Cancer modelling
Open Access
Issue
Math. Model. Nat. Phenom.
Volume 15, 2020
Cancer modelling
Article Number 45
Number of page(s) 27
DOI https://doi.org/10.1051/mmnp/2020001
Published online 24 September 2020
  1. Society AC. Cancer Facts & figures 2019. American Cancer Society, Atlanta, 2019. [Google Scholar]
  2. J. Adam and N. Bellomo, A Survey of Models for Tumor Immune Dynamics. Birkhauser, Boston (1997). [Google Scholar]
  3. R.P. Araujo and D.L.S. McElwain, A history of the study of solid tumor growth: the contribution of mathematical modelling. Bull. Math. Biol. 66 (2004) 1039–1091. [Google Scholar]
  4. S. Banerjee and S.S. Sarkar, Delay-induced model for tumor-immune interaction and control of malignant tumor growth. BioSystems 91 (2008) 268–288. [PubMed] [Google Scholar]
  5. J.J. Batzel and K. Kappel, Time delay in physiological systems: Analyzing and modeling its impact. Math Biosci. 234 (2011) 61–74. [Google Scholar]
  6. P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models. SIAM J. Appl. Dyn. Syst. 12 (2013) 1847–1888. [Google Scholar]
  7. P. Bi, S. Ruan and S. Zhang, Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays. Chaos 24 (2014) 023101. [Google Scholar]
  8. G. Caravagna and A. Graudenzi, Distributed delays in a hybrid model of tumor-immune system interplay. Math. Biosci. Eng. 10 (2013) 37–57. [PubMed] [Google Scholar]
  9. E. Coddington and N. Levinso Theory of ordinary differential equation. McGraw-Hill, New Delhi (1955). [Google Scholar]
  10. M. Cohn, Int. Immunol. 20 (2008) 1107–1118. [PubMed] [Google Scholar]
  11. K.L. Cooke and Z. Grossman, Discrete Delay, Distributed Delay and Stability Switches. J Math. Anal. Appl. 86 (1982) 592–627. [Google Scholar]
  12. P.S. Das, P. Das and A. Kundu, Delayed feedback controller based finite time synchronization of discontinuous neural networks with mixed time-varying delays. Neural Process Lett. 49 (2018) 693–709. [Google Scholar]
  13. P.S. Das, P. Das and S. Das, An investigation on Monod–Haldane immune response based tumor–effector–interleukin–2 interactions with treatments. Appl. Math. Comput. 361 (2019) 536–551. [Google Scholar]
  14. P.S. Das, S. Mukherjee and P. Das, An investigation on Michaelis-Menten kinetics based complex dynamics of tumor-immune interaction. Chaos Soliton Fractals 128 (2019) 197–305. [Google Scholar]
  15. P.S. Das, P. Das and S. Mukherjee, Stochastic dynamics of Michaelis-Menten kinetics based tumor-immune interactions. Physica A 541 (2020) 123603. [Google Scholar]
  16. L. De Pillis and A. Radunskaya, A mathemtical model with immune resistance and drug therapy: an optimal control approach. J. Thor. Med. 3 (2001) 79–100. [Google Scholar]
  17. Y. Dong, R. Miyazaki and Y. Takeuchi, Mathematical modelling on helper T-cells in a tumor immune system. Discrete Contin. Dyn. Syst. 19 (2014) 55–72. [Google Scholar]
  18. Y. Dong, G. Huang, R. Miyazaki and Y. Takeuchi. Dynamics in a tumor immune system with time delays. Appl. Math. Compt. 252 (2015) 99–113. [Google Scholar]
  19. A. D’Onofrioa, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumor-immune system interaction. Math. Comput. Model. 51 (2010) 572–591. [Google Scholar]
  20. C.W. Eurich, A. Thiel and L. Fahse, Distributed Delays Stabilize Ecological Feedback Systems. Phys. Rev. Lett. 94 (2005) 158104. [CrossRef] [PubMed] [Google Scholar]
  21. S. Feyissa and S. Banerjee, Delay-induced oscillatory dynamics in humoral mediated immune response with two time delays. Nonlinear Anal. Real World Appl. 14 (2013) 35–52. [Google Scholar]
  22. U. Forys, Stability and bifurcations for the chronic state in Marchuk’s model of an immune system. J Math. Anal. Appl. 352 (2009) 922–942. [Google Scholar]
  23. M. Galach, Dynamics of tumor-immune system comptition-the effect of the time delay. Int. J. Math. Comput. Sci. 13 (2003) 395–406. [Google Scholar]
  24. D. Ghosh, S. Khajanchi, S. Mangiarotti, F. Denis, S.K. Dana and C. Letellier, How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment? BioSystems 157 (2017) 17–30. [Google Scholar]
  25. J.K. Hale and S.M.A. Lunel, Introduction to functional Differential Equations. Springer-Verlag, New York (1993). [Google Scholar]
  26. B.D. Hassard, N.D. Kazarinoff and Y.H. Wan, Theory and Application of Hopf Bifurcation. University of Cambridge, Cambridge (1981). [Google Scholar]
  27. S. Khanjanchi, Bifurcation analysis of a delayed mathematical model for tumor growth. Chaos Solitons Fractals 77 (2015) 264–276. [Google Scholar]
  28. S. Khajanchi and S. Banerjee, Stability and bifurcation analysis of delay induced tumor immune interaction model. Appl. Math. Comput. 248 (2014) 652–671. [Google Scholar]
  29. D.E. Kirschner and J.C. Panetta, Modelling the immunotheraphy of tumor-immune interaction. J. Math. Biol. 37 (1998) 235–252. [CrossRef] [PubMed] [Google Scholar]
  30. V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor and A.S. Perelson, Non-linear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56 (1994) 295–321. [Google Scholar]
  31. H. Mayer, K. Zaenker and U. Heiden, A basic mathematical model of the immune response. Chaos 5 (1995) 155–161. [Google Scholar]
  32. M.J. Piotrowska and M. Bodnar, Influence of distributed delays on the dynamics of a generalized immune system cancerous cells interactions model. Commun. Nonlinear Sci. Numer. Simul. 54 (2018) 379–415. [Google Scholar]
  33. M.J. Piotrowska, M. Bodnar, J. Poleszczuk and U. Forys, Mathematical modelling of immune reaction against gliomas: sensivity analysis and influence of delays. Nonlinear Anal. Real World Appl. 14 (2013) 1601–1620. [Google Scholar]
  34. F.A. Rihan, D.H.A. Rahaman, S. Lakshmanan and A.S. Alkhajeh, A time delay model of tumor-immune system interactions: global dynamics, parameter estimation, sentivity analysis. Appl. Math. Comput. 232 (2014) 606–623. [Google Scholar]
  35. S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discret. Impuls Syst. Ser A 10 (2003) 863–874. [Google Scholar]
  36. M. Villasana and A. Radunskaya A delay differential equation model for tumor-growth. J. Math. Biol. 47 (2003) 270–294. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  37. X. Yang, L. Chen and J. Cheng, Parmanance and positive periodic solution for single-species non-autonomous delay diffusive model. Comput. Math. Appl. 32 (1996) 109. [Google Scholar]
  38. M. Yu, Y. Dong and Y. Takeuchi, Dual role of delay effects in a tumour- immune system. J Biol. Dyn. 11 (2017) 334–347. [CrossRef] [PubMed] [Google Scholar]

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