Dynamical Behavior of a Tumor-Immune Model With Vitamin Intervention

The interaction between the immune system and tumor cells is influenced by various biological and environmental factors. In this study, we develop a novel mathematical model to investigate how an effective vitamin-related intervention may modulate the immune response. The model describes the interaction between tumor cells and immune effector cells, incorporating the effects of this effective intervention on their dynamics. We establish fundamental mathematical properties of the model, including positivity, boundedness, and the existence of biologically relevant equilibria, and we analyze their stability. Furthermore, we explore the existence of periodic solutions and perform a bifurcation analysis. Taking the intrinsic tumor growth rate α as the bifurcation parameter, the model admits at most two equilibria: a tumor-free equilibrium and a coexistence equilibrium. Our analysis shows that the model undergoes a transcritical bifurcation at the critical value α∗, which depends directly on vitamin intervention and immune response parameters. This critical value defines the boundary separating stability of the tumor-free equilibrium from tumor persistence. Numerical simulations are carried out to illustrate and numerically corroborate the theoretical findings. The results suggest that vitamin intervention may enhance the immune response, potentially contributing to tumor suppression or delaying tumor progression. The model's novelty lies in combining a Michaelis–Menten saturable immune response with the simultaneous effects of an effective vitamin-related intervention on both immune activation and tumor inhibition. Under the assumptions of the model, this framework leads to a threshold R0 that explicitly links the effective vitamin intake rate to tumor control. This theoretical finding may motivate further biological and experimental investigations into the role of vitamin-related interventions in tumor-immune dynamics.