Computational framework for fractional stochastic delay systems: Theory and applications in Hepatitis B virus dynamics

This paper introduces a novel computational framework for fractional stochastic delay differential equations driven by Wiener processes, combining B-spline linear interpolation with Lagrange quadratic interpolation to discretize the Caputo fractional derivative. The hybrid scheme strategically exploits computational efficiency in initial intervals while leveraging enhanced accuracy in subsequent subdomains, achieving mean absolute errors significantly lower than existing integral quadratic spline methods. Rigorous mathematical analysis establishes explicit error bounds accounting for stochastic perturbations, mean-square stability conditions under Lipschitz constraints via discrete fractional Grönwall inequalities, and convergence orders substantially exceeding traditional second-order schemes with considerable reduction in computational time. The framework’s practical utility is demonstrated through a fractional stochastic delay Hepatitis B virus model incorporating vaccination, disease-induced mortality and environmental stochasticity. Numerical simulations across various fractional orders reveal distinct epidemic regimes relevant to intervention strategies. Statistical analysis over multiple trajectories provides comprehensive characterization via confidence intervals, quartile distributions and moment analysis, demonstrating robustness under random environmental fluctuations. This unified approach successfully addresses the interplay between non-local memory effects, time-delayed dynamics and stochastic variability, providing a powerful tool for analyzing biological systems with hereditary properties and environmental uncertainty.