Abstract
<title>Abstract</title> <p>This paper investigates the fundamental properties of the Caputo and Riemann-Liouville fractional derivatives and applies the obtained analytical results to a fractional-order cancer cell epidemic model. The theoretical framework covers linearity, fractional integrals and derivatives of power functions, correction-term relations, and inverse properties of both operators. A key finding is that the Caputo derivative of a constant vanishes while the Riemann-Liouville derivative produces a singular term near the origin, making the Caputo formulation more suitable for biological initial-value problems. These results are applied to a coupled fractional system governing healthy and cancer cell populations, incorporating logistic growth, inter-cellular competition, and immunotherapy control. Numerical simulations using the Grünwald-Letnikov scheme demonstrate the influence of the fractional order on memory effects, disease progression, and treatment response. Graphical comparisons confirm the theoretical distinctions between the two operators and illustrate the sub-diffusive behaviour induced by lower fractional orders. The presented analysis provides a rigorous and practically relevant foundation for the study of fractional differential equations in applied mathematical biology. Mathematics subject classification:26A33, 34A08, 34K37, 39A12, 92C50</p>