Stability Analysis and Optimal Control of a Fractional-Order Glioma-Immune Model under Combined Therapy

This study presents a novel fractional-order mathematical model to investigate the dynamics of glioma-immune interactions under therapeutic interventions. Building upon and extending previous models by de Pillis et al. and Pillay et al., we incorporate immune checkpoint inhibition and fractional-order derivatives to capture the memory-dependent behavior of immune responses. The model distinguishes between therapy-sensitive and resistant glioma cells and includes key immune components such as natural killer cells, cytotoxic T lymphocytes, cytokines, and exhausted effectors. An optimal control problem is formulated with three control variables representing immunotherapy, virus therapy, and checkpoint blockade. The forward-backward sweep method is employed to compute optimal treatment strategies over a 50-day horizon. Numerical simulations demonstrate that the fractional-order framework significantly influences treatment outcomes, with lower fractional orders delaying immune activation and reducing therapeutic efficacy. The proposed optimal control strategy achieves superior glioma suppression and immune activation compared to monotherapies and fixed-dose combinations. These findings highlight the importance of incorporating memory effects and multi-modal control in the design of effective glioma treatment protocols.