Improving passengers’ attitudes toward safety and unreliable train operations: analysis of a mathematical model of fractional order

In this study, we aimed to explore the dynamics of rail passengers’ negative attitudes that can be influenced by safety concerns and unreliable train operations. We mainly formulated and analyzed a mathematical model of fractional order and derived an optimal control problem considering the Caputo fractional order derivative. In the analysis part of the model, we proved that the solutions of the model for the dynamical system are non-negative and bounded, and determined the passengers’ negative attitude-free and negative attitude persistence equilibrium points of the model. Both the local and global stabilities of these equilibrium points were examined. Furthermore, we verified the conditions necessary for the existence of optimal control strategies. We then proceeded to analyze the proposed control strategies, which aim to prevent negative attitudes and improve the attitudes of passengers who have already developed negative attitudes. Finally, we conducted numerical simulations to examine the effects of these control strategies. The results revealed that protecting passengers from developing negative attitudes and improving the attitudes of those who have already developed such attitudes are crucial for improving the overall attitude of railway passengers. These measures can effectively address any negative experiences caused by safety concerns and unreliable train operations.