An Optimal Control Framework for Adaptive Virtual Reality Interventions in Drug Rehabilitation

Abstract

Substance use disorders remain a significant public health challenge, with relapse prevention continuing to be one of the central difficulties in rehabilitation. Traditional rehabilitation often faces barriers such as limited accessibility and difficulty recreating real world environments associated with cravings. As a result, researchers have begun to study virtual reality as a tool for drug rehabilitation. We introduce an Adaptive- Perturbed-Corrective-Relapse (APCR) compartmental model representing transitions between stable recovery, cue-induced perturbation, corrective intervention, and relapse states with an optimal control framework to model and optimize VR aided drug rehabilitation. We formulate the VR intervention intensity as a dynamic control variable u(t), allowing the timing and magnitude of therapeutic exposure to be optimized Using Pontryagin’s Minimum Principle, we derived an optimal intervention strategy that balances reduction of cue induced perturbation against the cost of VR implementation across a 60-minute period. We developed a baseline model that incorporates behavioral biomarkers collected from VR simulations, including gaze duration toward drug-associated cues, to parametrize time-dependent relapse triggers Numerical solutions obtained via a forward backward sweep method demonstrate the effectiveness of the optimal control strategy demonstrating the potential of mathematical optimization in the design of personalized VR rehabilitation while supporting the broader objective of improving health and wellbeing aligned with sustainable goal number three of the United Nations.