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<p>This study introduces a novel stochastic tumor-immune model with vitamins to demonstrate the impact of environmental noise and vitamin intake on tumor and immune cells. First, we establish the existence and uniqueness of a positive solution using the comparison theorem for Itô processes. Second, we show that the solutions are stochastically ultimately bounded. Next, we derive sufficient conditions for the extinction and non-persistence of tumor cells, CD4 T-helper cells and CD8 T-killer cells, as well as for the weak persistence of tumor cells and CD4 T-helper cells and the stochastic persistence of tumor cells. Moreover, we prove the existence of a unique ergodic stationary distribution of the system under certain conditions. Finally, we perform numerical simulations and discuss the biological significance of the findings.</p>

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Stationary distribution and persistence of a stochastic tumor-immune model with vitamins

  • 期刊论文

  • 发表日期：

    2025年6月30日

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  • Nabyl Bajja,
  • Driss Seghir

摘要

This study introduces a novel stochastic tumor-immune model with vitamins to demonstrate the impact of environmental noise and vitamin intake on tumor and immune cells. First, we establish the existence and uniqueness of a positive solution using the comparison theorem for Itô processes. Second, we show that the solutions are stochastically ultimately bounded. Next, we derive sufficient conditions for the extinction and non-persistence of tumor cells, CD4 T-helper cells and CD8 T-killer cells, as well as for the weak persistence of tumor cells and CD4 T-helper cells and the stochastic persistence of tumor cells. Moreover, we prove the existence of a unique ergodic stationary distribution of the system under certain conditions. Finally, we perform numerical simulations and discuss the biological significance of the findings.