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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 |
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 (): 371.
|
[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 (): 371.
|
[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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