In this paper, we develop a definite tumor-immune model considering Allee effect. The deterministic model is studied qualitatively by mathematical analysis method, including the positivity, boundness, and local stability of the solution. In addition, we explore the effect of random factors on the transition of the tumor-immune system from a stable coexistence equilibrium point to a stable tumor-free equilibrium point. Based on the method of stochastic averaging, we obtain the expressions of the steady-state probability density and the mean first-passage time. And we find that the Allee effect has the greatest impact on the number of cells in the system when the Allee threshold value is within a certain range; the intensity of random factors could affect the likelihood of the system crossing from the coexistence equilibrium to the tumor-free equilibrium.

中文翻译：

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Allee作用下肿瘤免疫系统的动力学分析

在本文中，我们考虑到Allee效应建立了确定的肿瘤免疫模型。通过数学分析方法对确定性模型进行定性研究，包括正解，有界性和解的局部稳定性。此外，我们探讨了随机因素对肿瘤免疫系统从稳定的共存平衡点到稳定的无肿瘤平衡点过渡的影响。基于随机平均的方法，我们得到了稳态概率密度和平均首次通过时间的表达式。并且我们发现，当Allee阈值在一定范围内时，Allee效应对系统中的单元数影响最大。