Threshold Behavior of an Age-structured Tumor Immune Model

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 $\mathfrak{R}_{0}$ for the existence of tumor-free steady state is derived. It is found that if $\mathfrak{R}_{0}<1$, the tumor-free steady state is not only locally stable but also globally stable. Moreover, numerical simulation shows that the threshold $\mathfrak{R}_{0}$ may be regarded as an index to reflect the ability of ``tumor immune surveillance", \ie, the smaller the $\mathfrak{R}_{0}$, the better the ability of tumor immune surveillance. If $\mathfrak{R}_{0}>1$, 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 $\mathfrak{R}_{0}>1$. In the end, we estimate the system parameters, verify the theoretical results and analyze some system  parameters' sensitivities.