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 \(α _i ∈ (0,1]\) 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 \(R_0\) , prove a transcritical bifurcation at \(R_0=1\) , perform global Sobol sensitivity analysis, and formulate a fractional optimal control problem with three realistic interventions. Principal findings reveal: (i) a transcritical bifurcation at \(R_0=1\) separating healthy recovery from chronic pathology; (ii) sensitivity analysis identifies the microbiota recovery rate \(δ _m\) (elasticity \(-1.667\) ) as the most potent target for preventing chronicity and the behavioral recovery rate \(δ _b\) ( \(S_i = 0.285\) ) 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 ( \(u_3\) ) essential for minimizing symptoms; and (iv) stronger memory effects (lower \(α\) ) significantly delay therapeutic recovery, confirming that ignoring memory fundamentally underestimates ASD symptom persistence.