Abstract
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.
Highlights
With the continuous improvement of living standards, individuals pay more attention to their own health
When m belongs to (− 1.2, − 0.5), the cell change trend is the most significant. is indicates that when the Allee threshold of tumor cells is within the range of − 1.2 to − 0.5, the Allee effect has the greatest influence on the number of cells in the system, and this proves that the Allee effect has an impact on the tumor-immune system
According to the calculation in the above section, we know that the steady-state probability density of tumor cells is present, which indicates that tumor cells will not be extinct under natural state, but the effects of random factors may change this. erefore, it is practical to study the average first-passage time β from the coexistence equilibrium point to the tumor-free equilibrium point, and the Pontryagin equation is satisfied that where m σ2
Summary
With the continuous improvement of living standards, individuals pay more attention to their own health. The research on tumor-immune system has obtained above results, and Katrin et al [17] showed that tumors have the Allee effect. Since the Allee effect has been shown to change the optimal control decisions and control costs and estimate the potential risks in ecology, Katrin et al [17] believed that it was crucial for tumor growth control. It has been neglected in tumor growth and persistence.
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