Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 873–886 | Cite as

The Extinction and Persistence of Tumor Evolution Influenced by External Fluctuations and Periodic Treatment

  • Dongxi LiEmail author
  • Fangjuan Cheng


This paper investigates the extinction and persistence of tumor evolution influenced by external fluctuations and periodic treatment. Firstly, a mathematical model describing the evolution of tumor cells with immunization under external fluctuations and periodic treatment is established based on stochastic differential equation. Then, making use of the methods of Ito’s formula, the sufficient conditions for extinction and persistence are derived by rigorous mathematical proofs. Finally, numerical simulations are applied to illustrate and verify the conclusions. The results of this work provide the theoretical basis for designing more effective and precise therapeutic strategies to eliminate cancer cells, especially for combining the immunotherapy and the traditional tools.


Extinction Persistence External fluctuations Periodic treatment 



This work was supported by the Shanxi Scholarship Council of China (Grant No. 2015-032), Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2015121) and Applied Basic Research Programs of Shanxi Province (Grant No. 201601D021013).


  1. 1.
    Parish, C.: Cancer immunotherapy: the past, the present and the future. Immunol. Cell Biol. 81, 106–113 (2003)CrossRefGoogle Scholar
  2. 2.
    Smyth, M., Godfrey, D., Trapani, J.: A fresh look at tumor immunosurveillance and immunotherapy. Nat. Immunol. 2, 293–299 (2001)CrossRefGoogle Scholar
  3. 3.
    Rosenberg, S., Spiess, P., Lafreniere, R.: A new approach to the adoptive immunotherapy of cancer with tumor-infiltrating lymphocytes. Science 233, 1318–1321 (1986)CrossRefGoogle Scholar
  4. 4.
    Lake, R., Robinson, B.: Immunotherapy and chemotherapy-a practical partnership. Nat. Rev. Cancer 5, 397–405 (2005)CrossRefGoogle Scholar
  5. 5.
    Kim, J., Tannock, I.: Repopulation of cancer cells during therapy: an important cause of treatment failure. Nat. Rev. Cancer 5, 516–525 (2005)CrossRefGoogle Scholar
  6. 6.
    Woo, M., Peterson, J., Liang, H., Bjornsti, M., Houghton, P.: Enhanced antitumor activity of irofulven in combination with irinotecan in pediatric solid tumor xenograft models. Cancer Chemother. Pharmacol. 55, 411–419 (2005)CrossRefGoogle Scholar
  7. 7.
    Zhong, W., Shao, Y., He, Z.: Pure multiplicative stochastic resonance of a theoretical anti-tumor model with seasonal modulability. Phys. Rev. E 73, 95–104 (2006)Google Scholar
  8. 8.
    Fiasconaro, A., Ochab-Marcinek, A., Spagnolo, B., Gudowska-Nowak, E.: Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment. Phys. Condens. Matter 65, 435–442 (2008)Google Scholar
  9. 9.
    Garay, R., Lefever, P.: A kinetic approach to the immunology of cancer: stationary states properties of effector-target cell reactions. J. Theor. Biol. 73, 417–438 (1978)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lefever, R., Horsthemke, W.: Bistability in fluctuating environments. Implications in tumor immunology. Bull. Math. Biol. 41, 469–490 (1979)CrossRefGoogle Scholar
  11. 11.
    Zhivkov, P., Waniewski, J.: Modelling tumour-immunity interactions with different stimulation functions. Guangdong J. Anim. Vet. Sci. 13, 307–315 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Lefever, R., Horsthemke, W.: Multiple transitions induced by light intensity fluctuations in illuminated chemical systems. Proc. Nat. Acad. Sci. 76, 2490–2494 (1979)CrossRefGoogle Scholar
  13. 13.
    Li, D., Xu, W., Guo, Y., Xu, Y.: Fluctuations induced extinction and stochastic resonance effect in a model of tumor growth with periodic treatment. Phys. Lett. A 375, 886–890 (2011)CrossRefGoogle Scholar
  14. 14.
    Fiasconaro, A., Spagnolo, B., Ochab-Marcinek, A., Gudowska-Nowak, E.: Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response. Phys. Rev. E 74, 159–163 (2006)CrossRefGoogle Scholar
  15. 15.
    Ochab-Marcinek, A., Gudowska-Nowak, E.: Population growth and control in stochastic models of cancer development. Phys. A Stat. Mech. Appl. 343, 557–572 (2004)CrossRefGoogle Scholar
  16. 16.
    Berke, G., Gabison, D., Feldman, M.: The frequency of effector cells in populations containing cytotoxic lymphocytes. Eur. J. Immunol. 5, 813–818 (2005)CrossRefGoogle Scholar
  17. 17.
    Ai, B., Wang, X., Liu, L.: Reply to comment on correlated noise in a logistic growth model. Phys. Rev. E 77, 241–251 (2003)Google Scholar
  18. 18.
    Li, D., Xu, W., Sun, C., Wang, L.: Stochastic fluctuation induced the competition between extinction and recurrence in a model of tumor growth. Phys. Lett. A 376, 1771–1776 (2012)CrossRefGoogle Scholar
  19. 19.
    Xu, Y., Feng, J., Li, J., Zhang, H.: Stochastic bifurcation for a tumor-immune system with symmetric levy noise. Phy. A 392, 4739–4748 (2013)CrossRefGoogle Scholar
  20. 20.
    Pankratov, A., Bernardo, S.: Suppression of timing errors in short overdamped josephson junctions. Phys. Rev. Lett. 93, 177001 (2004)CrossRefGoogle Scholar
  21. 21.
    Fiasconaro, A., Spagnolo, B., Boccaletti, S.: Signatures of noise-enhanced stability in metastable states. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 72, 061110 (2006)CrossRefGoogle Scholar
  22. 22.
    Liu, M., Wang, K.: Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment. J. Theor. Biol. 264, 934–944 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hallam, T., Ma, Z.: Persistence in population models with demographic fluctuations. Math. Biol. 24, 327–339 (1986)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schreiber, S., Benaim, M., Atchade, K.: Persistence in fluctuating environments. J. Math. Biol. 62(5), 655–683 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Meng, L., Bai, C.: Analysis of a stochastic tri-trophic food-chain model with harvesting. J. Math. Biol. 73(3), 597–625 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, D., Cheng, F.: Threshold for extinction and survival in stochastic tumor immune system. Commun. Nonlinear Sci. Numer. Simulat. 51, 1–12 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mao, X., Marion, G., Renshaw, E.: Environmental brownian noise suppresses explosions in populations dynamics. Stochastic Process. Appl. 97, 95–110 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Liu, M., Wang, K.: Persistence and extinction in stochastic non-autonomous logistic systems. J. Math. Anal. Appl. 375, 443–457 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Liu, M., Wang, K.: A note on stability of stochastic logistic equation. Appl. Math. Lett. 26, 601–606 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (1997)zbMATHGoogle Scholar
  31. 31.
    Evans, L.: An Introduction to Stochastic Differential Equations. Amer Mathematical Society, Providence (2014)Google Scholar
  32. 32.
    Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Liu, M., Yu, J., Mandal, P.: Dynamics of a stochastic delay competitive model with harvesting and Markovian switching. Appl. Math. Comput. 337, 335–349 (2018)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Liu, M., Zhu, Y.: Stationary distribution and ergodicity of a stochastic hybrid competition model with levy jumps. Nonlinear Anal. Hybrid Syst. 30, 225–239 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Liu, M., Fan, M.: Permanence of stochastic Lotka–Volterra systems. J. Nonlinear Sci. 27(2), 425–452 (2017)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Data ScienceTaiyuan University of TechnologyTaiyuanChina
  2. 2.Institute of AutomationChinese Academy of SciencesBeijingChina

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