Qualitative analysis of a simple tumor-immune system with time delay of tumor action

In this paper, we propose a simple tumor-immune system model with time delay of tumor action, where two kinds of effects of the tumor cells (i.e. stimulation and neutralization) on the effector cells are considered. The local stability of the model is obtained by analyzing the characteristic equations of the model at the corresponding equilibria, the sufficient conditions on the global stability are found by applying the Fluctuation Lemma and constructing the different convergent sequences. The obtained results show that, compared to the results for the model without time delay, the time delay of tumor action can affect the stability of tumor equilibrium of the model as the stimulation effect of the tumor cells is strong enough, while the delay is harmless for the stability of tumor equilibrium under the neutralization of tumor cells. For the appropriate neutralization of tumor cells on effector cells, the bistability of the tumor free equilibrium and the stronger tumor equilibrium can appear. In the case of stimulation of tumor cells, the sufficiently large time delay can lead to the appearance of a stable periodic solution by Hopf bifurcation, and the numerical simulation illustrates that the amplitude of the periodic orbit increases with time delay. We also discuss the dependence of the tumor equilibrium and the time delay threshold, determining the stability of the tumor equilibrium, on tumor action. The related conditions determining dynamics of the model are expressed by certain formulae with biological meanings.