Mathematical Analysis and Parameter Estimation of a Fractional-Order Addiction Model With Optimal Control and Cost-Effectiveness Evaluation: A Case Study on University Students

This study examines the transmission dynamics of various addictions among university students through an extensive fractional-order mathematical model. Using fractional calculus, the proposed framework can adequately account for the memory effects and hereditary properties of addictive behaviors. The model has been modified to accommodate relapse factors and also the role of persons in the recovery states. For empirical validity, the model’s parameters are estimated using field data collected through student surveys to complement the theoretical analyses with campus realities. Detailed mathematical analysis is done for the existence and uniqueness of solutions. The transmission threshold is identified by calculating the basic reproductive number, and the system’s stability at both the addiction-free and endemic equilibria is rigorously analyzed and proved. Bifurcation analysis is also performed to capture qualitative changes in the dynamics, and a parameter sensitivity analysis is conducted to identify the most sensitive parameters for the spread of addiction. To this end, an optimal control problem is formulated using time-dependent intervention strategies to minimize the prevalence of addiction within the university population. Numerical solutions of the fractional-order system are obtained using the Adams–Bashforth–Moulton predictor–corrector algorithm. Lastly, numerical simulations are given to prove the theoretical results, aided by an incremental cost-effectiveness ratio analysis to determine the most cost-effective strategies. This work contributes to the literature by providing a robust, data-driven decision-support tool that integrates fractional-order dynamics with economic evaluation for targeted public health interventions.