On Poor Academic Attitude: A Mathematical Model With Optimal Control

This study develops and analyzes a compartmental mathematical model to describe the spread of poor academic attitude among senior high school students. The population is divided into five compartments: susceptible (S), exposed (E), active poor attitude (A), partially reformed (Rp), and fully reformed (Rf). The poor attitude reproduction number R0 is derived using the next-generation matrix method. The poor attitude-free equilibrium is shown to be locally and globally asymptotically stable when R0 1. Bifurcation analysis revealed that the system exhibits a forward transcritical bifurcation at R0=1. Sensitivity analysis identifies the transmission rate and reform rate as the most influential parameters. An optimal control problem is formulated with two time-dependent controls representing sensitization programs and counseling interventions. Pontryagin's maximum principle is applied to derive the necessary optimality conditions. The findings provide a theoretical framework for understanding how poor academic attitudes spread through peer interactions and how interventions can be designed to manage them.