Global dynamics in a tumor-immune model with an immune checkpoint inhibitor

In this paper, we fill several key gaps in the study of the global dynamics of a highly nonlinear tumor-immune model with an immune check- point inhibitor proposed by Nikolopoulou et al. (Letters in Biomathematics, 5 (2018), S137-S159). For this tumour-immune interaction model, it is known that the model has a unique tumour-free equilibrium and at most two tumorous equilibria. We present sufficient and necessary conditions for the global stabil- ity of the tumour-free equilibrium or the unique tumorous equilibrium. The global dynamics is obtained by employing a new Dulac function to establish the nonexistence of nontrivial positive periodic orbits. Our analysis shows that we can almost completely classify the global dynamics of the model with two critical values CK0;CK1(CK0 > CK1) for the carrying capacity CK of tumour cells and one critical value dT0 for the death rate dT of T cells. Speciffically, the following are true. (i) When no tumorous equilibrium exists, the tumour- free equilibrium is globally asymptotically stable. (ii) When CK ≤ CK1 and dT > dT0, the unique tumorous equilibrium is globally asymptotically stable. (iii) When CK > CK1, the model exhibits saddle-node bifurcation of tumor- ous equilibria. In this case, we show that when a unique tumorous equilibrium exists, tumor cells can persist for all positive initial densities, or can be elim- inated for some initial densities and persist for other initial densities. When two distinct tumorous equilibria exist, we show that the model exhibits bistable phenomenon, and tumor cells have alternative fates depending on the positive initial densities. (iv) When CK > CK0 and dT = dT0, or dT > dT0, tumor cells will persist for all positive initial densities.