Threshold behavior of an age-structured tumor immune model

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.