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:
[1]

A. Albert, M. Freedman and A. S. Perelson, Tumor and the immune system: The effects of tumor growth modulator, Math. Biosci., 50 (1980), 25-58. doi: 10.1016/0025-5564(80)90120-0.

[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), 1312-1325. doi: 10.1093/annonc/mdg353.

[4]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[5]

C. DeLisi and A. Rescigno, Immune surveillance and neoplasia. I. A minimal mathematical model, Bull. Math. Biol., 39 (1977), 201-221.

[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), 7950-7958.

[7]

A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032.

[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), 572-591. doi: 10.1016/j.mcm.2009.11.005.

[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), 2-32. doi: 10.1007/s11538-010-9526-3.

[10]

P. Fortin and M. C. Mackey, Periodic chronicmyelogenous leukemia: Spectral analysis of blood cell counts and etiological implications, Brit. J. Haematol., 104 (1999), 336-345. doi: 10.1046/j.1365-2141.1999.01168.x.

[11]

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

[12]

M. Golubitsky and W. F. Langford, Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41 (1981), 375-415. doi: 10.1016/0022-0396(81)90045-0.

[13]

E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems, Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss., 94 (1942), 3-22.

[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), 61-89. doi: 10.1016/0022-247X(76)90278-X.

[15]

D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127.

[16]

D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Engin., 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573.

[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), 295-321.

[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), North-Holland Biomedical Press, (1978), 333-344.

[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), 649-662. doi: 10.1016/j.mcm.2007.02.026.

[20]

A.-H. Lin, A model of tumor and lymphocyte interactions, Discrete. Contin. Dynam. Systems Ser. B, 4 (2004), 241-266. doi: 10.3934/dcdsb.2004.4.241.

[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), 151-168. doi: 10.3934/dcdsb.2009.12.151.

[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), 239-259. doi: 10.3934/mbe.2007.4.239.

[23]

Z. Lu and Y. Luo, Two limit cycles in three-dimensional Lotka-Volterra systems, Comput. Math. Appl., 44 (2002), 51-66. doi: 10.1016/S0898-1221(02)00129-3.

[24]

L. Perko, "Differential Equations and Dynamical Systems,'' 3rd edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001.

[25]

A. B. Poore, On the theory and application of the Hopf-Friedrichs bifurcation theory, Arch. Rational Mech. Anal., 60 (1975/76), 371-393.

[26]

A. Rescigno and C. DeLisi, Immune surveillance and neoplasia. II. A two-stage mathematical model, Bull. Math. Biol., 39 (1977), 487-497.

[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), 293-299. doi: 10.1038/86297.

[29]

J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system, Immunol. Rev., 216 (2007), 213-231.

[30]

G. W. Swan, Immunological surveillance and neoplastic development, Rocky Mountain J. Math., 9 (1979), 143-148. doi: 10.1216/RMJ-1979-9-1-143.

[31]

R. Thomlinson, Measurement and management of carcinoma of the breast, Clin. Radiol., 33 (1982), 481-492. doi: 10.1016/S0009-9260(82)80153-0.

[32]

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

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), 25-58. doi: 10.1016/0025-5564(80)90120-0.

[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), 1312-1325. doi: 10.1093/annonc/mdg353.

[4]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[5]

C. DeLisi and A. Rescigno, Immune surveillance and neoplasia. I. A minimal mathematical model, Bull. Math. Biol., 39 (1977), 201-221.

[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), 7950-7958.

[7]

A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235. doi: 10.1016/j.physd.2005.06.032.

[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), 572-591. doi: 10.1016/j.mcm.2009.11.005.

[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), 2-32. doi: 10.1007/s11538-010-9526-3.

[10]

P. Fortin and M. C. Mackey, Periodic chronicmyelogenous leukemia: Spectral analysis of blood cell counts and etiological implications, Brit. J. Haematol., 104 (1999), 336-345. doi: 10.1046/j.1365-2141.1999.01168.x.

[11]

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

[12]

M. Golubitsky and W. F. Langford, Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41 (1981), 375-415. doi: 10.1016/0022-0396(81)90045-0.

[13]

E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differential systems, Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss., 94 (1942), 3-22.

[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), 61-89. doi: 10.1016/0022-247X(76)90278-X.

[15]

D. Kirschner and J. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127.

[16]

D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Engin., 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573.

[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), 295-321.

[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), North-Holland Biomedical Press, (1978), 333-344.

[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), 649-662. doi: 10.1016/j.mcm.2007.02.026.

[20]

A.-H. Lin, A model of tumor and lymphocyte interactions, Discrete. Contin. Dynam. Systems Ser. B, 4 (2004), 241-266. doi: 10.3934/dcdsb.2004.4.241.

[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), 151-168. doi: 10.3934/dcdsb.2009.12.151.

[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), 239-259. doi: 10.3934/mbe.2007.4.239.

[23]

Z. Lu and Y. Luo, Two limit cycles in three-dimensional Lotka-Volterra systems, Comput. Math. Appl., 44 (2002), 51-66. doi: 10.1016/S0898-1221(02)00129-3.

[24]

L. Perko, "Differential Equations and Dynamical Systems,'' 3rd edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 2001.

[25]

A. B. Poore, On the theory and application of the Hopf-Friedrichs bifurcation theory, Arch. Rational Mech. Anal., 60 (1975/76), 371-393.

[26]

A. Rescigno and C. DeLisi, Immune surveillance and neoplasia. II. A two-stage mathematical model, Bull. Math. Biol., 39 (1977), 487-497.

[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), 293-299. doi: 10.1038/86297.

[29]

J. Stark, C. Chan and A. J. T. George, Oscillations in the immune system, Immunol. Rev., 216 (2007), 213-231.

[30]

G. W. Swan, Immunological surveillance and neoplastic development, Rocky Mountain J. Math., 9 (1979), 143-148. doi: 10.1216/RMJ-1979-9-1-143.

[31]

R. Thomlinson, Measurement and management of carcinoma of the breast, Clin. Radiol., 33 (1982), 481-492. doi: 10.1016/S0009-9260(82)80153-0.

[32]

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

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