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.

This study develops and analyzes a fractional‐order compartmental model to describe the spread of gambling addiction within a community, incorporating Caputo derivatives to account for memory effects that influence individuals’ susceptibility over time. The model’s mathematical properties were rigorously investigated, confirming the boundedness and non‐negativity of solutions, identifying addiction‐free and endemic equilibrium points, and analyzing conditions for backward bifurcation when the effective reproduction number is below unity. Local and global stability of these equilibrium points were examined, and an optimal control problem was formulated to determine strategies that effectively reduce gambling prevalence while considering the costs and intensities of interventions. Using Pontryagin’s maximum principle, qualitative analyses were performed, and numerical simulations based on the fourth‐order Runge–Kutta method in MATLAB evaluated the dynamic responses of the system under different scenarios. Preventive and treatment strategies were assessed not only for their impact on addiction reduction but also for their cost‐effectiveness. The influence of two fractional orders κ = 1 (without memory) and κ = 0.7 (with memory) on system dynamics was also explored. Results indicate that a dual strategy—combining public awareness campaigns with targeted counseling and treatment—is the most effective and cost‐efficient approach for controlling gambling addiction in a community. This study provides novel insights into addiction dynamics and offers a practical framework for designing adaptive, economically viable intervention strategies.

This study presents a compartmental model that classifies employees into three categories: “indifferent,” “resistant,” and “adaptive,” to explore their transitions based on adaptation to workplace reform initiatives. The researchers rigorously assessed the model for well-posedness and stability of its steady states. Utilizing Pontryagin’s maximum principle alongside numerical simulations, the researchers identified effective strategies aimed at reducing the number of resistant employees, thereby cultivating a more supportive atmosphere for reform efforts. The findings indicate that such an environment encourages employees to remain indifferent or adaptive, fostering a positive outlook toward change. The optimal strategies identified include providing training sessions to enhance employees' skills for adapting to new processes and technologies, as well as ensuring clear communication regarding the rationale, benefits, and impacts of the reforms. Furthermore, the study examined memory effects by transforming the integer order model into a fractional order model, with graphical representations highlighting the significance of fractional derivatives in illustrating the evolution of employees' perceptions over time. This research contributes valuable insights into managing employee adaptation during organizational change.

The propagation of hate speech directed toward local public sector administrations in a community has become an issue of great concern. Hate speech not only undermines social harmony but also poses significant challenges to the smooth functioning of the public sector and the well-being of the community members. In this study, we formulated and analyzed a Caputo fractional order model with optimal control strategies on the propagation of hate speech, as an evolutionary system. The non-negativity and boundedness of the solutions of the fractional order model have been shown with the objective of making the evolutionary system meaningful. Both hate speech-free and hate speech-persistent equilibrium points were determined. Conditions for the backward bifurcation of the fractional order model were analyzed when the hate speech effective reproduction number is less than unity. The global asymptotic stability of hate speech-persistent equilibrium point has also been shown. Furthermore, we employed optimal control strategies and numerical simulations to examine the memory effect of the fractional order and the behavior of responses of the model. Protection and treatment of individuals engaged in propagating hate speech in the community has a significant impact in reducing and possibly exterminating hate speech propagation in the community.

Increasingly, rail passengers seek to experience higher quality services. Therefore, it is essential to identify and execute the most efficient improvement strategies that enhance passenger satisfaction. In order to achieve this goal, this research centers on formulating a compartmental model to examine passengers′ perceptions of rail services as a dynamic system and a corresponding optimal control problem. The researcher proves that the model has a unique, nonnegative, and bounded solution. Additionally, by applying Pontryagin′s maximum principle, the researcher derives optimal control strategies for the problem, enhancing operational efficiency and minimizing passenger dissatisfaction. The researcher also conducts numerical simulations using the ODE45 solver to analyze how the model solutions evolve over time. The numerical simulations demonstrate that the fractional derivative′s order significantly influences how passenger satisfaction changes over time. The findings reveal that the most effective strategies for enhancing passenger satisfaction include implementing measures such as improving the reliability and timeliness of train services as well as maintaining high safety and security standards. Rail operators are advised to prioritize these measures accordingly.