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Study on stationary probability density of a stochastic tumor-immune model with simulation by ANN algorithm

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Listed:
  • Li, Wei
  • Zhang, Ying
  • Huang, Dongmei
  • Rajic, Vesna

Abstract

In this paper, the theoretic analysis by stochastic dynamics and numerical simulation by ANN algorithm for tumor-immune models driven by random noises are studied. Firstly, a mathematical model about the competition between tumor cells and immune system is established based on the Michaelis-Menten enzyme kinetics. Secondly, the microenvironmental fluctuations are modeled as Gaussian white noises and Gaussian coloured noises, respectively. Then the corresponding Fokker-Planck equation (FPE) and the approximated Fokker-Planck equation (AFPE) are obtained to explore the stationary probability density (SPD) of the tumor cells. The innovation of this paper is that artificial neural network (ANN) algorithm is introduced to solve the SPD based on FPE or AFPE, which has higher robustness and accuracy. The SPDs of tumor cells show that the greater the intensity of Gaussian white noises, the more beneficial the prevention of tumor growth. In other words, the microenvironmental fluctuations could accelerate the extinction of tumor to a certain extent. For the case of Gaussian coloured noises, the existence of the cross-correlation time between multiplicative and additive Gaussian coloured noises lead the tumor to be large concentrations with probability of almost 1. All the theoretical results are examined by ANN algorithm and they are all in good agreement. In addition, we discuss the mean first passage time (MFPT) from the metastable state to the state of extinction of tumor cells, and discover the phenomena of the noise-enhanced stability (NES) as well as stochastic resonant activation (SRA). At last, the best parameters including penalty factors, the number of layers and nodes in ANN algorithm are also discussed in order to get the optimal accuracy.

Suggested Citation

  • Li, Wei & Zhang, Ying & Huang, Dongmei & Rajic, Vesna, 2022. "Study on stationary probability density of a stochastic tumor-immune model with simulation by ANN algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    • Handle: RePEc:eee:chsofr:v:159:y:2022:i:c:s0960077922003551
    DOI: 10.1016/j.chaos.2022.112145
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    References listed on IDEAS

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    1. Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse & Tuan Anh Nguyen, 2020. "A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-34, April.
    2. Han, Ping & Xu, Wei & Wang, Liang & Zhang, Hongxia & Ma, Shichao, 2020. "Most probable dynamics of the tumor growth model with immune surveillance under cross-correlated noises," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
    3. Sardanyés, Josep & Rodrigues, Carla & Januário, Cristina & Martins, Nuno & Gil-Gómez, Gabriel & Duarte, Jorge, 2015. "Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approach," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 484-495.
    4. Qian, Jiamin & Chen, Lincong, 2021. "Stochastic P-bifurcation analysis of a novel type of unilateral vibro-impact vibration system," Chaos, Solitons & Fractals, Elsevier, vol. 149(C).
    5. Xu, Pengfei & Jin, Yanfei, 2018. "Mean first-passage time in a delayed tristable system driven by correlated multiplicative and additive white noises," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 75-82.
    6. Duan, Wei-Long & Lin, Ling, 2021. "Noise and delay enhanced stability in tumor-immune responses to chemotherapy system," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    7. Jin, Yanfei & Wang, Heqiang, 2020. "Noise-induced dynamics in a Josephson junction driven by trichotomous noises," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    8. Liang, G.Y. & Cao, L. & Wu, D.J., 2004. "Approximate Fokker–Planck equation of system driven by multiplicative colored noises with colored cross-correlation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(3), pages 371-384.
    9. Sepehrian, Behnam & Radpoor, Marzieh Karimi, 2015. "Numerical solution of non-linear Fokker–Planck equation using finite differences method and the cubic spline functions," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 187-190.
    10. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    11. A. Fiasconaro & A. Ochab-Marcinek & B. Spagnolo & E. Gudowska-Nowak, 2008. "Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 65(3), pages 435-442, October.
    12. Yunying Zheng & Changpin Li & Zhengang Zhao, 2010. "A Fully Discrete Discontinuous Galerkin Method for Nonlinear Fractional Fokker-Planck Equation," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-26, December.
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    1. Tan, Yiping & Cai, Yongli & Sun, Xiaodan & Wang, Kai & Yao, Ruoxia & Wang, Weiming & Peng, Zhihang, 2022. "A stochastic SICA model for HIV/AIDS transmission," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).

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