Stability analysis and optimal control of a time-delayed panic-spreading model

In emergencies, the transmission of false and uncertain information from individual to individual causes group panic, which in turn leads to the spread of negative emotions in the group. To explore the process of panic spreading in groups, an improved panic-spreading model is constructed in this study. First, the groups are divided into the impatient group and the level-headed group, based on the theory of personality traits in psychology. Second, the logistic model is used to express the growth in the number of susceptible individuals subject to emergencies. Third, the delay effect of panic in the group can have an influence on the spread of panic. Therefore, a time-delayed panic-spreading model considering the epidemic model is established. The threshold value of the model is calculated, and the conditions for the local and global stability of the panic-free equilibrium and panic-permanent equilibrium are obtained by analyzing the dynamic behavior of the delayed-time panic model. On this basis, we choose the intensity of government measures as control variables and establish an optimal control model to minimize the spread scale. The existence and necessary conditions of the optimal solution are proved. Finally, the correctness of the conclusion is verified by numerical simulations.