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We highlight and study in this paper the phenomenon of the spread of addiction to electronic games, where the addict goes through stages before reaching the degree of addiction. In order to model this phenomenon, we have divided people into four groups, which are potential gamers, engaged gamers, addicted gamers, and gamers who have recovered from addiction. We propose a discrete mathematical model with control strategies using three controls that represent, respectively,

Electronic games are seen as a traded technological commodity because they are part of modern digital culture, which affects people in different ways. Electronic games can be a big problem if they are used by the child or adolescent in a very frequent way; they can become a reason for neglecting personal, family, or educational responsibilities. Electronic games can become addictive and 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. The statistics about gaming addiction increased with the increase of the distribution of these games and their connection to the Internet [

Descriptive epidemiology of gaming among the nine African countries where prevalence of gaming, mean hours of gaming per week, period from when participant considered himself a gamer, and type of device used for gaming purposes are described with age and sex [

Mathematical modeling and control theory are considered as two of the most necessary tools to represent, simulate, and control the evolution of some phenomena including ecological, social, and economic ones. These tools help convert the phenomenon into mathematical equations and also to formulate study, analyze, and interpret their results. For example, Brida and Cayssials [

In this research, we will adopt discrete-time modeling where statistical data are collected at a discrete time (day, week, month, and year). Therefore, it is more direct, more convenient, and more accurate to describe a phenomenon using discrete-time modeling compared to continuous-time modeling. Also, we mainly shed light on the category of addicts and make a differentiation between practitioners and addicts of electronic games. Eventually, we propose strategies for optimal control of the spread of this addiction. We add the cost-effectiveness ratio (ICER) to be able to choose between strategies based on obtaining better results at a lower cost.

In this work, we propose a new model that describes the dynamics of electronic game addiction. The population that we study consists of children and adolescents aged less than 24 years old within Morocco. The population under study is divided into four compartments, which are potential gamers, engaged gamers, addicted gamers, and recovered gamers. Our main objective is to propose an optimal control strategy that will minimize the number of addicted gamers. To achieve this objective, we adopt three controls that represent awareness through media, guidance on alternative educational and recreational methods, and creating rehabilitation centers for electronic game addiction.

This paper is organized as follows. In Section

We consider a discrete mathematical model PEAR that describes the dynamics of a population having gaming disorder. We divide the population into four compartments. The following illustration will show disease trends in the compartments in Figure

Illustration of movement between compartments.

It represents children and youth who are vulnerable to infection or who are more likely to become addicted to electronic games. This compartment is increased by the recruitment rate denoted by

It represents children and youth who are interested in electronic games and play more than four hours a day without secondary effects on the body and on the individual’s behaviour in the social environment. This compartment is increased by

It represents children and youth who are addicted to electronic games, suffer from gaming disorders, and have no control over their gaming habits. They prioritize gaming over other interests and activities and continue to game despite its negative consequences. This compartment is increased by

It represents children and youth recovering from their addiction to electronic games. This compartment is increased by

By adding the rates at which the steps of gaming disorder enter the compartment and also by subtracting the rates at which people leave a compartment, we obtain a system of difference equations for the rate at which patients change in each compartment during separate times. Therefore, we present the gaming disorder model with the following system of difference equations:

In order to demonstrate the efficiency of the model, we propose a numerical simulation (see Figure

Development of the number of the engaged gamers and the addicted gamers in a few months.

Figure

At present and in the light of the rapid development of computer technology and consequently the development and spread of electronic games, the world including Morocco has witnessed an increase in the number of addicts to electronic games. In fact, the addiction of children and young people leads to a sharp decline in the level of academic achievement, productivity at work and introspection, and a strong tendency to isolation. To address this phenomenon and mitigate its effects on individuals and society, we have proposed a set of practical strategies, which we present in the following paragraph.

Our objective in the proposed control strategy is to minimize the number of engaged gamers and the number of addicted gamers. Therefore, in model (

Thus, the controlled mathematical system is given by the following system of difference equations:

Then, the problem is to minimize the objective functional:

They are selected to weigh the relative importance of

The sufficient condition for the existence of the optimal controls

There exist the optimal controls

Since the coefficients of the state equations are bounded and there are a finite number of time steps,

Finally, due to the finite-dimensional structure of system (

In order to derive the necessary condition for the optimal controls, we use the discrete version of Pontryagin’s maximum principle [

Given the optimal controls

With the transversality conditions at time

Furthermore, for

The Hamiltonian at time step

In this section, we present the results obtained by numerically solving the optimality system. As explained in [

The description of the parameters data used for systems (

Parameter | Description | Estimated value | Source |
---|---|---|---|

The number of children and youth in Morocco | 15000000 | [ | |

Initial number of potential gamers | 9585000 | [ | |

Initial number of engaged gamers | 1290000 | [ | |

Initial number of addicted gamers | 4125000 | [ | |

Initial number of recovered gamers | 0 | Assumed | |

The incidence of potential gamers | 7988 | [ | |

The rate of the potential gamers who become engaged gamers | 0.36 | [ | |

The rate of the engaged gamers who become addicted gamers | 0.76 | [ | |

The rate of the addicted gamers who become recovered gamers | 0.1 | Assumed | |

The rate of the | 0.46 | [ | |

The rate of the P who become engaged gamers by contact with addicted gamers | 0.26 | [ | |

The rate of the recovered gamers who become engaged gamers | 0.2 | Assumed | |

The rate of the addicted gamers who become engaged gamers | 0.1 | Assumed | |

The rate of the recovered gamers who become potential gamers | 0.4 | Assumed |

In this section, we offer a digital simulation to highlight the effectiveness of our strategy to combat the impact of addiction to electronic games on children and young people and reduce their addiction. Different simulations can be carried out using various values of parameters as shown in Table

In this scenario, we simulate the case where we apply a single control

Development of the number of the addicted gamers with one control in a few months.

Development of the number of the engaged gamers with one control in a few months.

In this scenario, applying the control

In this scenario, we apply the control

In this scenario, we combine Scenarios

Development of the number of the addicted gamers with the combination of two controls among the controls

Development of the number of the engaged gamers with the combination of two controls among the controls

In this scenario, we combine Scenarios

In this scenario, we combine all the three controls

Finally, in Figures

The evolution of the number of the addicted gamers with combination of the three controls.

The evolution of the number of the engaged gamers with combination of the three controls.

The evolution of all controls

In this section, we analyze the cost-effectiveness ratio (ICER) of the previous seven scenarios by comparing them to determine the most cost-effectiveness ratio (ICER). Following the method as applied in several studies [

Total costs and total averted infections for all scenarios.

Strategy | Total averted infections (TA) | Total cost (TC) |
---|---|---|

2 | ||

4 | ||

1 | ||

6 | ||

3 | ||

7 | ||

5 |

First, we compared the cost-effectiveness of Scenarios

We note that ICER (2) is higher than ICER (4). This means that Scenario

Second, we compared the cost-effectiveness of Scenarios

By comparing Scenarios

Next, Scenario

Since

Next, we compare the cost-effectiveness of Scenario

The comparison of ICER (3) and ICER (6) indicates that Scenario

Now, we compare the cost-effectiveness of Scenario

The lower ICER obtained for Scenario

Finally, Scenario

The comparison reveals that Scenario

In this research, we proposed a new mathematical model that describes the dynamics of gaming disorder. The model will allow us to understand the phenomenon of the spread of this disorder and to deal with it. Thus, we applied three control strategies. As a result, we were able to study and combine several scenarios in order to see the impact and effect of each of these controls on the spread of this disorder. To obtain results that relatively simulate reality, we calculated transactions

No data were used to support this study.

The authors declare that they have no conflicts of interest.