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Nonlinear dynamics in tumor-immune system interaction models with delays

In memory of my Ph.D. thesis supervisor Professor Herbert I. Freedman (1940 - 2017)

Research was partially supported by National Science Foundation grant (DMS-1853622)

Abstract Full Text(HTML) Figure(16) / Table(1) Related Papers Cited by
  • In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

    Mathematics Subject Classification: Primary: 34K18, 92C37; Secondary: 37N25.


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  • Figure 1.  Three processes in cancer immunoediting: (a) Elimination corresponds to immunosurveillance; (b) Equilibrium represents the process by which the immune system iteratively selects and/or promotes the generation of tumor cell variants with increasing capacities to survive immune attack; (c) Escape is the process wherein the immunologically sculpted tumor expands in an uncontrolled manner in the immunocompetent host. Adapted from Dunn [44]

    Figure 2.  Scheme of the essential mechanisms of interaction between the tumor cells and immune effector cells

    Figure 3.  (a) Solution trajectories converge to the stable equilibrium $ P_2 = (92.1911, 1.3344); $ (b) Periodic solutions bifurcated from the positive equilibrium when $ \tau = 2.0>\tau_0 $

    Figure 4.  The stability regions of the equilibria (a) $ P^1_2(8.18971, 1.6092) $ and (b) $ P^3_2(447.134, 0.17298) $ with blue dashed lines in the $ (B_2, \tau) $-parameter plane

    Figure 5.  The bifurcation diagram for system (40). (a) $ \alpha_2>0. $ (b) $ \alpha_2<0. $

    Figure 6.  Phase portraits for the case VIa in Table 7.5.2 of [58]: (a) Bifurcation diagram in $ (\mu_1,\mu_2) $; (b) Phase portraits of (81)

    Figure 7.  The bifurcation diagram of (81) on the $ (\mu_1, \mu_2) $-plane

    Figure 8.  (a) The periodic solution $ (x(t), y(t)) $ bifurcated from the microscopic equilibrium $ (8.18971, 1.6092) $ with $ \tau = 0.333814. $ (b) The corresponding solution $ x(t) $ in terms of time $ t. $ (c) The periodic solution $ (x(t), y(t)) $ bifurcated from the macroscopic equilibrium $ (447.134, 0.172977) $ as $ \tau = 2.08803. $ (d) The corresponding solution $ x(t) $ in terms of time $ t. $

    Figure 9.  Graphs of $ f(T) $ for three different parameter sets: (a) $ u = v = 1 $; (b) $ u = v = 5>1 $; (c) $ u = \frac{1}{2}<1<v = 2. $

    Figure 10.  The phase portraits of system (87): (a) with parameter set (I); (b) with parameter set (II)

    Figure 11.  (a) The converging solutions of system (87) in terms of $ t $ when $ \tau = 4, \delta = 20; $ (b) The solution trajectories of system (87) spiral toward the positive equilibrium in the $ (T, E) $-plane when $ \tau = 4, \delta = 20 $; (c) The periodic solutions of system (87) in terms of $ t $ when $ \tau = \tau_0 $; (d) The periodic trajectories of system (87) in the $ (T, E) $-plane

    Figure 12.  Stability diagram of system (87) on the ($ \tau, \delta $)-delay parameter space

    Figure 13.  (a) The locations of $ f(\omega) $ and $ h(\omega) $ when $ a = 0.1, b = 0.1 $. (a) $ 0.2 < \omega < 1.5; $ (b) $ 0.26 < \omega < 0.3. $

    Figure 14.  (a) The stable solutions of system (87) when $ \tau = 3.8, \delta = 13.5,\ \Delta = 2; $ (b) The solution trajectory of system (87) converges to the positive equilibrium in the $ (T,E) $ plane; (c) The periodic solutions $ T(t) $ and $ E(t) $ of system (87) in terms of $ t $ when $ \tau = \tau_0 = 3.8641,\ \delta = \delta_0 = 15.1378,\ \Delta = \Delta_0 = 2.0592 $; (d) The periodic trajectories of system (87) in the $ (T,E) $ plane

    Figure 15.  The stability diagram of the positive equilibrium for system (87) in the ($ \tau, \delta,\ \Delta $) parameter space

    Figure 16.  (a)(b) The regular periodic oscillations in system (87) with $ \tau = 0.5, \delta = 5, \Delta = 8; $ (c)(d) The irregular long periodic oscillations in system (87) with $ \tau = 0.5, \delta = 15, \Delta = 8; $ (e)(f) The chaotic solutions in system (87) with $ \tau = 0.5, \delta = 50, \Delta = 38. $

    Table 1.  Existence and stability chart for the ODE model (Gałach [56])

    Sign of $\zeta$ Conditions $P_0$ $P_1$ $P_2$
    $\zeta > 0$ $\alpha \delta< \sigma$ stable
    $\alpha \delta> \sigma$ unstable stable
    $\zeta< 0$ $\alpha(\beta\delta-\zeta)^2+4\beta\zeta\sigma<0$ stable
    $\alpha(\beta\delta-\zeta)^2+4\beta\zeta\sigma>0$ $\alpha \delta> \sigma$
    $\zeta + \beta \delta<0$
    unstable stable
    $\alpha \delta< \sigma$} stable unstable stable
     | Show Table
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  • [1] I. Abdulrashid, A. A. M. Alsammani and X. Han, Stability analysis of a chemotherapy model with delays, Discrete Contin. Dyn. Syst.-Ser. B, 24 (2019), 989-1005. doi: 10.3934/dcdsb.2019002.
    [2] J. A. Adam, Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features, Math. Comp. Modelling, 23 (1996), 1-10. doi: 10.1016/0895-7177(96)00016-7.
    [3] J. Adam and N. Bellomo, A Survey of Models on Tumor Immune Systems Dynamics, Birkhäuser, Boston, 1996.
    [4] M. Adimy, F. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays., J. Theor. Biol., 242 (2006), 288-299. doi: 10.1016/j.jtbi.2006.02.020.
    [5] A. Albert, M. Freedman and A. S. Perelson, Tumors and the immune system: The effects of a tumor growth modelator, Math. Biosci., 50 (1980), 25-58. doi: 10.1016/0025-5564(80)90120-0.
    [6] L. K.. Andersen and M. C. Mackey, Resonance in periodic chemotherapy: a case study of acute myelogenous leukemia, J. Theor. Biol., 209 (2001), 113-130. doi: 10.1006/jtbi.2000.2255.
    [7] A. R. A. Anderson and P. K. Maini, Mathematical oncology, Bull. Math. Biol., 80 (2018), 945-953. doi: 10.1007/s11538-018-0423-5.
    [8] J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discrete Contin. Dyn. Syst.-Ser. B, 4 (2004), 39-58. doi: 10.3934/dcdsb.2004.4.39.
    [9] O. Arino, A. Bertuzzi, A. Gandolfi, E. Sánchez and C. Sinisgalli, The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations, J. Math. Anal. Appl., 302 (2005), 521-542. doi: 10.1016/j.jmaa.2004.08.024.
    [10] A. L. Asachenkov, G. I. Marchuk, R. R. Mohler and S. M. Zuev, Immunology and disease control: A systems approach, IEEE Trans. Biomed. Eng., 41 (1994), 943-53. doi: 10.1109/10.324526.
    [11] S. Banerjee and R. R. Sarkar, Delay-induced model for tumor–immune interaction and control of malignant tumor growth, BioSystems 91 (2008), 268-288. doi: 10.1016/j.biosystems.2007.10.002.
    [12] M. V. Barbarossa, C. Kuttler and J. Zinsl, Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells, Math Biosci. Engin., 9 (2012), 241-257. doi: 10.3934/mbe.2012.9.241.
    [13] P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Appl. Dynam. Syst., 12 (2013), 1847-1888. doi: 10.1137/120887898.
    [14] P. Bi, S. Ruan and X. Zhang, Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays, Chaos, 24 (2014), 023101. doi: 10.1063/1.4870363.
    [15] P. Bi and H. Xiao, Hopf bifurcation for tumor-immune competition systems with delay, Electr. J. Differ. Equ., 2014, No. 27, pp. 1-13.
    [16] P. Bi, Z. Liu, M. D. Muthoni and J. Pang, The dynamical behaviors for a class of immunogenic tumor model with delay, Comput. Math. Meth. Med., 2017, Article ID 1642976, 1-9. doi: 10.1155/2017/1642976.
    [17] F. Billya, J. Clairambaultt, O. Fercoq, S. Gaubertt, T. Lepoutre, T. Ouillon and S Saito, Synchronisation and control of proliferation in cycling cellpopulation models with age structure, Math. Comput. Simul., 96 (2014), 66-94. doi: 10.1016/j.matcom.2012.03.005.
    [18] M. Bodnar and U. Foryś, Delays do not cause oscillations in a corrected model of humoral mediated immune response, Appl. Math. Comput., 289 (2016), 7-21. doi: 10.1016/j.amc.2016.05.006.
    [19] R. Brady and H. Enderling, Mathematical models of cancer: when to predict novel therapies, and when not to, Bull. Math. Biol., 81 (2019), 3722-3731. doi: 10.1007/s11538-019-00640-x.
    [20] F. B. Brikci, J. Clairambault, B. Ribba and B. Perthame, An age-and-cyclin-structured cell population model for healthy and tumoral tissues, J. Math. Biol., 57 (2008), 91-110. doi: 10.1007/s00285-007-0147-x.
    [21] P.-L. Buono and J. Bélair, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations, 189 (2003), 234-266. doi: 10.1016/S0022-0396(02)00179-1.
    [22] N. Buric, M. Mudrinic and N. Vasovic, Time delay in a basic model of the immune response, Chaos, Solitons Fractals, 12 (2001), 483-489. doi: 10.1016/S0960-0779(99)00205-2.
    [23] F. M. Burnet, Cancer – a biological approach, Brit. Med. J., 1 (1957), 841-847. doi: 10.1136/bmj.1.5023.841.
    [24] H. M. Byrne, S. M. Cox and C. E. Kelly, Macrophage-tumor interactions: In vivo dynamics, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 81-98. doi: 10.3934/dcdsb.2004.4.81.
    [25] S. A. Campbell and J. Bélair, Resonant codimension two bifurcation in the harmonic oscillator with delayed forcing., Canad. Appl. Math. Quart., 7 (1999), 218-238.
    [26] S. A. Campbell and Y. Yuan, Zero singularities of codimension two and three in delay differential equations, Nonlinearity, 21 (2008), 2671-2691. doi: 10.1088/0951-7715/21/11/010.
    [27] Y. Choi and V. G. LeBlanc, Toroidal normal forms for bifurcations in retarded functional differential equations I: Multiple Hopf and transcritical/multiple Hopf interaction, J. Differential Equations, 227 (2006), 166-203. doi: 10.1016/j.jde.2005.12.003.
    [28] J. Clairambault, B. Perthame and A. Q. Maran, Analysis of a system describing proliferative-quiescent cell dynamics, Chin. Ann. Math. Ser. B, 39 (2018), 345-356. doi: 10.1007/s11401-018-1068-2.
    [29] J. Couzin-Frankel, Cancer immunotherapy, Science, 342 (2013), 1432-1433. doi: 10.1016/0022-247X(82)90243-8.
    [30] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1126/science.342.6165.1432.
    [31] V. Cristini et al., Nonlinear modeling and simulation of tumor growth, in "Selected Topics in Cancer Modeling", Birkhäuser, Boston, 2008, pp. 1-69. doi: 10.1007/978-0-8176-4713-1_6.
    [32] G. R. Dagenais, D. P. Leong, S. Rangarajan, et al., Variations in common diseases, hospital admissions, and deaths in middle-aged adults in 21 countries from five continents (PURE): A prospective cohort study, Lancet, 395 (2019), 7-13. doi: 10.1016/S0140-6736(19)32007-0.
    [33] C. DeLisi and A. Rescigno, Immune surveillance and neoplasia-I: A minimal mathematical model, Bull. Math. Biol., 39 (1977), 201-221. doi: 10.1007/bf02462859.
    [34] L. G. dePillis, A. Eladdadi and A. E. Radunskaya, Modeling cancer-immune responses to therapy, J. Pharmacokinet. Pharmacodyn., 41 (2014), 461-478. doi: 10.1007/s10928-014-9386-9.
    [35] 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), 7950-7958. doi: 10.1158/0008-5472.CAN-05-0564.
    [36] Y. Dong, G. Huang, R. Miyazaki and Y. Takeuchi, Dynamics in a tumor immune system with time delays, Appl. Math. Comput., 252 (2015), 99-113. doi: 10.1016/j.amc.2014.11.096.
    [37] A. d'Onofrio, A general framework for modeling tumour-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Phys. D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032.
    [38] A. d'Onofrio, Tumour-immune system interaction: Modeling the tumour-stimulated proliferation of effectors and immunotherapy, Math. Models Methods Appl. Sci., 16 (2006), 1375-1401. doi: 10.1142/S0218202506001571.
    [39] A. d'Onofrio, Metamodeling tumour-immune system interaction, tumour evasion and immunotherapy, Math. Comput. Modelling, 47 (2008), 614-637. doi: 10.1016/j.mcm.2007.02.032.
    [40] A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Math. Med. Biol., 26 (2009), 63-95. doi: 10.1093/imammb/dqn024.
    [41] A. d'Onofrio, P. Cerrai and A. Gandolfi, New Challenges for Cancer Systems Biomedicine, SIMAI Springer Series. Springer, Milano, 2012. doi: 10.1007/978-88-470-2571-4.
    [42] A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour immune system interaction, Math. Comput. Modelling, 51 (2010), 572-591. doi: 10.1016/j.mcm.2009.11.005.
    [43] H. Dritschel, S. L. Waters, A. Roller and H. M. Byrne, A mathematical model of cytotoxic and helper T cell interactions in a tumour microenvironment, Letters Biomath., 5 (2018), S36-S68. doi: 10.30707/LiB5.2Dritschel.
    [44] G. P. Dunn, A. T. Bruce, H. Ikeda, L. J. Old and R. D. Schreiber, Cancer immunoediting: From immunosurveillance to tumor escape, Nature Immunol., 3 (2002), 991-998. doi: 10.1038/ni1102-991.
    [45] G. P. Dunn, L. J. Old and R. D. Schreiber, The three Es of cancer immunoediting, Annu. Rev. Immunol., 22 (2004), 329-360. doi: 10.1146/annurev.immunol.22.012703.104803.
    [46] J. Dyson, R. Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Math. Biosci., 177 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9.
    [47] R. Eftimie, J. L. Bramson and D.J.D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2-32. doi: 10.1007/s11538-010-9526-3.
    [48] A. Eladdadi, P. Kim and D. Mallet (eds.), Mathematical Models of Tumor-Immune System Dynamics, Springer Proceedings in Mathematics & Statistics Vol 107, Springer, New York, 2014. doi: 10.1007/978-1-4939-1793-8.
    [49] T. Faria and L. T. Magalhães, Normal forms for retarded functional differential equations and applications to Hopf bifurcation, J. Differential Equations, 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144.
    [50] T. Faria and L. T. Magalhães, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145.
    [51] 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. doi: 10.1016/j.nonrwa.2012.05.001.
    [52] F. Frascoli, E. Flood and P. S. Kim, A model of the effects of cancer cell motility and cellular adhesion properties on tumour-immune dynamics, Math. Med. Biol., 34 (2017), 215-240. doi: 10.1093/imammb/dqw004.
    [53] H. I. Freedman, Modeling cancer treatment using competition: a survey, In "Mathematics for Life Science and Medicine" (eds. Y. Takeuchi, Y. Iwasa and K. Sato), Springer, Berlin, 2007, pp. 207-223. doi: 10.1007/978-3-540-34426-1_9.
    [54] A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-160. doi: 10.3934/dcdsb.2004.4.147.
    [55] P. Gabriel, S. P. Garbett, V. Quaranta, D. R. Tyson and G. F. Webb, The contribution of age structure to cell population responses to targeted therapeutics, J. Theor. Biol., 311(2012), 19-27. doi: 10.1016/j.jtbi.2012.07.001.
    [56] M. Gałach, Dynamics of the tumor-immune system competition: The effect of time delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395-406.
    [57] Z. Grossman and G. Berke, Tumor escape from immune elimination, J. Theor. Biol., 83 (1980), 267-296. doi: 10.1016/0022-5193(80)90293-3.
    [58] J. Guckhenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.
    [59] S. Guo, Y. Chen and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, J. Differential Equations, 244 (2008), 444-486. doi: 10.1016/j.jde.2007.09.008.
    [60] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.
    [61] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. Math. Biol., 28 (1990), 671-694. doi: 10.1007/BF00160231.
    [62] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, {Appl. Math. Sci.}, 99, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
    [63] B. D. HassardN. D. Kazarinoff and  Y. H. WanTheory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. 
    [64] X. Hu and S. R.-J. Jang, Dynamics of tumor-CD4$^+$-cytokine-host cells interactions with treatments, Appl. Math. Comput., 321 (2018), 700-720. doi: 10.1016/j.amc.2017.11.009.
    [65] G. Huisman and R. J. De Boer, A formal derivation of the Beddington functional response, J. Theor. Biol., 185 (1997), 389-400. doi: 10.1006/jtbi.1996.0318.
    [66] A.-V. Ion, An example of Bautin-type bifurcation in a delay differential equation, J. Math. Anal. Appl., 329 (2007), 777-789. doi: 10.1016/j.jmaa.2006.06.083.
    [67] H. Jiang, J. Jiang and Y. Song, Normal form of saddle-node-Hopf bifurcation in retarded functional differential equations and applications, Intl. J. Bif. Chaos, 26 (2016), 1650040, 24 pp. doi: 10.1142/S0218127416500401.
    [68] W. Jiang and H. Wang, Hopf-transcritical bifurcation in retarded functional differential equations, Nonlinear Anal., 73 (2010), 3626-3640. doi: 10.1016/j.na.2010.07.043.
    [69] S. Khajanchi and S. Banerjee, Stability and bifurcation analysis of delay induced tumor immune interaction model, Appl. Math. Comput., 248 (2014), 652-671. doi: 10.1016/j.amc.2014.10.009.
    [70] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor - immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127.
    [71] C. M. Koebel, W. Vermi, J. B. Swann, N. Zerafa, S. J. Rodig, L. J. Old, M. J. Smyth and R. D. Schreiber, Adaptive immunity maintains occult cancer in an equilibrium state, Nature, 450 (2007), 903-907. doi: 10.1038/nature06309.
    [72] A. Konstorum, A. T. Vella, A. J. Adler and R. C. Laubenbacher, Addressing current challenges in cancer immunotherapy with mathematical and computational modelling, J. R. Soc. Interface, 14 (2017), 20170150. doi: 10.1098/rsif.2017.0150.
    [73] Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, Boca Raton, FL, 2016.
    [74] V. Kuznetsov, I. Makalkin, M. Taylor and A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimationestimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
    [75] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.
    [76] A. K. Laird, Dynamics of tumor growth, Br. J. Cancer, 18 (1964), 490-502. doi: 10.1038/bjc.1964.55.
    [77] V. G. LeBlanc, Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators, J. Differential Equations, 254 (2013), 637-647. doi: 10.1016/j.jde.2012.09.008.
    [78] O. Lejeune, M. A. J. Chaplaina and I. El Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours, Math. Comput. Modelling, 47 (2008), 649-662. doi: 10.1016/j.mcm.2007.02.026.
    [79] D. Liu, S. Ruan and D. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete Contin. Dynam. Syst.-Ser. B, 12 (2009), 151-168. doi: 10.3934/dcdsb.2009.12.151.
    [80] D. Liu, S. Ruan and D. Zhu, Stable periodic oscillations in a two-stage cancer model of tumor-immune interaction, Math. Biosci. Eng., 9 (2012), 347-368. doi: 10.3934/mbe.2012.9.347.
    [81] W. Liu, T. Hillen and H. I. Freedman, A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse, Math. Biosci. Engin., 4 (2007), 239-259. doi: 10.3934/mbe.2007.4.239.
    [82] Z. Liu, J. Chen, J. Pang, P. Bi and S. Ruan, Modeling and analysis of a nonlinear age-structured model for tumor cell populations with quiescence, J. Nonlinear Sci., 28 (2018), 1763-1791. doi: 10.1007/s00332-018-9463-0.
    [83] P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Appl. Math. Sci. 201, Springer, Cham, 2018. doi: 10.1007/978-3-030-01506-0.
    [84] G. E. Mahlbacher, K. C. Reihmer and H. B. Frieboes, Mathematical modeling of tumor-immune cell interactions, J. Theor. Biol., 469 (2019), 47-60. doi: 10.1016/j.jtbi.2019.03.002.
    [85] M. Marušsić, Ž. Bajzer, J. P. Freyer and S. Vuk-Pavlović, Analysis of growth of multicellular tumour spheroids by mathematical models, Cell Prolif., 27 (1994), 73-94. doi: 10.1111/j.1365-2184.1994.tb01407.x.
    [86] H. Matsushita, M. D. Vesely, D. C. Koboldt et al., Cancer exome analysis reveals a T-cell-dependent mechanism of cancer immunoediting, Nature, 482 (2012), 400-404. doi: 10.1038/nature10755.
    [87] H. Mayer, K. Zaenker and U. an der Heiden, A basic mathematical model of the immune response, Chaos, 5 (1995), 155-161. doi: 10.1063/1.166098.
    [88] C. J. M. Melief and R. S. Schwartz, Immunocompetence and malignancy, in "Cancer: A Comprehensive Treatise" (eds. F. F. Becker), Springer, New York, 1975, pp. 121-159.
    [89] I. Mellman, G. Coukos and G. Dranoff, Cancer immunotherapy comes of age, Nature, 480 (2011), 480-489. doi: 10.1038/nature10673.
    [90] J. P. Mendonça, I. Gleria and M. L. Lyra, Delay-induced bifurcations and chaos in a two-dimensional model for the immune response, Physica A, 517 (2019), 484-490. doi: 10.1016/j.physa.2018.11.039.
    [91] M. Mohme, S. Riethdorf and K. Pantel, Circulating and disseminated tumour cells - mechanisms of immune surveillance and escape, Nat. Rev. Clin. Oncol., 14 (2017), 155-167. doi: 10.1038/nrclinonc.2016.144.
    [92] F. Nani and H. I. Freedman, A mathematical model of cancer treatment by immunotherapy, Math. Biosci., 163 (2000), 159-199. doi: 10.1016/S0025-5564(99)00058-9.
    [93] E. Nikolopoulou, L. R. Johnson, D. Harris, J. D. Nagy, E. C. Stites and Y. Kuang, Tumour-immune dynamics with an immune checkpoint inhibitor, Letters Biomath., 5 (2018), (sup1), S137-S159. doi: 10.30707/LiB5.2Nikolopoulou.
    [94] M. Owen and J. Sherratt, Modeling the macrophage invasion of tumors: Effects on growth and composition, Math. Med. Biol., 15 (1998), 165-185. doi: 10.1093/imammb/15.2.165.
    [95] D. Pardoll, Does the immune system see tumors as foreign or self?, Annu. Rev. Immunol., 21 (2003), 807-839.
    [96] M. J. Piotrowska, An immune system-tumour interactions model with discrete time delay: Model analysis and validation, Commun. Nonlinear Sci. Numer. Simulat., 34 (2016), 185-198 doi: 10.1016/j.cnsns.2015.10.022.
    [97] M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180-203. doi: 10.1016/j.jmaa.2011.04.046.
    [98] D. L. Porter, B. L. Levine, M. Kalos, A. Bagg and C. H. June, Chimeric antigen receptor modified T c ells in chronic lymphoid leukemia, N. Egnl. J. Med., 365 (2011), 725-733. doi: 10.1056/NEJMoa1103849.
    [99] M. Qomlaqi, F. Bahrami, M. Ajami and J. Hajati, An extended mathematical model of tumor growth and its interaction with the immune system, to be used for developing an optimized immunotherapy treatment protocol, Math. Biosci., 292 (2017), 1-9. doi: 10.1016/j.mbs.2017.07.006.
    [100] M. Robertson-Tessi, A. El-Kareh and A. Goriely, A mathematical model of tumor-immune interactions, J. Theor. Biol., 294 (2012), 56-73. doi: 10.1016/j.jtbi.2011.10.027.
    [101] D. Rordriguez-Perez, O. Sotolongo-Grau, R. Espinosa Riquelme, O. Sotolongo-Costa, J. A. Santos Miranda and J. C. Antoranz, Assessment of cancer immunotherapy outcome in terms of the immune response time features, Math. Med. Biol., 24 (2007), 287-300. doi: 10.1093/imammb/dqm003.
    [102] S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173. doi: 10.1090/qam/1811101.
    [103] S. Ruan, On nonlinear dynamics of predator-prey models with discrete delay, Math. Model. Nat. Phenom., 4 (2009), 140-188. doi: 10.1051/mmnp/20094207.
    [104] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam. Contin. Discrete Impuls. Syst. Ser. A, 10 (2003), 863-874.
    [105] S. Ruan, J. Wei and D. Xiao, On the distribution of zeros of a third-degree exponential polynomial with applications to delayed biological systems, J. Nanjing Univ. Information Sci. Tech., 9 (2017), 381-390.
    [106] R. D. Schreiber, L. J. Old and M. J. Smyth, Cancer immunoediting: Integrating immunity's roles in cancer suppression and promotion, Science, 331 (2011), 1565-1570. doi: 10.1126/science.1203486.
    [107] O. Sotolongo-Costa, L. Morales-Molina, D. Rodriguez-Perez, J. C. Antonranz and M. Chacon-Reyes, Behavior of tumors under nonstationary therapy, Physica D, 178 (2003), 242-253. doi: 10.1016/S0167-2789(03)00005-8.
    [108] L. Spinelli, A. Torricelli, P. Ubezio and B. Basse, Modelling the balance between quiescence and cell death in normal and tumour cell populations, Math. Biosci., 202 (2006), 349-370. doi: 10.1016/j.mbs.2006.03.016.
    [109] N. Stepanova, Course of the immune reaction during the development of a malignant tumor, Biophysics, 24 (1980), 917-923.
    [110] Z. Szymańska, M. Cytowski, E. Mitchell, C.K. Macnamara and M.A. Chaplain, Computational modelling of cancer development and growth: modelling at multiple scales and multiscale modelling, Bull. Math. Biol., 80 (2018), 1366-1403. doi: 10.1007/s11538-017-0292-3.
    [111] F. Takens, Singularities of vector fields, Publ. Math. IHES, 43 (1974), 47-100. doi: 10.1007/BF02684366.
    [112] L. Thomas, Discussion, in "Cellular and Humoral Aspects of the Hypersensitive States" (ed. H. S. Lawrence), Hoeber-Harper, New York, 1959, pp. 529-532.
    [113] K. Vermeulen, D. R. Van Bockstaele and Z. N. Berneman, The cell cycle: A review of regulation, deregulation and therapeutic targets in cancer, Cell Prolif., 36 (2003), 131-149. doi: 10.1046/j.1365-2184.2003.00266.x.
    [114] M. D. Vesely, M. H. Kershaw, R. D. Schreiber and M. J. Smyth, Natural innate and adaptive immunity to cancer, Annu. Rev. Immunol., 29 (2011), 235-271. doi: 10.1146/annurev-immunol-031210-101324.
    [115] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol., 47 (2003), 270-294. doi: 10.1007/s00285-003-0211-0.
    [116] G. F. Webb, Resonance phenomena in cell population chemotherapy models, Rocky Mountain J. Math., 20 (1980), 1195-1210. doi: 10.1216/rmjm/1181073070.
    [117] J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D, 130 (1999), 255-272. doi: 10.1016/S0167-2789(99)00009-3.
    [118] T. E. Wheldon, Mathematical Models in Cancer Research, Adam Hilger, Bristol, 1988.
    [119] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Springer-Verlag, New York, 2003.
    [120] K. P. Wilkie, A review of mathematical models of cancer-immune interactions in the context of tumor dormancy, in " Systems Biology of Tumor Dormancy" (eds. H. Enderling et al.), Springer, New York, 2013, pp. 201-234. doi: 10.1007/978-1-4614-1445-2_10.
    [121] World Health Organization (WHO), The top 10 causes of death, 24 May 2018. https://www.who.int/news-room/fact-sheets/detail/the-top-10-causes-of-death.
    [122] X. P. Wu and L. Wang, Zero-Hopf singularity for general delayed differential equations, Nonlinear Dynam., 75 (2014), 141-155. doi: 10.1007/s11071-013-1055-9.
    [123] X. P. Wu and L. Wang, Zero-Hopf bifurcation analysis in delayed differential equations with two delays, J. Franklin Inst., 354 (2017), 1484-1513. doi: 10.1016/j.jfranklin.2016.11.029.
    [124] X. P. Wu and L. Wang, Normal form of double-Hopf singularity with 1: 1 resonance for delayed differential equations, Nonlinear Anal. Model. Control, 24 (2019), 241-260. doi: 10.15388/NA.2019.2.6.
    [125] D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations, 176 (2001), 494-510. doi: 10.1006/jdeq.2000.3982.
    [126] C. Yu and J. Wei, Stability and bifurcation analysis in a basic model of the immune response with delays, Chaos Solitons Fractals, 41 (2009), 1223-1234. doi: 10.1016/j.chaos.2008.05.007.
    [127] M. Yu, Y. Dong and Y. Takeuchi, Dual role of delay effects in a tumour–immune system, J. Biol. Dynam., 11 (2017) (S2), 334-347. doi: 10.1080/17513758.2016.1231347.
    [128] M. Yu, G. Huang, Y. Dong and Y. Takeuchi, Complicated dynamics of tumor-immune system interaction model with distributed time delay, Discrete Contin. Dynam. Syst. Ser. B, 25 (2020), 2391-2406. doi: 10.3934/dcdsb.2020015.
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