 C 2 C 3 ...C n Q to describe and analyze the dynamics of the population of excessive alcohol consumers and their health complications. We divided the population into several compartments, including potential drinkers P(t), moderate drinkers M(t), heavy drinkers and heavy drinkers with different complications C i (t) and quitters of drinking Q(t).

Figure1: Schematic diagram of the fourten drinking classes in the model
We consider the following system of six non-linear differential equations: Potential drinkers P: The potential drinkers P(t) represents individuals who are older than the age of majority, is increased by the recruitment rate denote b and decreased by the rates β 1 PM N and µP , where, µ is the natural death rate, β 1 is the transmission rate from P to M .
Moderate drinkers M: Heavy drinkers H: Heavy drinkers with complication C i : The heavy drinkers with liver complications associated with prolonged and excessive alcohol consumption (alcoholic hepatitis, fibrosis and cirrhosis), is increased by the rate α i H and decreased by the rates γ i C i , µC i and µδ i . where, δ i is the death rate induced by the i complication of heavy drinkers.  The total population size at time t is denoted by N(t) with N (t)=P(t)+M(t)+H(t)+C 1 (t)+ C 2 (t) + ..... +C n (t) + Q(t).
We consider system (1) with the following parameter values.

Invariant Region.
It is necessary to prove that all solutions of system (1) with positive initial data will remain positive for all times t > 0. This will be established by the following lemma.  Proof. From the system equation(1) Where N(0) is the initial value of total number of people, thus, lim Hence, for the analysis of model (1), we get the region which is given by the set: Which is a positively invariant set for (1), so we only need to consider dynamics of system on the set Ω non-negative of solutions. Proof. From the second equation of system (1), we have:

Positivity
Similarly, From the third equation of system (1), we have: Similarly, From the forth equation of system (1), we have: Similarly, From the sixth equation of system (1), we have: and Q(t) > 0 ∀t ≥ 0, this completes the proof.
Since the first tree equations in system (1) are independent of the variables C i and Q, it is sufficient to consider the following reduced system:

MODEL ANALYSIS
4.1. Equilibrium states. We first find the equilibrium of the PMH model, by setting the righthand side of the system (14) to zero, we get two equilibrium states, namely the drinking-free state E 0 b µ , 0, 0 and the endemic state E * (P * , M * , H * ). Where R 0 is the basic reproduction number.

The drinking-free equilibrium.
In this section, we analyze the local stability of the drinking-free equilibrium Theorem 3. The drinking-free equilibrium E 0 b µ , 0, 0 of the system (16) is asymptotically stable if R 0 < 1 and unstable if R 0 > 1.
Proof. We now consider the stability local of the drinking-free equilibrium, for the system defined by (14), the matrix Jacobian is given by: The Jacobian matrix for the drinking-free equilibrium is given by: The characteristic equation of this matrix is given by det(J(E 0 ) − λ I 3 ) = 0 where I 3 is a square identity matrix of order 3.
The following eigenvalues where obtained: Therefore, all the Eigenvalues of the characteristic equation are negatives if R 0 < 1. Therefore, we conclude the drinking-free equilibrium is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

Endemic equilibrium.
In this section, we analyze the local stability of the endemic equilibrium.
To find the drinking present equilibrium of the system of equation (16) setting dP(t) dt = 0, dM(t) dt = 0 and dH(t) dt = 0. provided that at least one of the infected compartments is non zero. We evaluate the equilibrium of system (14) by setting the right-hand side of equation of system (14) to zero and then solve for P * , M * and H * .
We obtained system (23) : From the fourth equation in the system (23), we have From the second equation in the system (23), we have From the first equation in the system (23), we have Theorem 4. if R 0 > 1, E * is locally asymptotically stable .
Proof. We present E * (P * , M * , H * ) as endemic equilibrium of system (23) and P * = 0, M * = 0, The Jacobian matrix is The characteristic equation of this matrix is given by det(J(E * ) − λ I 2 ) = 0, where I 2 is a square identity matrix of order 2.

Global stability.
4.3.1. Global stability of the drinking-free equilibrium. We will investigate the global stability of E 0 when R 0 ≤ 1: Proof. Consider the following Lyapunov function [...], Hence, by Lasalle's invariance principle [23], E 0 is globally asymptotically stable.

4.3.2.
Global stabilty of the endemic equilibrium. Our final result in this section is for the global stability of E * .
Proof. Consider the Lyapunov function V : where c 1 and c 2 are positive constant to be chosen latter and Γ = {(P, M) ∈ Γ/P > 0, M > 0} Then, the time derivative of the Lyapunov function is given by Then, the time derivative of the Lyapunov function is given by For c 1 = c 2 = 1, we have Also, we obtain Hence by LaSalle's invariance principle [23] the free equilibrium point E * is globally asymptotically stable on Γ.

Sensitivity
Analysis of R 0 . To examine the sensitivity of R 0 to each of its parameters, following normalized forward sensitivity index with respect to each of the parameters is computed [3].
Using the approach in Chitnis et al. [24], we calculate the normalized forward sensitivity indices of R 0 . Let denote the sensitivity index of R 0 with respect to the parameter m. We get From the above discussion we observe that the basic reproduction number R 0 is most sensitive to changes in β 1 . if β 1 will increase R 0 will also increase with same proportion and if β 1 will decrease in same proportion, µ and β 2 have an inversely proportional relationship with R 0 .
We conclude that the basic reproduction number (R 0 ) is most sensitive to changes in β 1 . An increase in β 1 will cause an increase in R 0 with same proportion and a decrease in β 1 will cause a decrease in R 0 in same proportion. µ and β 2 have an inversely proportional relationship with R 0 , so an increase in any of them will bring about a decrease in R 0 . However, the size of the decrease will be proportionally smaller. Recall that µ is the natural death rate of the population. It is clear that increase in either of these rates is neither ethical nor practical. Thus we choose to focus on one parameters: β 1 the transmission rate from potential drinker to moderate drinker. Given R 0 's sensitivity to β 1 , it seems sensible to focus efforts on the reduction of β 1 .
In other words, this sensitivity analysis tells us that prevention is better than cure. Efforts to increase prevention are more effective in controlling the spread of habitual drinkers than efforts to increase the numbers of individuals accessing treatment.

NUMERICAL SIMULATIONS
This section includes the numerical simulation of the model proposition 1 describing the dynamics of excessive alcohol consumption and its complications, the resolution of the system (1) was created using the technique of implicit finite differences of the Gauss-Seidel type developed by Gumel et al [28], presented in [29] and noted GSS1 method. We start with a graphical representation of the equilibrium without consumption E0 = (1000;0;0;0;0), using the estimated values of the parameters shown in Table 1, R 0 = 0 and R 0 ≺0 and the state variables initial are chosen as P + M + H + Ci + Q = 1000.
We consider system (1) with the following parameter values. The cure rate of C3 and R and Q 0.0015 Assumed Therefore, the solution converges to the equilibrium E 0 (P(0), 0, 0, 0, 0, 0). It is clearly globally asymptotically stable as soon as R0 < 1, this numerical verification confirms the result stated in model 1 concerning the stability.  (ii) the number of moderate drinkers decreases rapidly at first, then increases slightly and approaches the value M* = 400 (see Figure 3(b)) (iii) the number of heavy drinkersdecreases and approaches the value H*=460 (see Figure   2(c)).
(iv) the number of heavy drinkers with complications of liver disease decreases and approaches the value C1*=120 (see Figure 2 Therefore, the solution curves to the equilibrium E*(P*, M*, H*, C i *, R*, Q*) when R0>1.

DISCUSSION
In this work, we formulated a continuous mathematical model that describes the population dynamics of heavy drinkers and their complications. We studied the stability of this model using the Routh-Hurwitz criteria to analyze the local stability, we proved that the equilibrium point E0 is local asymptotic stable if R 0 <1. We also examined the global stability at l using the Lyapunov function and we also demonstrated that if R>0 then the equilibrium point with alcohol consumption is globally asymptotically stable. We calculated the basic reproduction number R 0 and studied the sensitivity analysis of the model parameters to determine which parameters have a high impact on the reproduction number. Numerical simulations are carried out by MATHLAB to illustrate the theoretical results. These findings could be used to inform public policies and public health interventions aimed at reducing excessive alcohol consumption and its health consequences.