The influence of time delay in a chaotic cancer model.

The tumor-immune interactive dynamics is an evergreen subject that continues to draw attention from applied mathematicians and oncologists, especially so due to the unpredictable growth of tumor cells. In this respect, mathematical modeling promises insights that might help us to better understand this harmful aspect of our biology. With this goal, we here present and study a mathematical model that describes how tumor cells evolve and survive the brief encounter with the immune system, mediated by effector cells and host cells. We focus on the distribution of eigenvalues of the resulting ordinary differential equations, the local stability of the biologically feasible singular points, and the existence of Hopf bifurcations, whereby the time lag is used as the bifurcation parameter. We estimate analytically the length of the time delay to preserve the stability of the period-1 limit cycle, which arises at the Hopf bifurcation point. We also perform numerical simulations, which reveal the rich dynamics of the studied system. We show that the delayed model exhibits periodic oscillations as well as chaotic behavior, which are often indicators of long-term tumor relapse.