Population Dynamics on Fractional Tumor System Using Laplace Transform and Stability Analysis

Modeling is an effective way of using mathematical concepts and tools to represent natural systems and phenomena. Fractional calculus is an essential part of modeling a biological system. Recently, many researchers have been interested in modeling real-time problems mathematically and analyzing them. In this paper, the tumor system under fractional order is considered, and it comprises normal cells, tumor cells and effector-immune cells. By taking chemotherapy drugs into account, the toxicity of the drug and concentration of the drug is also studied in the model. The main objective of this work is to establish the solution for the model using Laplace transform and analyze the stability of the model. Laplace transform, a simple and efficient method, is used in solving the system that proves the existence and uniqueness of the solution. The boundedness of the system is also verified using the Lipschitz condition. Further, the system is solved for numerical values, and the population dynamics of cells are provided for different values of $\alpha$ as a graphical representation. Also, after analyzing the effect of chemotherapy drugs on tumor cells for different $\alpha$'s, which signifies that $\alpha$ = 0.9 provides a sufficient decrease in the dynamics of tumor cells. The main and significant part of this work is presenting that the usage of chemotherapy drugs reduces the number of tumor cells. The importance of the work is that apart from the immune system, chemotherapy drugs play a significant role in destroying tumor cells. The Hyers Ulam stability has a significant application that one need not find the exact solution to when analyzing a Hyers Ulam stable system. Thus, the stability of this tumor model under Caputo fractional order is presented using Hyers-Ulam stability and Hyers-Ulam-Rassias stability.