THE INFLUENCE OF DELAYS AND INJECTION TREATMENTS ON THE DYNAMICS OF TUMOR-IMMUNE SYSTEM

We propose a tumor-CD4+-cytokine-host system with two delays and injection treatments, where two delays denote activation delays for CD4+ T cells and cytokines respectively and host cells compete with tumor cells. First we prove the positivity and boundedness of solutions without injection treatments, the positivity means that tumor cells always exist, the boundedness means that immune cells can control the increase of tumor cells, then we discuss the influence of two delays on the dynamics of system, that is as any of the delays is sufficiently small, the corresponding interior positive equilibrium point is locally stable, as the delay increases to critical value, Hopf bifurcation occurs and periodic solution bifurcates from the equilibrium point, which means the relapse of tumor cells occurs, furthermore base on center manifold and normal form, we study the property of Hopf bifurcation, at last, in absence of delay, we study the influence of relative parameters on the volume of tumor cells and feasibility of injection treatments of CD4+ T cells and cytokines on eradication of tumor cells, that is the tumor cells can be eradicated if the dose of injection treatment of CD4+ T cells or cytokines is sufficiently large. All the results are supported by simulations.