Global Dynamics of the model of tumor-immune interaction

In this paper, we consider the model of the tumor and immune system, proposed by Han, He and Kuang (Discrete and Contin. Dyn. Syst. Ser. S 13:2347–2363, 2020). We give the necessary and sufficient conditions for the model to have no tumor equilibrium, one, two or three tumor equilibria respectively. Moreover, we prove that the model has no periodic solution, and give its global dynamics in the first quadrant. We find that the magnitude of the tumor reproduction number $ R_0 $ relative to 1 can describe the final state of tumor and the level of the immune system. If $ R_0\leq1 $, the tumor will be cleaned out by immune system in the end. However, if $ R_0>1 $, the model appears monostable or bistable, i.e., the tumor coexists with the immune system and they are ultimately in a stable steady state.