Complex dynamics of a tumor-immune system with antigenicity

With the effect of antigenicity in consideration, we propose and analyze a conceptual model for the tumor-immune interaction. The model is described by a system of two ordinary differential equations. Though simple, the model can have complicated dynamical behaviors. Besides the tumor-free equilibrium, there can be up to three tumor-present equilibria, which can be a saddle or stable node/focus. Sufficient conditions on the nonexistence of nonconstant periodic solutions are provided. Bifurcation analysis including Hopf bifurcation and Bogdanov-Takens bifurcation is carried out. The theoretical results are supported by numerical simulations. Numerical simulations reveal the complexity of the dynamical behaviors of the model, which includes the subcritical/supercritical Hopf bifurcation, homoclinic bifurcation, saddle-node bifurcation at a nonhyperbolic periodic orbit, the appearance of two limit cycles with a singular closed orbit, and so on. Some biological implications of the theoretical results and numerical simulations are also provided. (C) 2021 Elsevier Inc. All rights reserved.