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doi: 10.3934/dcdsb.2020282

Nonlinear dynamics in tumor-immune system interaction models with delays

1. 

Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA

2. 

Sylvester Comprehensive Cancer Center, University of Miami Miller School of Medicine, Miami, FL 33136, USA

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

Received  December 2019 Revised  August 2020 Published  September 2020

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

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.

Citation: Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020282
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

J. Adam and N. Bellomo, A Survey of Models on Tumor Immune Systems Dynamics, Birkhäuser, Boston, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

F. M. Burnet, Cancer – a biological approach, Brit. Med. J., 1 (1957), 841-847. doi: 10.1136/bmj.1.5023.841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Couzin-Frankel, Cancer immunotherapy, Science, 342 (2013), 1432-1433. doi: 10.1016/0022-247X(82)90243-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[60]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[63] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[73]

Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, Boca Raton, FL, 2016.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[76]

A. K. Laird, Dynamics of tumor growth, Br. J. Cancer, 18 (1964), 490-502. doi: 10.1038/bjc.1964.55.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[89]

I. Mellman, G. Coukos and G. Dranoff, Cancer immunotherapy comes of age, Nature, 480 (2011), 480-489. doi: 10.1038/nature10673.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[95]

D. Pardoll, Does the immune system see tumors as foreign or self?, Annu. Rev. Immunol., 21 (2003), 807-839. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[109]

N. Stepanova, Course of the immune reaction during the development of a malignant tumor, Biophysics, 24 (1980), 917-923. Google Scholar

[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.  Google Scholar

[111]

F. Takens, Singularities of vector fields, Publ. Math. IHES, 43 (1974), 47-100. doi: 10.1007/BF02684366.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[116]

G. F. Webb, Resonance phenomena in cell population chemotherapy models, Rocky Mountain J. Math., 20 (1980), 1195-1210. doi: 10.1216/rmjm/1181073070.  Google Scholar

[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.  Google Scholar

[118]

T. E. Wheldon, Mathematical Models in Cancer Research, Adam Hilger, Bristol, 1988. Google Scholar

[119]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

J. Adam and N. Bellomo, A Survey of Models on Tumor Immune Systems Dynamics, Birkhäuser, Boston, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

F. M. Burnet, Cancer – a biological approach, Brit. Med. J., 1 (1957), 841-847. doi: 10.1136/bmj.1.5023.841.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Couzin-Frankel, Cancer immunotherapy, Science, 342 (2013), 1432-1433. doi: 10.1016/0022-247X(82)90243-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[60]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[63] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[73]

Y. Kuang, J. D. Nagy and S. E. Eikenberry, Introduction to Mathematical Oncology, Chapman and Hall/CRC, Boca Raton, FL, 2016.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[76]

A. K. Laird, Dynamics of tumor growth, Br. J. Cancer, 18 (1964), 490-502. doi: 10.1038/bjc.1964.55.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[89]

I. Mellman, G. Coukos and G. Dranoff, Cancer immunotherapy comes of age, Nature, 480 (2011), 480-489. doi: 10.1038/nature10673.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[95]

D. Pardoll, Does the immune system see tumors as foreign or self?, Annu. Rev. Immunol., 21 (2003), 807-839. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[109]

N. Stepanova, Course of the immune reaction during the development of a malignant tumor, Biophysics, 24 (1980), 917-923. Google Scholar

[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.  Google Scholar

[111]

F. Takens, Singularities of vector fields, Publ. Math. IHES, 43 (1974), 47-100. doi: 10.1007/BF02684366.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[116]

G. F. Webb, Resonance phenomena in cell population chemotherapy models, Rocky Mountain J. Math., 20 (1980), 1195-1210. doi: 10.1216/rmjm/1181073070.  Google Scholar

[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.  Google Scholar

[118]

T. E. Wheldon, Mathematical Models in Cancer Research, Adam Hilger, Bristol, 1988. Google Scholar

[119]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed., Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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
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