Comparative study of fractional-order tumour–immune–drug interaction models using singular and nonsingular kernels

Abstract

This study develops a fractional-order reaction–diffusion framework for tumour–immune–drug interactions, incorporating Caputo, Caputo–Fabrizio, and Atangana–Baleanu derivatives to capture memory-dependent biological processes. The novel approach integrates and compares these three fractional operators within a unified system. Theoretical analysis rigorously establishes positivity, boundedness, equilibrium points, and the basic reproduction number. Tailored numerical schemes are constructed for each operator, with detailed stability and convergence analyses. Simulations demonstrate that nonsingular kernels significantly enhance tumour suppression, immune response, and drug diffusion compared to classical singular-kernel models. These findings provide deeper insights into memory effects in cancer progression and support the use of fractional-order models to optimise therapeutic strategies.