MATHEMATICAL MODELING AND STABILITY ANALYSIS OF ENDEMIC EQUILIBRIUM POINT OF GAMING DISORDER

In this article, we propose a PEAR mathematical model that describes the dynamic of a population that reacts in the spread of the E-game infection. By using Routh-Hurwitz criteria and constructing Lyapunov functions, the local stability and the global stability of endimec equilibrium point are obtained. Since there are usually errors in data collection and assumed parameter values, we also study the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number R0. The stability analysis of the model that they proposed shows that the system is locally and globally asymptotically stable at endemic equilibrium point when R0 > 1. Finally, some numerical simulations are performed to verify the theoretical analysis using Matlab software.


INTRODUCTION
Electronic games are widespread among children and youth and are now a lucrative business and are an important part of modern digital culture that affects people in many different ways.
Electronic games can be a big problem if used frequently by a child or teenager, they can become a cause of neglect of personal, family or educational responsibilities. Electronic games can become addictive. trying to prohibit the child or teenager from using these games, will make them either sad or angry, and they may want to spend more time playing these games, it should be noted that problems with electronic games are very common, as the start of addiction statistics for these games was underway. In the early eighties, these statistics increased with the increase of the distribution of these games and their connection to the Internet. The problem of addiction to electronic games has been classified as a major problem that can lead to destruction of public health. The concept of electronic games has been linked to many health issues such as obesity, laziness, convergence and social illnesses, as it also affects physical and mental health, which occurs due to frequent use of these games. It also affects children's behaviour, as a study was conducted by a group of children aged 8 to 18 over a period of three years, and the results of this study indicated that violent electronic games stimulate aggressive behaviour, in children due to the large number of children witnessing violence [1]. Dr. Douglas Gentili, an expert on violence in video games and doctor of evolutionary psychology at Iowa State University, says that electronic games affect everybody, whether they are aggressive in nature or not [2].
The World Health Organization has officially classified (International Classification of Diseases ICD-11; WHO, 2018) continuing to play video or electronic games as an addiction leading to a mania, and has announced that people with this mania have certain characteristics, as the inability to 'stop gambling on winning, and according to the organization, a person is classified as having this disease if their addictive behaviour persists for 12 months. However, the diagnosis can be confirmed in a shorter period of time if it is certain that all symptoms are present, the warning about the severity of electronic gaming addiction is not new, but the formal classification of this addiction as a pathological obsession by the World Health Organization can be a major impetus to raise awareness of this disease and take serious action in families and societies to counter it [3] [4] . For years, research has been going on to study and understand the effect of electronic games on the behavior and health of children, for example, we mention. In 2000, Jeanne B. Funk et Al [5] , this study examined relationships among time commitment, gender, prefirence for violent games, and self-concept in 364 fourth and fifth graders. Jeanne B. Funk et Al (2002) [6], correlations were examined between the preference for violent electronic gaming and adolescents' self-perception of emotional behaviors and feelings. In 2003, Barbara Krahé et Al [7], study to sample widely used electronic games varying in violent content,231 adolescents in Germany reported their use of and attraction to violent electronic games. Significant relationships were found between attraction to violent electronic games and the acceptance of norms condoning physical aggression. In 2007, L. Rowell Huesmann [8], the impact of media violence was compared to other known threats to society in order to estimate the importance of the threat to consider. Tom Baranowski et Al (2012) [9], This article explains the basic charac-  [11], diagnostic specificity to distinguish between gaming addiction and high engagement. Lemmens, J.S et Al (2016) [12], Examining the relationship between game genres and Internet Gaming Disorder.
In this work, and primarily based on the research done on the topic, we present a new approach in addressing the phenomenon by defining the degrees of the impact of electronic games on children and adolescents and satisfying them to a potential category that interests all children and adolescents between the ages of 0 and 16 years, including a group interested and plays less than 4 hours a day, and a class that has reached addiction level. Mathematical modeling is one of the most necessary functions that contribute to the representation and simulation of ecological, social and economic phenomena, and convert them into mathematical equations formulated, studied, analyzed, and interpreted their results as examples; Juan Gabriel Brida et Al [13], generalizes the classical model of determination of production prices for two commodities by introducing a dynamics generated by the possibility that the prot rate can be computed using prices of different stages. Zeyang Wang et Al [14], considers and describes the class of cooperative differential games with non-transferable utility and the process of construction of the optimal Pareto strategy with continuous updating. Many studies and research in the social sciences have focused on this subject ( [15][16][17]). However, mathematical studies and research on this subject are still limited and most have focused on the statistical aspect of the phenomenon, for example the study completed by F. A. E tindele Sosso et Al (2020) [18] on insomnia, sleepiness, anxiety and depression among different types of players in African countries. In this work, we propose a new model that describes the dynamics of electronic game addiction. By using Routh-Hurwitz criteria and constructing Lyapunov functions, the local stability and the global stability of endimec equilibrium point are obtained. Since there are usually errors in data collection and assumed parameter values, we also study the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number R 0 . The stability analysis of the model that they proposed shows that the system is locally and globally asymptotically stable at endemic equilibrium point when R 0 > 1.
The paper is organized as follows. in Section 2, we propose a PEAR mathematical model that describes the dynamic of a population that reacts in the spread of the E-game infection. In Section 3, we give some basic properties of the model. In Section 4, we analyze the local stability and global stability and the problem of parameters' sensitivity and some numerical simulations through Matlab software are given. Finally, we conclude the paper in Section 5. Potential Gamers (P), represent children and youth who are vulnerable to infection or who are more likely to become addicted to electronic games. This compartment increasing by the recruitment rate denoted Λ. It is decreasing by µ the rate of people who are older than the age limit set for people concerned with the study. Also It is decreasing by an effective contact with engaged gamers at rate β 1 ( the rate of patients who become engaged gamers because of the bad contact with the other engaged gamers) and with addicted gamers at rate β 2 ( the rate of patients who become engaged gamers because of the bad contact with the addicted gamers ).

A MATHEMATICAL MODEL
Engaged Gamers (E), represent children and youth interested in electronic games and plays more than four hours a day. This compartment is increasing by β 1 and β 2 , and decreasing by the rate µ and by α 1 that represent a rate of the engaged gamers who have become addicted gamers.
Also it is decreasing by γ the rate of the engaged gamers who have become the recovered gamers.
Addicted Gamers (A), represent children and youth who are addicted to electronic games and who suffer from gaming disorders and who have no control over their gaming habits, prioritize gaming over other interests and activities, and continue to game despite its negative consequences. This compartment increasing by α 1 . The compartment of addicted gamers A is decreasing by the rate µ . Also it is decreasing by the rate of the addicted gamers who have become the recovered gamers denoted α 2 .
Recovered Gamers (R), represent children and youth recovering from their addiction to electronic games. This compartment increasing by γ and α 2 .
As a deterministic model, we suppose that, the compartments depend on time. The total population of individuals, N(t) at time t is given as; N(t) = P(t) + E(t) + A(t) + R(t). The following illustration will present disease trends in the compartments (Figure 2). FIGURE 2. Description diagram of the gaming disorder dynamics 2.2. Model Equations. By adding the rates at which the compartment and also by subtracting the rates at which people leave compartment, we obtain a differential equations for the rate at which patients change in each compartmeent during separate times. Therefore, we present the gaming disorder model with the following system of differential equations: where P(0) ≥ 0, E(0) ≥ 0, A(0) ≥ 0, and R(0) ≥ 0 are given initial states. Proof.:

THE MODEL ANALYSIS
Then The both sides in the last inequality are multiplied by exp(´t 0 F(s)ds).
We obtain integrating this inequality from 0 to t gives : So the solution P(t) is positive.
For the positivity of E(t) and A(t) we have: We conclude that E(t) ≥ 0 and A(t) ≥ 0 Similarly, from the fourth equation of system(1), we have Therefore,we can see that P(t) ≥ 0, E(t) ≥ 0, A(t) ≥ 0 and R(t) ≥ 0 ∀t ≥ 0.

Invariant Region.
It is necessary to prove that all solutions of system(1) with positive initial data will remain positive for all times t. we obtained the invariant region, in whith the model solution is bounded. This will be established by the following lemma. Proof: By adding all equations in system (1), one has Then all possible solutions of the system(1) enter the region Ω. It inplies that Ω is a positively invariant set for the system (1). Therefore the basic model is well posed epidemiologically and mathematically. Hence, it is sufficient to study the dynamics of the basic model in Ω.

STABILITY ANALYSIS AND SENSITIVITY OF THE MODEL PARAMETERS
In this section, we will study the stability behavior of system(1) at an Disease an Endemic Equilibrium point. The first three equations in system(1) are independent of the variable R.
Hence, the dynamics of equation system(1) is equivalent to the dynamics of the equation system: where R 0 is the basic reproduction number given by:

The Basic Reproductive Number.
In our work, the basic reproduction number R 0 is defined as the average number of secondary infections produced by an infected individual in a completely potential gamers. To obtain the basic reproduction number, we used the nextgeneration matrix method formulated in [21,22].
Through the model equations system (2),then by the principle of next generation matrix, we obtained: The inverse of V is given by

Local Stability of The Endemic Equilibrium point.
Theorem 2: The endemic equilibrium point is locally asymptotically stable if R 0 > 1 and unstable otherwise.
So, in order to determine the stability of the endemic equilibrium point, we discuss the roots of the following equation λ 3 + aλ 2 + bλ + c = 0.
Obviously we see that a, b, c and ab-c to be positive, (R 0 − 1) must be positive, which leads to R 0 > 1.
So the endemic equilibrium point is locally asymptotically stable if R 0 > 1 and unstable if R 0 < 1.

Global Stability of The Endemic Equilibrium point.
Theorem 3: If R 0 > 1 then The endemic equilibrium point of the system is globally asymptotically stable on Ω.
Proof: Consider the lyapunov fonction L v * : Ω → R given by: L v * = c 1 (P − P * ln( P P * )) + c 2 (E − E * ln( E E * )) Where c 1 and c 2 are positive constants to be chosen latter. Then the time derivative of the lyapunov fonction is given by : Also we obtain dL v * dt = 0 ⇐⇒ P = P * Hence by la salle's invariance principle [23], the endemic equilibrium point is globally asymptotically stable on Ω.

Numerical Simulation.
In this section, we present some numerical solutions of system (1) for different values of the parameters. The resolution of system (1) was created using the Gauss-Seidel-like implicit finite-difference method developed by Gumel and al. [24], presented in [25] and denoted the GSS1 method. the different initial values for each variable of state, and the following parameters: Λ = 20000, µ = 0.01, α 1 = 0.3, α 2 = 0.2, β 1 = 0.5, β 2 = 0.7, γ = 0.4, we have the Endemic Equilibrium point (1.93 * 10 6 , 2.1 * 10 4 , 3.062 * 10 4 , 1.838 * 10 4 ) and R 0 = 2.1127 > 1 . In this case, we obtained the following remarks: Over time, we notice that the number of potential gamers is close to 1.93 * 10 6 . We also note that the number of the engaged gamers is close to 2.1 * 10 4 , and the number of the addicted gamers are close to 3.062 * 10 4 (see Figure 3).
Therefore the solution curves to the endemic equilibruim point when R 0 > 1. To achieve a sensitivity analysis of the model (1) , we followed the technique described by [26].
This technique develops a formula to obtain the sensitivity index of all the basic parameters, defined as , for x represents all the basic parameters. The sensitivity index of R 0 with respect to the remaining parameters are computed as follows: µ + α 1 + γ Their sensitivity indices obtained and evaluated are in Table 1: Interpretation of Sensitivity indexes : The sensitivity indexes of the basic reproductive number with regard to the main parameters are arranged in order in Table 1

CONCLUSION
In this research a mathematical epidemic model for the human population of the gaming disorder is considered. In the pattern of transmission of infection spreads from the engaged or addicted gamers to the sensitive individual population, In the article we proceeded as , after the introduction and related literature in the first, we form a mathematical model that describes the dynamics of gaming disorder. The qualitative analysis of the model shows that the solution of the model is bounded and positive and also by using Routh-Hurwitz criterian and constructing Lyapunov functions, the local stability and the global stability of endemic equilibrium point are obtained. We have found R 0 = β 1 (µ + α 2 ) + β 2 α 1 (µ + α 2 )(µ + α 1 + γ) as basic reproduction number of system (1), which helps us to determine the dynamical behavior of the system. We also studied the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number R 0 . In our future work, we expect to obtain some more results based on real data from known gaming disorder disease to illustrate the validity of our theoretical results, such as how to predict the occurrence of gaming disorder disease, in which way do bifurcations, chaos and strange attractors impact on the dynamics of disease. In addition gain insight into the appropriate intervention strategies to prevent and control the spread and behavior of the gaming disorder described by model.