Modeling Placement-Based Student Pathways in Higher Education: A Compartmental Framework of Flow from Placement to Assessment

Student progression through placement-based course pathways can be represented as a continuous-time dynamical system. We utilize a theoretical three-compartment framework to model institutional student flow in placement-exam-determined courses, with movement through placement (P: students placed but not yet engaged), active engagement (E: students participating in coursework), and assessment (A: students reaching the assessment stage). The model consists of ordinary differential equations describing recruitment, stage-to-stage transitions, stage-specific attrition, and remediation that returns students who failed assessments to re-enrollment in the course. We establish the existence and uniqueness of solutions, characterize the equilibria, and derive conditions for local stability. The system is solved numerically using an adaptive hybrid block method, and parameters are examined under five simulated scenarios: high recruitment, low placement, high attrition, slow progression, and low pass rates. The result suggests that high placement recruitment increases enrollment and may not be proportional to success at the assessment stage; low placement-to-engagement transitions can limit access, slow progression constrains the conversion of enrolled students into successful completers, low pass rates generate remediation cycles that may create additional resource demands, and high attrition produces substantial losses across the pathway. Future work should focus on empirical validation using real-world educational datasets to calibrate the model.