Design and stability analysis of a multi-delay tumor-immune model with adaptive nonlinear feedback control

Abstract

We introduce a tumor-immune interaction model with three biologically motivated and mutually independent delays, representing effector-cell activation, delayed tumor growth regulation, and treatment-induced immune stimulation. These delays correspond to experimentally observed time scales ranging from hours to days. Using fixed point theory and inequality methods, we prove the existence, uniqueness, positivity, and boundedness of solutions. A complete parameter table with biological sources and units is provided to ensure reproducibility. Stability conditions and Hopf bifurcation thresholds are derived using the Routh-Hurwitz criterion and delay-dependent bifurcation analysis. To control bifurcation onset and extend the stability domain, we design an adaptive nonlinear feedback controller with clinically interpretable gain parameters. Numerical simulations illustrate the effectiveness of the controller and demonstrate how multiple delays influence tumor-immune dynamics. Negative control experiments confirm the specific contribution of the proposed adaptive law.