Threshold behavior of an age-structured tumor immune model^*

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, PR China

^** Corresponding author: hbliuzijian@126.com

Received: 29 September 2022
Accepted: 3 January 2023

Abstract

In this paper, we present and analyze an age-structured tumor immune model. Based on the fact that tumor cells of different ages tend to exhibit different physiological behaviors, we consider the age structure of tumor cells, age-based proliferation function and age-dependent death function in the model. The threshold R₀ for the existence of tumor-free steady state is derived. It is found that if R₀ > 1, the tumor-free steady state is not only locally stable but also globally stable. Moreover, numerical simulation shows that the threshold R₀ may be regarded as an index to reflect the ability of “tumor immune surveillance”, that is, the smaller the R₀, the better the ability of tumor immune surveillance. If R₀ > 1, it is proved that the tumor steady state is existent and uniformly persistent. The local stability of the tumor steady state is investigated under some further conditions besides R₀ > 1. In the end, we estimate the system parameters, verify the theoretical results and analyze some system parameters’ sensitivities.