# American Institute of Mathematical Sciences

October  2021, 26(10): 5707-5722. doi: 10.3934/dcdsb.2021041

## Survival analysis for tumor growth model with stochastic perturbation

 College of Data Science, Taiyuan University of Technology, Taiyuan, 030024, China

* Corresponding author: Dongxi Li

Received  October 2019 Revised  December 2020 Published  October 2021 Early access  February 2021

Fund Project: 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)

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.

Citation: Dongxi Li, Ni Zhang, Ming Yan, Yanya Xing. Survival analysis for tumor growth model with stochastic perturbation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5707-5722. doi: 10.3934/dcdsb.2021041
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##### References:
The potential $U(x)$ as a function of $x$ for different value $\beta$ with $\theta = 0.25$
The steady probability distribution function of tumor cells population $x(t)$ for model (4) with $\theta = 0.25$
The valid regions as a function of $\theta(t)$ and $\beta(t)$
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 $x_0 = 0.5$
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 $x_0 = 0.5$
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 $x_0 = 0.5$
Mean time to extinction (MET) as a function of the noise intensity for different values of $\beta$ at $\theta = 0.25$
 [1] Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Mesoscopic model for tumor growth. Mathematical Biosciences & Engineering, 2007, 4 (4) : 687-698. doi: 10.3934/mbe.2007.4.687 [2] Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101 [3] Rudolf Olach, Vincent Lučanský, Božena Dorociaková. The model of nutrients influence on the tumor growth. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021150 [4] Cristina Anton, Alan Yong. Stochastic dynamics and survival analysis of a cell population model with random perturbations. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1077-1098. doi: 10.3934/mbe.2018048 [5] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2729-2749. doi: 10.3934/dcdss.2020457 [6] Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129 [7] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [8] Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011 [9] T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 187-201. doi: 10.3934/dcdsb.2004.4.187 [10] J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263 [11] Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173 [12] Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925 [13] Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005 [14] Xingwang Yu, Sanling Yuan. Asymptotic properties of a stochastic chemostat model with two distributed delays and nonlinear perturbation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2373-2390. doi: 10.3934/dcdsb.2020014 [15] Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335 [16] Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5641-5660. doi: 10.3934/dcdsb.2020371 [17] Shaoyong Lai, Yulan Zhou. A stochastic optimal growth model with a depreciation factor. Journal of Industrial & Management Optimization, 2010, 6 (2) : 283-297. doi: 10.3934/jimo.2010.6.283 [18] Francisco Montes de Oca, Liliana Pérez. Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2663-2690. doi: 10.3934/dcdsb.2015.20.2663 [19] Krzysztof Fujarewicz, Krzysztof Łakomiec. Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1131-1142. doi: 10.3934/mbe.2016034 [20] Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787

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