One of the simplest model of immune surveillance and neoplasia was proposed by Delisi and Resigno [6]. Later Liu et al [10] proved the existence of non-degenerate Takens-Bogdanov (BT) bifurcations defining a surface in the whole set of five positive parameters. In this paper we prove the existence of Bautin bifurcations completing the scenario of possible codimension two bifurcations that occur in this model. We give an interpretation of our results in terms of the Immuno Edition Theory (IET) of three phases: elimination, equilibrium and escape.

This article is concerned with the mathematical modeling of cancer virotherapy, emphasizing the impact of Allee effects on tumor cell growth. We propose a modeling framework that describes the complex interaction between tumor cells and oncolytic viruses. The efficacy of this therapy against cancer is mathematically investigated. The analysis involves linear and logistic growth scenarios coupled with different Allee effects, including weak, strong, and hyper Allee forms. Critical points are identified, and their existence and stability are analyzed using dynamical system theories and bifurcation techniques. Also, bifurcation diagrams and numerical simulations are utilized to verify and extend analytical results. It is observed that Allee effects significantly influence the stability of the system and the conditions necessary for tumor control and eradication.

We present a mathematical model that explores the influence of Allee effects on tumor cell dynamics in the course of chemotherapy and immunotherapy. Our model includes both weak and strong Allee effects to account for nonlinear tumor growth behaviors. Using bifurcation analysis, we identify critical thresholds and transitions between cancer states—elimination, persistence, and escape—under different treatment scenarios. We analyze key phenomena, including saddle-node, Hopf, and Bogdanov-Takens bifurcations, to characterize system stability and dynamic behaviors. Our bifurcation diagrams offer preliminary insights that may inform the optimization of treatment strategies, by suggesting parameter ranges under which tumor eradication could be feasible. This work presents a conceptual framework that might guide the development of combined therapeutic regimens and highlights the potential value of incorporating biological principles—such as the Allee effect—into mechanistic cancer treatment models.