Bidirectional fractional-order dynamics of the microbiota-gut-brain axis in autism spectrum disorder featuring memory effects and inflammation thresholds

Conventional models of the microbiota-gut-brain axis (MGBA) in Autism Spectrum Disorder (ASD) overlook the long-range memory effects inherent to neuroinflammation and gut dysbiosis, leading to underestimation of symptom persistence. This study introduces a bidirectional fractional-order dynamical system with compartment-specific fractional orders [Formula: see text] to capture these memory-dependent dynamics across five interconnected compartments: dietary antigens, gut dysbiosis, gut inflammation, neuroinflammation, and behavioral symptoms. We establish existence, uniqueness, boundedness, and stability via Matignon's criterion, derive a novel inflammation threshold parameter [Formula: see text], prove a transcritical bifurcation at [Formula: see text], perform global Sobol sensitivity analysis, and formulate a fractional optimal control problem with three realistic interventions. Principal findings reveal: (i) a transcritical bifurcation at [Formula: see text] separating healthy recovery from chronic pathology; (ii) sensitivity analysis identifies the microbiota recovery rate [Formula: see text] (elasticity [Formula: see text]) as the most potent target for preventing chronicity and the behavioral recovery rate [Formula: see text] ([Formula: see text]) as the dominant driver of steady-state symptom severity; (iii) optimal control simulations demonstrate that triple combination therapy outperforms all double combinations, with direct behavioral intervention ([Formula: see text]) essential for minimizing symptoms; and (iv) stronger memory effects (lower α) significantly delay therapeutic recovery, confirming that ignoring memory fundamentally underestimates ASD symptom persistence.