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Survival analysis for tumor growth model with stochastic perturbation

  • * Corresponding author: Dongxi Li

    * Corresponding author: Dongxi Li 
This work was supported by the National Natural Science Foundation of China (Grant No.11571009), and Applied Basic Research Programs of Shanxi Province (Grant No. 201901D111086)
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  • In this paper, we investigate the dynamical behavior of extinction and survival in tumor growth model with immunization under stochastic perturbation. Firstly, the model describing the growth of cancer cells monitored by immune cells is established. Then, the steady probability distribution of tumor cells for different noise intensities and immune parameter intensities, and necessary conditions for extinction and different survival of cancer cells are obtained by numerical and theoretical method. Besides, it is found that the extinction and survival of cancer cells rely on the state of immunization and noise. Finally, stochastic simulations are taken to test the theoretical analytical results. The results of our work are beneficial to discover the evolution mechanism and design effective immunotherapy of tumor.

    Mathematics Subject Classification: Primary: 34F05, 92D25, 60H10.


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  • Figure 1.  The potential $ U(x) $ as a function of $ x $ for different value $ \beta $ with $ \theta = 0.25 $

    Figure 2.  The steady probability distribution function of tumor cells population $ x(t) $ for model (4) with $ \theta = 0.25 $

    Figure 3.  The valid regions as a function of $ \theta(t) $ and $ \beta(t) $

    Figure 4.  Solutions of extinction of tumor cells for $ (a):\sigma^2(t) = 0.02+0.004\sin t $, $ \beta(t) = 3+\sin t $; $ (b):\sigma^2(t) = 1+0.8\sin t $, $ \beta(t) = 3+\sin t $; $ (c):\sigma^2(t) = 1+0.8\sin t $, $ \beta(t) = 6+\sin t $, with the initial value $ x_0 = 0.5 $

    Figure 5.  Solutions of strong persistence in the mean of tumor cells for $ (d):\sigma^2(t) = 0.002+0.001\sin t $, $ \beta(t) = 0.97+0.009\sin t $, with the initial value $ x_0 = 0.5 $

    Figure 6.  Solutions of strong persistence in the mean of tumor for $ (e):\sigma^2(t) = 0.02+0.002\sin t $, $ \beta(t) = 0.8+0.008\sin t $, with the initial value $ x_0 = 0.5 $

    Figure 7.  Mean time to extinction (MET) as a function of the noise intensity for different values of $ \beta $ at $ \theta = 0.25 $

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