Impact of tumor anti-immunity: Dynamics of a simple tumor-immune model with two-stage lymphocytes

To investigate the impact of tumor anti-immunity on tumor development, we develop a simple tumor-immune model which incorporates the two-stage development of T lymphocytes (naive and effector cells). With qualitative analysis, we not only derive threshold conditions for the stability of the tumor-free equilibrium and for the existence of tumor-present equilibria, but also obtain some sufficient conditions to describe the dependence of the stability of tumor-present equilibria on the intensity of tumor anti-immunity. Combining these results with quantitative analysis, we find that tumor anti-immunity significantly enriches the dynamical behaviors which are absent in models without tumor anti-immunity. These behaviors include at most two tumor-present equilibria, stability switching phenomena, Hopf bifurcations (both supercritical and subcritical), homoclinic bifurcations, and saddle-node bifurcations of periodic solutions. Quantitative bifurcation analyses with the intensities of tumor anti-immunity and anti-tumor immunity as bifurcation parameters not only uncover complex dynamic behaviors of the model but also demonstrate discontinuous transitions in asymptotic states, such as transitions from the tumor-free equilibrium to a tumor-present equilibrium and from an equilibrium to a periodic oscillation. The results obtained highlight the significance of incorporating tumor anti-immunity response in comprehensively capturing the tumor-immune dynamics. They also provide new theoretical insights into tumor heterogeneity, immune escape, and potential therapeutic strategies targeting tumor immunosuppression.