MATHEMATICAL MODELING OF KNOWLEDGE TRANSMISSION IN A CLASSROOM USING THE SIR FRAMEWORK

Abstract

This study presents a mathematical analysis of a Susceptible–Informed–Recovered (SIR) model to describe the spread of knowledge in a classroom environment. The model divides students into three compartments: susceptible students who have not yet understood a concept, informed students who have learned the concept and can assist others, and recovered students who have fully mastered the concept. The model incorporates learning transmission and knowledge mastery rates to represent realistic classroom interactions. The existence and uniqueness of solutions are established using the Lipschitz condition, ensuring that the model is mathematically well-posed. Positivity of solutions is proved to guarantee that all state variables remain non-negative over time. An invariant region is derived to demonstrate that the total number of students remains bounded. The knowledge-free equilibrium is determined and analyzed as the baseline state where no students have learned the concept. A threshold parameter analogous to the basic reproduction number  is computed to measure the effectiveness of knowledge transmission in the classroom. Local stability of the equilibrium is examined through Jacobian matrix and eigenvalue analysis, showing stability when the threshold parameter is less than unity. Furthermore, a Lyapunov function is constructed to establish global stability under the same condition. The results provide theoretical insights into how knowledge spreads through peer interaction and instruction. Overall, the model offers a useful mathematical framework for understanding learning dynamics and improving teaching strategies in educational settings.