| Citation: |
Jingnan Wang, Shengnan Liu. PERSISTENCE AND EXTINCTION OF THE TUMOR-IMMUNE STOCHASTIC MODEL WITH EFFECTOR CELLS AND CYTOKINES[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 655-670. doi: 10.11948/20210464
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PERSISTENCE AND EXTINCTION OF THE TUMOR-IMMUNE STOCHASTIC MODEL WITH EFFECTOR CELLS AND CYTOKINES
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Abstract
To investigate the effects of microenvironment on the tumor growth and the loss rates of effector cells and cytokine, we present a stochastic tumor-immune model with the treatment response of effector cells assisted by cytokine to tumor growth. By using the comparison theorem, the Itô formula and the law of large numbers, we prove the existence of globally unique positive solution and obtain the sufficient conditions for the extinction and the persistence of tumor cells. Moreover, using our theoretical results, we perform some numerical simulations to show that different noise intensities lead to different states of tumor cells, including tumor extinction and tumor persistence, which confirms the obtained theoretical results and is useful for theoretical guidance of inhibiting tumor growth in clinical medicine.
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Keywords:
- Stochastic responses /
- Itô formula /
- tumor-immune model /
- extinction /
- persistent in mean
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References
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Jingnan Wang, Shengnan Liu. PERSISTENCE AND EXTINCTION OF THE TUMOR-IMMUNE STOCHASTIC MODEL WITH EFFECTOR CELLS AND CYTOKINES[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 655-670. doi: 10.11948/20210464
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Figure 1.
Waveform plots of effector cell
$ x(t) $ $ y(t) $ $ z(t) $ $ \rho_{i}=0, i=1, 2, 3 $ $ \rho_{1}=0.1, \rho_{2}=1.15, \rho_{3}=0.1 $ -
Figure 2.
Waveform plots of effector cell
$ x(t) $ $ y(t) $ $ z(t) $ $ \rho_{1}=0.1, \rho_{3}=0.1 $ $ \rho_{2}=0.1 $ $ \rho_{2}=0.7 $ -
Figure 3.
Waveform plots of effector cell
$ x(t) $ $ y(t) $ $ z(t) $ $ \rho_{2}=0.7 $ $ \rho_{1}=1.15, \rho_{3}=0.1 $ $ \rho_{1}=0.1, \rho_{3}=1.15 $ -
Figure 4.
Waveform plots of effector cell
$ x(t) $ $ y(t) $ $ z(t) $ $ \rho_{1}=1.15, \rho_{3}=1.15 $ $ \rho_{2}=0.7 $ $ \rho_{2}=1.15 $