Delay-induced oscillatory dynamics in humoral mediated immune response with two time delays

A mathematical model for the quantitative analysis of cancer immune interaction, considering the role of humoral (antibody) mediated immune response with two time delays, namely maturation and interaction delays has been proposed in this paper. The aim of this work is to assess the effect of time delays on the interaction between cancerous cells and the antibodies. After categorizing the parametric plane into different regions based on the existence of equilibria, we investigate both analytically and through simulations, the stability of equilibria and the onset of sustained oscillations through Hopf bifurcations. The direction and stability of the Hopf bifurcation which occurs at the positive interior equilibrium point of the system have also been studied. It is observed that both the delays play an important role in stability switching. Appropriate therapy with a proper choice of system parameters are suggested to obtain cancer free equilibrium.