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 checkpoint 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 stability 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 $ C_{K0}, C_{K1} (C_{K0}>C_{K1}) $ for the carrying capacity $ C_K $ of tumour cells and one critical value $ d_{T0} $ for the death rate $ d_{T} $ of T cells. Specifically, the following are true. (ⅰ) When no tumorous equilibrium exists, the tumour-free equilibrium is globally asymptotically stable. (ⅱ) When $ C_K \leq C_{K1} $ and $ d_T>d_{T0} $, the unique tumorous equilibrium is globally asymptotically stable. (ⅲ) When $ C_K >C_{K1} $, the model exhibits saddle-node bifurcation of tumorous 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 eliminated 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 $ C_K > C_{K0} $ and $ d_T = d_{T0} $, or $ d_T>d_{T0} $, tumor cells will persist for all positive initial densities.