Stability and Hopf bifurcation of a tumor–immune system interaction model with an immune checkpoint inhibitor

In this paper we study a three-dimensional tumor–immune system interaction model consisted of tumor cells, activated T cells, and immune checkpoint inhibitor anti-PD-1. Based on the uncontrollable character of tumor cells in the absence of immune response and treatment, the growth of tumor cells is assumed to be exponential. We discuss the distribution of equilibria qualitatively and study the stability of all possible equilibria with and without anti-PD-1 drug. When no drug is applied, the model has a tumor-free equilibrium and at most one tumorous equilibrium. Biologically, there exists a threshold d_T1 for the death rate d_T of T cells: when d_T≥d_T1 tumor cells will keep growing; when d_T<d_T1 tumor cells may be eradicated for some positive initial values and keep growing for some other positive initial values. For the case with anti-PD-1 treatment, the model has at most five tumor-free equilibria and two interior equilibria. Our analysis indicates that there exists a threshold γ_A1 for the intravenous continuous injection γ_A: when γ_A≤γ_A1 the fate of tumor cells is the same as the case with no drug applied; when γ_A>γ_A1 the model may exhibit bistable phenomena and periodic orbits. Furthermore, we establish the existence of local Hopf bifurcation around the interior equilibrium and determine the stability of the bifurcating periodic orbits. Our simulations show that the model exhibits a stable periodic orbit which implies the long term coexistence and balance of the tumor and immune system.