Stability and bifurcation analysis of a delayed fractional-order tumor-immune model

This paper investigates the stability and Hopf bifurcation of a delayed fractional-order tumor-immune interaction model involving immune effector cells, tumor cells, and interleukin-2. The time delay is introduced to describe the delayed activation and proliferation of effector cells induced by immune stimulation. For the commensurate-order case, the existence of equilibrium points is discussed, and the local stability of the positive equilibrium is analyzed by deriving the corresponding characteristic equation. Taking the delay as the bifurcation parameter, sufficient conditions are established for stability switching and the occurrence of Hopf bifurcation at the critical delay. Numerical simulations are carried out to verify the theoretical results and to illustrate the effects of the fractional order and time delay on the dynamic behavior of the system. The results show that increasing the delay can destabilize the positive equilibrium and lead to sustained oscillations, while the fractional order affects the critical bifurcation threshold. These findings provide theoretical insight into the dynamic mechanisms of tumor-immune interactions and offer a qualitative reference for further mathematical modeling of immunotherapy-related processes.