Asymptotic Behavior Analysis of a Fractional-Order Tumor-Immune Interaction Model with Immunotherapy

Wenbin Yang,Shuhui Liang,Xiaojuan Wang,Xiaozhou Feng

doi:10.1155/2020/7062957

Abstract

A fractional-order tumor-immune interaction model with immunotherapy is proposed and examined. The existence, uniqueness, and nonnegativity of the solutions are proved. The local and global asymptotic stability of some equilibrium points are investigated. In particular, we present the sufficient conditions for asymptotic stability of tumor-free equilibrium. Finally, numerical simulations are conducted to illustrate the analytical results. The results indicate that the fractional order has a stabilization effect, and it may help to control the tumor extinction.

Highlights

Tumor or tumour is a term used to describe the name for a swelling or lesion formed by an abnormal growth of cells
Cancer cells can spread to other parts of the body through blood and lymph systems [1], and so cancer is known as the leading cause of death in the world
adoptive cellular immunotherapy (ACI) refers to the injection of cultured immune cells that have antitumor reactivity into the tumor-bearing host, which is typically achieved in conjunction with large amounts of IL-2 by using the following two methods: lymphokine-activated killer (LAK) therapy and tumor-infiltrating lymphocytes (TIL) therapy

Summary

Introduction

Tumor or tumour is a term used to describe the name for a swelling or lesion formed by an abnormal growth of cells. Is is because fractional-order differential equations are naturally related to systems with memory [8]. Motivated by the above considerations, in this paper, we study a fractional-order tumor-immune interaction system by extending the integer order model (3) as follows:. 2. Existence, Uniqueness, and Nonnegativity is section studies the existence, uniqueness, and nonnegativity of the solutions of the fractional-order model (4). To prove the existence and uniqueness of the solution for model (4), we need the following lemma. For each initial condition X0 (u0, v0, w0) ∈ Ω, there exists a unique solution of the fractional-order model (4), which is defined for all t ≥ 0. Us, F(X) satisfies the Lipschitz condition with respect to X It follows from Lemma 1 that there exists a unique solution of model (4).

Equilibria Analysis and Asymptotic Stability

Numerical Simulation

Concluding Remarks