Numerical Solution of Tumor-Immune Model with Targeted Chemotherapy by Multi Step Differential Transformation Method

  • Biplab Dhar
  • Praveen Kumar GuptaEmail author
Conference paper
Part of the Learning and Analytics in Intelligent Systems book series (LAIS, volume 12)


In this paper, a tumor-immune model having four compartments; population of tumor cells, CD8 T killer cells, CD4 T-helper cells and amount of targeted chemotherapeutic drug are considered. We have depicted a qualitative analysis for the proposed model, which includes the existence and the boundedness. The dynamics of the proposed model is presented by examining the stability and admissibility of the model at tumor-free and co-existing equilibrium points. The situation for local stability of all equilibrium points are derived by using Jacobian matrix and Routh-Hurwitz criterion. Numerical calculations are presented to verify the theoretical results so obtained for tumor free equilibrium points. The calculations are carried out with a new method known as multi step differential transformation method (MsDTM). We have expressed that the model is fit for removing large initial population or size of tumor, with passage of time.


Tumor-immune Stability analysis Targeted chemotherapy MsDTM 



The authors would like to thank TEQIP III for financial support in the proceedings of this article. The first author also gratefully thanks to the same for supporting his visit.


  1. 1.
    De Pillis, L., Gu, W., Radunskaya, A.: Mixed immunotherapy and chemotherapy of tumors: modelling, applications and biological interpretations. J. Theor. Biol. 238(4), 841–862 (2006)CrossRefGoogle Scholar
  2. 2.
    De Pillis, L.G., Radunskaya, A.: A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. Comput. Math. Methods Med. 3(2), 79–100 (2001)zbMATHGoogle Scholar
  3. 3.
    De Pillis, L.G., Radunskaya, A.E., Wiseman, C.L.: A validated mathematical model of cell-mediated immune response to tumor growth. Can. Res. 65(17), 7950–7958 (2005)CrossRefGoogle Scholar
  4. 4.
    Ghosh, D., Khajanchi, S., Mangiarotti, S., Denis, F., Dana, S.K., Letellier, C.: How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment? Biosystems 158, 17–30 (2017)CrossRefGoogle Scholar
  5. 5.
    Gupta, P., Dhar, B.: Dynamical behaviour of fractional order tumor-immune model with targeted chemotherapy treatment. Int. J. Eng. Technol. 7(2.28), 6–9 (2018)CrossRefGoogle Scholar
  6. 6.
    Kirschner, D., Panetta, J.C.: Modeling immunotherapy of the tumor immune interaction. J. Math. Biol. 37(3), 235–252 (1998)CrossRefGoogle Scholar
  7. 7.
    Kuznetsov, V.A., Makalin, I.A., Taylor, M., Perelson, A.: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56(4), 295–321 (1994)CrossRefGoogle Scholar
  8. 8.
    Liao, K.-L., Bai, X.-F., Friedman, A.: Mathematical modeling of interleukin-27 induction of anti-tumor t cells response. PLoS ONE 9(3), e91844 (2014)CrossRefGoogle Scholar
  9. 9.
    Liu, P., Liu, X.: Dynamics of a tumor-immune model considering targeted chemotherapy. Chaos, Solitons Fractals 98, 7–13 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Odibat, Z., Bertelle, C., Aziz-Alaouni, M., Duchamp, G.: A multi-step differential transform method and application to non-chaotic or chaotic systems. Comput. Math Appl. 59(4), 1462–1472 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sharma, S., Samanta, G.: Dynamical behaviour of a tumor-immune system with chemotherapy and optimal control. J. Nonlinear Dyn. 2013, 13 (2013)zbMATHGoogle Scholar
  12. 12.
    Sharma, S., Samanta, G.: Analysis of the dynamics of a tumor–immune system with chemotherapy and immunotherapy and quadratic control. Diff. Equat. Dyn. Syst. 24(2), 149–171 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Valle, P.A., Starkov, K.E., Coria, L.N.: Global stability and tumor clearance conditions for a cancer chemotherapy system. Commun. Nonlinear Sci. Numer. Simul. 40, 206–215 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wodarz, D.: Viruses as antitumor weapons. Can. Res. 61(8), 3501–3507 (2001)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsNIT SilcharSilcharIndia

Personalised recommendations