TITLE:
Dynamical Analysis of a Class of Tumor-Immune Models with Effector Cell Action
AUTHORS:
Juanlan Wang
KEYWORDS:
Tumor-Immune Model, Equilibrium, Stability, Routh-Hurwitz Criterion, Transcritical Bifurcation, Hopf Bifurcation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.7,
July
31,
2026
ABSTRACT: Tumor cells interact with the immune system via multiple nonlinear feedbacks, which can trigger various clinically relevant dynamical outcomes including tumor clearance, immune escape, long-term tumor dormancy, and recurrent periodic oscillations. This study investigates a two-stage T-lymphocyte-tumor model. First, the positivity and boundedness of solutions with nonnegative initial values are rigorously established. Next, the existence conditions and local stability criteria for the tumor-free equilibrium and positive equilibria are derived. Then, the Sotomayor theorem is applied to analyze the transcritical bifurcation at the boundary equilibrium. The Poincare-Andronov-Hopf bifurcation theory is further used to obtain the conditions for a Hopf bifurcation at a positive equilibrium. Moreover, normal form theory and the center manifold theorem are employed to characterize the direction of the Hopf bifurcation and the stability of the resulting periodic solutions. Finally, MATLAB simulations verify the stability of the equilibria and the Hopf bifurcation.