# American Institute of Mathematical Sciences

2012, 9(2): 347-368. doi: 10.3934/mbe.2012.9.347

## Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions

 1 Department of Mathematics, Xidian University, Xi’an, Shaanxi 710071, China 2 Department of Mathematics, University of Miami, Coral Gables, FL 33124, United States 3 Department of Mathematics, East China Normal University, Shanghai 200062

Received  January 2011 Revised  October 2011 Published  March 2012

This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions between lymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions. We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerate Hopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated from the interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained for the two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parameters and demonstrate the phenomenon of long-term tumor relapse in the model.
Citation: Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Mathematical Biosciences & Engineering, 2012, 9 (2) : 347-368. doi: 10.3934/mbe.2012.9.347
##### References:
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Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Res., 65 (2005), 7950. Google Scholar [7] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032. Google Scholar [8] 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. doi: 10.1016/j.mcm.2009.11.005. Google Scholar [9] 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 (2010), 2. doi: 10.1007/s11538-010-9526-3. Google Scholar [10] P. Fortin and M. C. Mackey, Periodic chronicmyelogenous leukemia: Spectral analysis of blood cell counts and etiological implications,, Brit. J. Haematol., 104 (1999), 336. doi: 10.1046/j.1365-2141.1999.01168.x. Google Scholar [11] K. O. Friedrichs, "Advanced Ordinary Differential Equations,", Notes by P. Berg, (1965). Google Scholar [12] M. Golubitsky and W. F. Langford, Classification and unfoldings of degenerate Hopf bifurcations,, J. Differential Equations, 41 (1981), 375. doi: 10.1016/0022-0396(81)90045-0. Google Scholar [13] E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems,, Ber. Verh. Saechs. Akad. Wiss. Leipzig, 94 (1942), 3. Google Scholar [14] I. D. Hsü and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model,, J. Math. Anal. Appl., 55 (1976), 61. doi: 10.1016/0022-247X(76)90278-X. Google Scholar [15] D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar [16] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy,, Math. Biosci. Engin., 6 (2009), 573. doi: 10.3934/mbe.2009.6.573. Google Scholar [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), 295. Google Scholar [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), (1978), 333. Google Scholar [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), 649. doi: 10.1016/j.mcm.2007.02.026. Google Scholar [20] A.-H. Lin, A model of tumor and lymphocyte interactions,, Discrete. Contin. Dynam. Systems Ser. B, 4 (2004), 241. doi: 10.3934/dcdsb.2004.4.241. Google Scholar [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), 151. doi: 10.3934/dcdsb.2009.12.151. Google Scholar [22] 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. doi: 10.3934/mbe.2007.4.239. Google Scholar [23] Z. Lu and Y. Luo, Two limit cycles in three-dimensional Lotka-Volterra systems,, Comput. Math. Appl., 44 (2002), 51. doi: 10.1016/S0898-1221(02)00129-3. Google Scholar [24] L. Perko, "Differential Equations and Dynamical Systems,'', 3rd edition, 7 (2001). Google Scholar [25] A. B. Poore, On the theory and application of the Hopf-Friedrichs bifurcation theory,, Arch. Rational Mech. Anal., 60 (): 371. Google Scholar [26] A. Rescigno and C. DeLisi, Immune surveillance and neoplasia. II. A two-stage mathematical model,, Bull. Math. Biol., 39 (1977), 487. Google Scholar [27] H. L. Smith, "Monotone Dynamical Systems: An Introduction to Theory of Competitive and Coorperative Systems,'', Math. Surveys Monogr., 41 (1995). Google Scholar [28] M. J. Smyth, D. I. Godfrey and J. A. Trapani, A fresh look at tumor immunosurveillance and immunotherapy,, Nat. Immunol., 4 (2001), 293. doi: 10.1038/86297. Google Scholar [29] J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system,, Immunol. Rev., 216 (2007), 213. Google Scholar [30] G. W. Swan, Immunological surveillance and neoplastic development,, Rocky Mountain J. Math., 9 (1979), 143. doi: 10.1216/RMJ-1979-9-1-143. Google Scholar [31] R. Thomlinson, Measurement and management of carcinoma of the breast,, Clin. Radiol., 33 (1982), 481. doi: 10.1016/S0009-9260(82)80153-0. Google Scholar [32] J. Zhang, "The Geometric Theory and Bifurcation Problem of Ordinary Differential Equation,'', Peking University Press, (1987). Google Scholar

show all references

##### References:
 [1] A. Albert, M. Freedman and A. S. Perelson, Tumor and the immune system: The effects of tumor growth modulator,, Math. Biosci., 50 (1980), 25. doi: 10.1016/0025-5564(80)90120-0. Google Scholar [2] P. Bi and S. Ruan, Bifurcations in tumor and immune system interaction models with delay effect,, submitted., (). Google Scholar [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), 1312. doi: 10.1093/annonc/mdg353. Google Scholar [4] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill Book Company, (1955). Google Scholar [5] C. DeLisi and A. Rescigno, Immune surveillance and neoplasia. I. A minimal mathematical model,, Bull. Math. Biol., 39 (1977), 201. Google Scholar [6] 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. Google Scholar [7] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032. Google Scholar [8] 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. doi: 10.1016/j.mcm.2009.11.005. Google Scholar [9] 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 (2010), 2. doi: 10.1007/s11538-010-9526-3. Google Scholar [10] P. Fortin and M. C. Mackey, Periodic chronicmyelogenous leukemia: Spectral analysis of blood cell counts and etiological implications,, Brit. J. Haematol., 104 (1999), 336. doi: 10.1046/j.1365-2141.1999.01168.x. Google Scholar [11] K. O. Friedrichs, "Advanced Ordinary Differential Equations,", Notes by P. Berg, (1965). Google Scholar [12] M. Golubitsky and W. F. Langford, Classification and unfoldings of degenerate Hopf bifurcations,, J. Differential Equations, 41 (1981), 375. doi: 10.1016/0022-0396(81)90045-0. Google Scholar [13] E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems,, Ber. Verh. Saechs. Akad. Wiss. Leipzig, 94 (1942), 3. Google Scholar [14] I. D. Hsü and N. D. Kazarinoff, An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model,, J. Math. Anal. Appl., 55 (1976), 61. doi: 10.1016/0022-247X(76)90278-X. Google Scholar [15] D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar [16] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy,, Math. Biosci. Engin., 6 (2009), 573. doi: 10.3934/mbe.2009.6.573. Google Scholar [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), 295. Google Scholar [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), (1978), 333. Google Scholar [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), 649. doi: 10.1016/j.mcm.2007.02.026. Google Scholar [20] A.-H. Lin, A model of tumor and lymphocyte interactions,, Discrete. Contin. Dynam. Systems Ser. B, 4 (2004), 241. doi: 10.3934/dcdsb.2004.4.241. Google Scholar [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), 151. doi: 10.3934/dcdsb.2009.12.151. Google Scholar [22] 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. doi: 10.3934/mbe.2007.4.239. Google Scholar [23] Z. Lu and Y. Luo, Two limit cycles in three-dimensional Lotka-Volterra systems,, Comput. Math. Appl., 44 (2002), 51. doi: 10.1016/S0898-1221(02)00129-3. Google Scholar [24] L. Perko, "Differential Equations and Dynamical Systems,'', 3rd edition, 7 (2001). Google Scholar [25] A. B. Poore, On the theory and application of the Hopf-Friedrichs bifurcation theory,, Arch. Rational Mech. Anal., 60 (): 371. Google Scholar [26] A. Rescigno and C. DeLisi, Immune surveillance and neoplasia. II. A two-stage mathematical model,, Bull. Math. Biol., 39 (1977), 487. Google Scholar [27] H. L. Smith, "Monotone Dynamical Systems: An Introduction to Theory of Competitive and Coorperative Systems,'', Math. Surveys Monogr., 41 (1995). Google Scholar [28] M. J. Smyth, D. I. Godfrey and J. A. Trapani, A fresh look at tumor immunosurveillance and immunotherapy,, Nat. Immunol., 4 (2001), 293. doi: 10.1038/86297. Google Scholar [29] J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system,, Immunol. Rev., 216 (2007), 213. Google Scholar [30] G. W. Swan, Immunological surveillance and neoplastic development,, Rocky Mountain J. Math., 9 (1979), 143. doi: 10.1216/RMJ-1979-9-1-143. Google Scholar [31] R. Thomlinson, Measurement and management of carcinoma of the breast,, Clin. Radiol., 33 (1982), 481. doi: 10.1016/S0009-9260(82)80153-0. Google Scholar [32] J. Zhang, "The Geometric Theory and Bifurcation Problem of Ordinary Differential Equation,'', Peking University Press, (1987). Google Scholar
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