The numerical methods, including the fractional Runge–Kutta scheme (Section 2.4), model equations (Section 2.1–2.2), and parameter estimation procedures (Section 2.4), are described in sufficient detail to enable independent implementation in any scientific computing environment. The fractional step size formula κ=hα/Γ(α+1), the complete system of fractional differential equations, and all time-dependent parameter functions are explicitly provided. Researchers may independently reproduce our results using these specifications with standard numerical computing software such as MATLAB, Python, Mathematica, or R.

© 2026 The Authors. Published by Elsevier B.V.