Mathematical modeling, sensitivity analysis, and optimal control of students awareness in mathematics education

Abstract

This article presents a continuous-time mathematical model, EFSMK, integrating various educational categories, serving as a valuable tool for understanding students interactions with mathematics. The paper begins by exploring the necessary preliminaries to describe the model. Subsequently, we examine the model’s well-posedness, focusing on the positivity and boundedness of solutions. The basic reproduction number is then derived using the next-generation matrix method. The model exhibits two stable states: the mathematics-free equilibrium point and the learning equilibrium point for students who struggle with mathematics. The study shows that when the basic reproduction number is less than one, the mathematics-free equilibrium point is globally asymptotically stable. Conversely, when the basic reproduction number exceeds one, the learning equilibrium point for students who struggle with mathematics is globally asymptotically stable. Numerical simulations validate the theoretical findings. In the last part of the paper, we present a structured model involving two distinct strategies aimed at improving the mathematical skills of students who encounter difficulties in this subject. The first strategy consists of offering individualized or small-group tutoring sessions, focusing on challenging topics and using alternative teaching methods. The second strategy entails adjusting the overall program and pedagogical approach to better respond to diverse learning needs. This approach uses Pontryagin’s maximum principle and iterative analysis to determine optimal controls. The effectiveness of these strategies is evaluated using numerical simulations in various scenarios.

Keywords

• Optimal control
• Global stability
• Forward-backward sweep numerical method
• Lyapunov’s function
• Sensitive analysis