From Discrete to Distributed Delay in a Tumor–Immune Model: Stability, Hopf Bifurcation, and the Shape of Immune Memory

This paper revisits a delayed tumor–immune model by replacing the discrete delay with weak and strong Gamma distributed memories, reflecting the realistic spread of immune-response times. Because the Gamma kernels are normalized, the biologically relevant equilibria of the reference model are preserved. The local stability problem, however, changes substantially: a careful linearization shows that the characteristic equation contains both the first and the second power of the memory transfer function, since delayed immune and tumor variables enter coupled feedback terms. Consequently, the weak Gamma chain leads to a quintic characteristic polynomial, whereas the strong Gamma chain leads to a seventh-degree polynomial. Routh–Hurwitz conditions and explicit Hopf bifurcation tests are derived for both memory structures, including simplicity and transversality requirements. Numerical simulations performed with the parameter sets of the reference study show that distributed memory reproduces the main biological regimes while shifting stability thresholds and modifying transient oscillations. The results indicate that not only the mean immune-response time but also the shape of its distribution can influence tumor–immune dynamics.