Optimal control of an online game addiction model with positive and negative media reports

In the spread of infectious diseases, media reports have played a positive role. However, in the process of game communication, there are not only positive but also negative media reports. Therefore, in this paper, we establish a model with positive and negative media reports to analyze the role of media reports in the process of game communication. First, we study its positivity and boundedness, and calculate the basic reproduction number \(R_{0}\) under three different incidence rates. The existence and stability of the equilibria are proved. Secondly, the optimal control problem is studied by adding two dynamic variables of media reports. Finally, in the simulation we simulate the stability of the equilibria, so as to verify the correctness of the theory. Then the influence of media parameters on \(R_{0}\) is analyzed. The numerical results of optimal control are simulated by forward and backward sweep method. By comparing the results of the optimal control and without control, the media coverage should be controlled according to the optimal control measures shown in this paper. The serious situation of game addiction will be greatly alleviated, which can reduce the infection rate by at least 90%.