Stable periodic oscillations in a twostage cancer model of tumor and immune system interactions
Pages: 347  368,
Issue 2,
April
2012
doi:10.3934/mbe.2012.9.347 Abstract
References
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Dan Liu  Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, China (email)
Shigui Ruan  Department of Mathematics, University of Miami, Coral Gables, FL 33124, United States (email)
Deming Zhu  Department of Mathematics, East China Normal University, Shanghai 200062, China (email)
1 
A. Albert, M. Freedman and A. S. Perelson, Tumor and the immune system: The effects of tumor growth modulator, Math. Biosci., 50 (1980), 2558. 

2 
P. Bi and S. Ruan, Bifurcations in tumor and immune system interaction models with delay effect, submitted. 

3 
P. Boyle, A. d'Onofrio, P. Maisonneuve, G. Severi, C. Robertson, M. Tubiana and U. Veronesi, Measuring progress against cancer in Europe: Has the 15% decline targeted for 2000 come about?, Ann. Oncol., 14 (2003), 13121325. 

4 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGrawHill Book Company, Inc., New YorkTorontoLondon, 1955. 

5 
C. DeLisi and A. Rescigno, Immune surveillance and neoplasia. I. A minimal mathematical model, Bull. Math. Biol., 39 (1977), 201221. 

6 
L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth, Cancer Res., 65 (2005), 79507958. 

7 
A. d'Onofrio, A general framework for modeling tumorimmune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220235. 

8 
A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delayinduced oscillatory dynamics of tumour immune system interaction, Math. Comput. Modelling, 51 (2010), 572591. 

9 
R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of nonspatial mathematical models, Bull. Math. Biol., 73 (2010), 232. 

10 
P. Fortin and M. C. Mackey, Periodic chronicmyelogenous leukemia: Spectral analysis of blood cell counts and etiological implications, Brit. J. Haematol., 104 (1999), 336345. 

11 
K. O. Friedrichs, "Advanced Ordinary Differential Equations," Notes by P. Berg, W. Hirsch and P. Treuenfels, Gordon and Breach Science Publishers, New YorkLondonParis, 1965. 

12 
M. Golubitsky and W. F. Langford, Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41 (1981), 375415. 

13 
E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems, Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss., 94 (1942), 322. 

14 
I. D. Hsü and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of the FieldNoyes model, J. Math. Anal. Appl., 55 (1976), 6189. 

15 
D. Kirschner and J. Panetta, Modeling immunotherapy of the tumorimmune interaction, J. Math. Biol., 37 (1998), 235252. 

16 
D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Engin., 6 (2009), 573583. 

17 
V. A. Kuznetsov, I. A. Makalkin, M. A. Talor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameters estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295321. 

18 
R. Lefever and R. Garay, Local discription of immune tumor rejection, in "Biomathematics and Cell Kinetics'' (eds. A. J. Valleron and P. D. M. Macdonald), NorthHolland Biomedical Press, (1978), 333344. 

19 
O. Lejeune, M. A. J. Chaplain and I. El Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours, Math. Computer Model., 47 (2008), 649662. 

20 
A.H. Lin, A model of tumor and lymphocyte interactions, Discrete. Contin. Dynam. Systems Ser. B, 4 (2004), 241266. 

21 
D. Liu, S. Ruan and D. Zhu, Bifurcation analysis in models of tumor and immune system interactions, Discrete Contin. Dynam. Systems Ser. B, 12 (2009), 151168. 

22 
W. Liu, T. Hillen and H. I. Freedman, A mathematical model for Mphase specific chemotherapy including the $G_0$phase and immunoresponse, Math. Biosci. Engin., 4 (2007), 239259. 

23 
Z. Lu and Y. Luo, Two limit cycles in threedimensional LotkaVolterra systems, Comput. Math. Appl., 44 (2002), 5166. 

24 
L. Perko, "Differential Equations and Dynamical Systems,'' 3^{rd} edition, Texts in Applied Mathematics, 7, SpringerVerlag, New York, 2001. 

25 
A. B. Poore, On the theory and application of the HopfFriedrichs bifurcation theory, Arch. Rational Mech. Anal., 60 (1975/76), 371393. 

26 
A. Rescigno and C. DeLisi, Immune surveillance and neoplasia. II. A twostage mathematical model, Bull. Math. Biol., 39 (1977), 487497. 

27 
H. L. Smith, "Monotone Dynamical Systems: An Introduction to Theory of Competitive and Coorperative Systems,'' Math. Surveys Monogr., 41, Amer. Math. Soc., Providence, RI, 1995. 

28 
M. J. Smyth, D. I. Godfrey and J. A. Trapani, A fresh look at tumor immunosurveillance and immunotherapy, Nat. Immunol., 4 (2001), 293299. 

29 
J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system, Immunol. Rev., 216 (2007), 213231. 

30 
G. W. Swan, Immunological surveillance and neoplastic development, Rocky Mountain J. Math., 9 (1979), 143148. 

31 
R. Thomlinson, Measurement and management of carcinoma of the breast, Clin. Radiol., 33 (1982), 481492. 

32 
J. Zhang, "The Geometric Theory and Bifurcation Problem of Ordinary Differential Equation,'' Peking University Press, Beijing, 1987. 

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