Mathematical Modeling of Knowledge Transfer between Students and Mentors to Estimate Ideal Quantities of Mentors for Student Populations Using Optimal Control Theory

Abstract

A shortage of educators has always been a concern across all educational institutes in the US. In education, the main goal is to progress students adequately and evenly; however, it is inevitable for some to fall behind. To combat this, upwards of 30% of students graciously volunteer their time as mentors. This population is limited; therefore, it is crucial to optimize the mentor assignments maximizing students benefited and minimizing mentors employed. In this project, a compartmental model of differential equations was used to describe the interactions between students and mentors. The compartments of a Susceptible-Exposed-Infected-Recovered (SEIR) model are modified to describe the positive propagation of knowledge. As a result, the Amendable-Learning-Informed-Unlearned (ALIUM) model describes the spread of information, where instead of an Infected category, the Informed compartment holds the population of students that were exposed to information through other students, students in the process of learning, and mentors each with unique transmission rates (β₁, β ₂, β₃). The M variable is used to keep track of the percentage of Informed students that are required as mentors. To optimize the M variable, Optimal Control Theory is carried out using Pontryagin’s Maximum Principle and the Forward-Backward Sweep Algorithm with the aim to minimize the number of tutors necessary and maximize the informed population. Preliminary results show that, with a nonlinear control, 30% of the Informed population must be employed as tutors. Mentors’ high employment rate is needed during the first quarter of the whole learning period, before gradually declining to 0% of the Informed population by the time learning is finished. Future research hopes to explore heterogeneous learning speeds for students. Also, a further step is to shape the model according to real world data using Physics Informed Neural Networks (PINNs). This work aligns with UN Sustainability Goal #4: Quality Education.