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In this paper, we investigate the various responses in cancer immune therapy through bifurcation theory. Our results characterize the influence of key parameters on different treatment responses and illustrate the response regions through bifurcation diagrams using parameters tuned by therapy. In particular, we examine a periodic outcome with a delayed therapy response followed by a cancer relapse. The repetitive pattern in this periodic solution is formed by two quasi-steady cancer states connected by two fast transitions. Our results suggest that the transient behavior of the delayed response, which is the fast transition after the cancer progression state, is induced by a periodic solution exhibiting an exploding period near a saddle–node bifurcation. Another transient behavior, which shows as a fast relapse after a long period of dormancy state, is caused by the slow change near a saddle-type equilibrium. Our findings have the potential to guide and inform future therapeutic research. • Various responses in cancer immune therapy are studied through bifurcation theory. • Key parameters on different treatment responses are characterized. • Different response regions are illustrated through bifurcation diagrams. • Transient dynamics in delayed responses are explained mathematically. • Our findings have the potential to guide and inform future therapeutic research. [ABSTRACT FROM AUTHOR]

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 T 1 for the death rate d T of T cells: when d T ≥ d T 1 tumor cells will keep growing; when d T < d T 1 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 A 1 for the intravenous continuous injection γ A : when γ A ≤ γ A 1 the fate of tumor cells is the same as the case with no drug applied; when γ A > γ A 1 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. • Study a tumor–immune system interaction model with an immune checkpoint inhibitor. • Discuss the number and stability of all possible equilibria. • Establish the existence of Hopf bifurcation and the stability of the periodic orbit. • Carry out numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]