Analysis of a caputo fractional-order modeling approach to dietary-gut-brain interactions in autism spectrum disorder

Autism spectrum disorder (ASD) affects approximately 1 in 36 children worldwide, yet the complex, memory-dependent dynamics of the gut–brain–behavior axis remain poorly quantified. Existing integer-order models cannot capture the prolonged inflammatory memory characteristic of ASD, while fractional-order formulations, though promising, have never been applied to this pathway. Moreover, many fractional models suffer from dimensional inconsistency and improperly defined disease-free equilibria. This study addresses these gaps by developing a dimensionally consistent Caputo fractional-order system (orders \(\alpha _i\in (0,1]\) ) with bilinear coupling \(bFG\) , ensuring a genuine inflammation-free equilibrium. Using next-generation matrix analysis, we derive the reproduction number \(\mathcal {R}_0 = bI_f/(\mu _f\delta _g)+c/\delta _g\) , which decomposes into antigen-driven and intrinsic inflammatory components. Rigorous proofs of existence, uniqueness, non-negativity, boundedness, and Matignon stability are provided, along with local and global Sobol’ sensitivity analyses. The gut resolution rate \(\delta _g\) shows unitary negative elasticity, while \(\delta _b\) , \(\delta _g\) and \(I_f\) explain over 94% of behavioral variance in the model. Optimal control theory (Pontryagin’s principle for fractional systems) and numerical simulations (Adams–Bashforth–Moulton) reveal that within the model’s assumptions only triple combination therapy (dietary restriction, gut barrier repair, and behavioral support) pushes \(\mathcal {R}_0<1\) and reduces symptom severity by 98% in silico. This work proposes a mathematically rigorous fractional-order framework for the ASD gut–brain axis. Because the model is theoretical and exploratory, all findings require empirical validation before any clinical application.