A mathematical model of cancer with delayed activation of immune cells

Abstract

In this paper, we investigate a three-dimensional nonlinear cancer model describing the interactions among cancer cells, normal cells, and immune cells, incorporating a time delay to account for the delayed activation of the immune response. We first establish the biological feasibility of the model by proving the boundedness of its solutions. The equilibrium points of the system are determined, and their local stability is analyzed. Treating the time delay as a bifurcation parameter, we derive conditions for the occurrence of Hopf bifurcation, demonstrating that an increase in the delay can destabilize the system and induce oscillatory behavior. Numerical simulations are performed to validate the analytical findings and to illustrate rich dynamical phenomena, including limit cycle oscillations, period-doubling bifurcations, and chaotic dynamics. These results highlight the significant role of immune activation delay in shaping cancer dynamics.