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Original Articles

Dual role of delay effects in a tumour–immune system

, &
Pages 334-347 | Received 31 Jan 2016, Accepted 26 Aug 2016, Published online: 20 Sep 2016

References

  • J. Arciero, T. Jackson, and D. Kirscher, A mathematical model of tumor–immune evasion and siRNA treatment, Discrete Contin. Dyn. Syst. Ser. B 4 (2004), pp. 39–58.
  • 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), pp. 1847–1888. doi: 10.1137/120887898
  • L.G. de Pillis, W. Gu, and A.E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, application and biological interpretations, J. Theor. Biol. 238 (2006), pp. 841–862. doi: 10.1016/j.jtbi.2005.06.037
  • L.G. de Pillis, A.E. Radunskaya, and C.L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res. 65 (2005), pp. 7950–7958.
  • K.E. de Visser, A. Eichten, and L.M. Coussens, Paradoxical roles of the immune system during cancer development, Nat. Rev. Cancer 6 (2006), pp. 24–37. doi: 10.1038/nrc1782
  • Y. Dong, H. Gang, R. Miyazaki, and Y. Takeuchi, Dynamics in a tumor immune system with time delays, Appl. Math. Comput. 252 (2015), pp. 99–113.
  • Y. Dong, R. Miyazaki, and Y. Takeuchi, Mathematical modeling on helper T cells in a tumor immune system, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), pp. 55–72. doi: 10.3934/dcdsb.2014.19.55
  • A. d'Onofrio, A general framework for modeling tumor–immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D 208 (2005), pp. 220–235. doi: 10.1016/j.physd.2005.06.032
  • A. d'Onofrio, Tumor evasion from immune system control: Strategies of a MISS to become a MASS, Chaos, Solitons and Fractals 31 (2007), pp. 261–268. doi: 10.1016/j.chaos.2005.10.006
  • A. d'Onofrio, Metamodeling tumor–immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model. 47 (2008), pp. 614–637. doi: 10.1016/j.mcm.2007.02.032
  • A. d'Onofrio, F. Gatti, P. Cerrai, and L. Freschi, Delay-induced oscillatory dynamics of tumor–immune system interaction, Math. Comput. Model. 51 (2010), pp. 572–591. doi: 10.1016/j.mcm.2009.11.005
  • G.B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002.
  • M. Galach, Dynamics of the tumor–immune system competition: The effect of time delay, Int. J. Appl. Math. Comput. Sci. 13 (2003), pp. 395–406.
  • I. Kareva, F. Berezovskaya, and C. Castillo-Chavez, Myeloid cells in tumour–immune interactions, J. Biol. Dynam. 4 (2010), pp. 315–327. doi: 10.1080/17513750903261281
  • D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor–immune interaction, J. Math. Biol. 37 (1998), pp. 235–252. doi: 10.1007/s002850050127
  • V. Kuznetsov, I. Makalkin, M. Taylor, and A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol. 2 (1994), pp. 295–321. doi: 10.1007/BF02460644
  • O. Lejeune, M. Chaplain, and I. Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumors, Math. Comput. Model. 47 (2008), pp. 649–662. doi: 10.1016/j.mcm.2007.02.026
  • D. Liu, S. Ruan, and D. Zhu, Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions, Math. Biosci. Eng. 9 (2012), pp. 347–368. doi: 10.3934/mbe.2012.9.785
  • F.A. Rihan, D.H.A. Rahmana, S. Lakshmanana, and A.S. Alkhajeh, A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput. 232 (2014), pp. 606–623.
  • M. Saleem, T. Agrawal, and A. Anees, A study of tumour growth based on stoichiometric principles: A continuous model and its discrete analogue, J. Biol. Dynam. 8 (2014), pp. 117–134. doi: 10.1080/17513758.2014.913718