Dynamical Analysis of a Simple Tumor-Immune Model With Two-Stage Lymphocytes

The growth of tumor cells involves complex interactions with the immune response. We propose a simple two-stage model that describes the interaction between tumor cells and lymphocytes, where it is assumed that lymphocytes undergo two stages of development (immature and mature) and that only mature lymphocytes can kill tumor cells. The model incorporates a linear function to represent the effect of tumor antigen stimulation and a logistic model to describe the tumor growth in the absence of immune response. We analyze the oscillatory behavior of tumor levels from three perspectives: the intrinsic growth rate of tumor, the killing rate of lymphocytes against tumor cells, and the stimulation effect of tumor antigens on the immune system. Supported by theoretical analysis of Hopf bifurcation, we observe distinct differences among these factors. The oscillation occurs between two critical values for the intrinsic growth rate and the killing rate of lymphocytes, while for the stimulation effect of tumor antigens, there is a single critical value that triggers the oscillation. Numerical simulations show that strong tumor antigen stimulation can induce long-term dormancy in tumor growth. Furthermore, we establish the equivalence between the local and global stability of the tumor-free equilibrium using the fluctuation lemma and derive a sufficient condition on the global attractivity of the tumor-present equilibrium by constructing auxiliary convergent sequences.