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In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [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]

• Obtain threshold values and formulas of the Hopf bifurcation. • Show the interesting chaotic attractors and the periodic oscillation. • Use bifurcation theory to explain the chaotic attractor formation. • Obtain the biomedical significance corresponding to model dynamics. In this paper, a tumor and Lymphatic immune system interaction model with two time delays is discussed in which the delays describe the proliferation of tumor cells and the transmission from immature T lymphocytes to mature T lymphocytes respectively. Conditions for the asymptotic stability of the equilibrium and the existence of Hopf bifurcations are obtained by analyzing the roots of a characteristic equation. The computing formulas for the stability and the direction of the Hopf bifurcating periodic solutions are given. Numerical simulation show that different values of time delays can generate different behaviors, including the stable-state, the periodic oscillation and the chaotic attractors, as well as the coexistence of two periodic oscillations. These theoretical and numerical results not only can be useful for explaining the occurrence of chaotic attractors, but also can help for understanding the biomedical significance corresponding to the interaction dynamics of tumor cells and T lymphocytes. [ABSTRACT FROM AUTHOR]