Analysis of time-fractional cancer-tumor immunotherapy model using modified He-Laplace algorithm

Cancer encompasses various diseases characterized by the uncontrolled growth of abnormal cells, which can invade healthy tissues and spread throughout the body, making it the second leading cause of death worldwide. This study presents a fractional cancer treatment model with immunotherapy to enhance understanding of cancer's mathematical framework and behavior. The model comprises fractional differential equations analyzed using the Caputo-fractional derivative, aiming to control cancer growth while considering cell population metrics. A framework integrating various homotopies and Laplace transforms is developed to explore cancer's complexities. Simultaneous solution profiles for effector immune cells and tumor cells illustrate their mutual influence. The model examines parameters such as the death rate of immune cells, natural tumor growth rate, rate of immune cells killing fractional tumor cells and numerous others graphically for clarity. The fractional parameter β is visually represented through 2D, 3D, and contour plots. This comprehensive analysis validates the proposed approach, suggesting its applicability to other complex cancer treatment models for better decision-making in cancer treatment.