Effects of delayed immune-activation in the dynamics of tumor-immune interactions

Parthasakha Das¹ ; Pritha Das¹ ; Samhita Das¹

¹ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur Howrah, West Bengal, India.

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 45.

Résumé

This article presents the impact of distributed and discrete delays that emerge in the formulation of a mathematical model of the human immunological system describing the interactions of effector cells (ECs), tumor cells (TCs) and helper T-cells (HTCs). We investigate the stability of equilibria and the commencement of sustained oscillations after Hopf-bifurcation. Moreover, based on the center manifold theorem and normal form theory, the expression for direction and stability of Hopf-bifurcation occurring at tumor presence equilibrium point of the system has been derived explicitly. The effect of distributed delay involved in immune-activation on the system dynamics of the tumor is demonstrated. Numerical simulations are also illustrated for elucidating the change of dynamic behavior by varying system parameters.

Détail

DOI : 10.1051/mmnp/2020001

Bibliographie

[1] Society AC. Cancer Facts figures 2019. American Cancer Society, Atlanta, 2019.

[2] J. Adam and N. Bellomo, A Survey of Models for Tumor Immune Dynamics. Birkhauser, Boston (1997).

[3] R.P. Araujo, D.L.S. Mcelwain A history of the study of solid tumor growth: the contribution of mathematical modelling Bull. Math. Biol 2004 1039 1091

[4] S. Banerjee, S.S. Sarkar Delay-induced model for tumor-immune interaction and control of malignant tumor growth BioSystems 2008 268 288

[5] J.J. Batzel, K. Kappel Time delay in physiological systems: Analyzing and modeling its impact Math Biosci 2011 61 74

[6] P. Bi, S. Ruan Bifurcations in delay differential equations and applications to tumor and immune system interaction models SIAM J. Appl. Dyn. Syst. 2013 1847 1888

[7] P. Bi, S. Ruan, S. Zhang Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays Chaos 2014 023101

[8] G. Caravagna, A. Graudenzi Distributed delays in a hybrid model of tumor-immune system interplay Math. Biosci. Eng. 2013 37 57

[9] E. Coddington and N. Levinso Theory of ordinary differential equation. McGraw-Hill, New Delhi (1955).

[10] M. Cohn Int. Immunol 2008 1107 1118

[11] K.L. Cooke, Z. Grossman Discrete Delay, Distributed Delay and Stability Switches J Math. Anal. Appl. 1982 592 627

[12] P.S. Das, P. Das, A. Kundu Delayed feedback controller based finite time synchronization of discontinuous neural networks with mixed time-varying delays Neural Process Lett. 2018 693 709

[13] P.S. Das, P. Das, S. Das An investigation on Monod–Haldane immune response based tumor–effector–interleukin–2 interactions with treatments. Appl. Math. Comput. 2019 536 551

[14] P.S. Das, S. Mukherjee, P. Das An investigation on Michaelis-Menten kinetics based complex dynamics of tumor-immune interaction. Chaos Soliton Fractals 2019 197 305

[15] P.S. Das, P. Das, S. Mukherjee Stochastic dynamics of Michaelis-Menten kinetics based tumor-immune interactions. Physica A 2020 123603

[16] L. De Pillis, A. Radunskaya A mathemtical model with immune resistance and drug therapy: an optimal control approach J. Thor. Med 2001 79 100

[17] Y. Dong, R. Miyazaki, Y. Takeuchi Mathematical modelling on helper T-cells in a tumor immune system Discrete Contin. Dyn. Syst 2014 55 72

[18] Y. Dong, G. Huang, R. Miyazaki, Y. Takeuchi. Dynamics in a tumor immune system with time delays Appl. Math. Compt 2015 99 113

[19] A. D’Onofrioa, F. Gatti, P. Cerrai, L. Freschi Delay-induced oscillatory dynamics of tumor-immune system interaction Math. Comput. Model 2010 572 591

[20] C.W. Eurich, A. Thiel, L. Fahse Distributed Delays Stabilize Ecological Feedback Systems. Phys. Rev. Lett. 2005 158104

[21] S. Feyissa, S. Banerjee Delay-induced oscillatory dynamics in humoral mediated immune response with two time delays Nonlinear Anal. Real World Appl. 2013 35 52

[22] U. Forys Stability and bifurcations for the chronic state in Marchuk’s model of an immune system J Math. Anal. Appl 2009 922 942

[23] M. Galach Dynamics of tumor-immune system comptition-the effect of the time delay Int. J. Math. Comput. Sci 2003 395 406

[24] D. Ghosh, S. Khajanchi, S. Mangiarotti, F. Denis, S.K. Dana, C. Letellier How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment? BioSystems 2017 17 30

[25] J.K. Hale and S.M.A. Lunel, Introduction to functional Differential Equations. Springer-Verlag, New York (1993).

[26] B.D. Hassard, N.D. Kazarinoff and Y.H. Wan, Theory and Application of Hopf Bifurcation. University of Cambridge, Cambridge (1981).

[27] S. Khanjanchi Bifurcation analysis of a delayed mathematical model for tumor growth Chaos Solitons Fractals 2015 264 276

[28] S. Khajanchi, S. Banerjee Stability and bifurcation analysis of delay induced tumor immune interaction model. Appl. Math. Comput. 2014 652 671

[29] D.E. Kirschner, J.C. Panetta Modelling the immunotheraphy of tumor-immune interaction J. Math. Biol 1998 235 252

[30] V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson Non-linear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 1994 295 321

[31] H. Mayer, K. Zaenker, U. Heiden A basic mathematical model of the immune response. Chaos 1995 155 161

[32] M.J. Piotrowska, M. Bodnar Influence of distributed delays on the dynamics of a generalized immune system cancerous cells interactions model Commun. Nonlinear Sci. Numer. Simul 2018 379 415

[33] M.J. Piotrowska, M. Bodnar, J. Poleszczuk, U. Forys Mathematical modelling of immune reaction against gliomas: sensivity analysis and influence of delays Nonlinear Anal. Real World Appl 2013 1601 1620

[34] F.A. Rihan, D.H.A. Rahaman, S. Lakshmanan, A.S. Alkhajeh A time delay model of tumor-immune system interactions: global dynamics, parameter estimation, sentivity analysis. Appl. Math. Comput. 2014 606 623

[35] S. Ruan, 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 2003 863 874

[36] M. Villasana, A. Radunskaya A delay differential equation model for tumor-growth. J. Math. Biol. 2003 270 294

[37] X. Yang, L. Chen, J. Cheng Parmanance and positive periodic solution for single-species non-autonomous delay diffusive model. Comput. Math. Appl. 1996 109

[38] M. Yu, Y. Dong, Y. Takeuchi Dual role of delay effects in a tumour- immune system J Biol. Dyn. 2017 334 347