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Use of quasinormal form to examine stability of tumorfree equilibrium in a mathematical model of bcg treatment of bladder cancer
1.  Ariel University Centerof of Samaria, Mathematics Department, Ariel, Israel, Israel 
References:
[1] 
A. Algaba, E. Friere and E. Gamero, Characterizing and computing normal forms using Lie transforms: A survey,, Dyn. Continuous, 8 (2001), 449. 
[2] 
V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,", Springer Verlag, (1982). 
[3] 
J. Archuleta, P. Mullens and T. P. Primm, The relationship of temperature to desiccation and starvation tolerance of the Mycobacterium avium complex,, Arch. Microbiol., 178 (2002), 311. doi: 10.1007/s002030020455x. 
[4] 
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics,, in, (1993). 
[5] 
R. F. M. Bevers, K. H. Kurth and D. J. H. Schamhart, Role of urothelial cells in BCG immunotherapy for superficial bladder cancer,, Brit. J. Cancer, 91 (2004), 607. 
[6] 
Y. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,", Lecture Notes in Mathematics, 702 (1979). 
[7] 
G. D. Birkhoff, "Dynamical Systems,", New York, (1927). 
[8] 
A. Bohle and S. Brandau, Immune mechanisms in bacillus CalmetteGuerin immunotherapy for superficial bladder cancer,, J. Urol., 170 (2003). doi: 10.1097/01.ju.0000073852.24341.4a. 
[9] 
A. D. Bruno, "Local Methods in Nonlinear Diff. Equations,", Springer Verlag, (1989). 
[10] 
S. BunimovichMendrazitsky, E. Shochat and L. Stone, Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer,, Bull. Math. Biol., 69 (2007), 1847. doi: 10.1007/s115380079195z. 
[11] 
S. BunimovichMendrazitsky, H. M. Byrne and L. Stone, Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer,, Bull. Math. Biol., 70 (2008), 2055. doi: 10.1007/s115380089344z. 
[12] 
C. W. Cheng, M. T. Ng, S. Y. Chan and W. H. Sun, Low dose BCG as adjuvant therapy for superficial bladder cancer and literature review,, Anz Journal of Surgery, 74 (2004), 569. doi: 10.1111/j.14452197.2004.02941.x. 
[13] 
S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory,", New York: Springer, (1982). 
[14] 
S. N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcations of Planar Vector Fields,", Cambridge University Press, (1994). doi: 10.1017/CBO9780511665639. 
[15] 
Y. M. Goltser, On the strong stability of resonance systems with parametrical perturbations,, Applied Mathematics and Mechanics (PMM), 41 (1977), 251. 
[16] 
Y. M. Goltser, Bifurcation and stability of neutral systems in the neighborhood of third order resonance,, Applied Mathematics and Mechanics (PMM), 43 (1979), 429. 
[17] 
Y. M. Goltser, On the extent of proximity of neutral systems to internal resonance,, Applied Mathematics and Mechanics (PMM), 50 (1986), 945. 
[18] 
Y. M. Goltser, Some bifurcation problems of stability,, Nonlinear Analysis, 30 (1997), 1461. doi: 10.1016/S0362546X(97)000448. 
[19] 
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, (1983). 
[20] 
G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications,", World Scientific, (1992). 
[21] 
A. Jemal, T. Murray, E. Ward, A. Samuels, R. C. Tiwari, A. Ghafoor, E. J. Feuer and M. J. Thun, Cancer Statistics,, CA Cancer. J. Clin., 55 (2005), 10. doi: 10.3322/canjclin.55.1.10. 
[22] 
F. A. Kelley, The stable, center stable, center, center unstable, and unstable manifolds,, J. Diff. Eqns, 3 (1967), 546. doi: 10.1016/00220396(67)900162. 
[23] 
V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumours: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol., 56 (1994), 295. 
[24] 
A. M. Liapunov, "Probléme géné ral de la stabilité du Mouvement,", Princeton University Press, (1947). 
[25] 
I. G. Malkin, "Theory of Stability of Motion,", Translated by Atomic Energy Commission, (1952), 92. 
[26] 
J. J Patard, F. Saint, F. Velotti, C. C. Abbou and D. K. Chopin, Immune response following intravesical bacillus CalmetteGuerin instillations in superficial bladder cancer: A review,, Urol. Res., 26 (1998), 155. doi: 10.1007/s002400050039. 
[27] 
V. A Pliss, The reduction principle in the theory of the stability motion,, Izv.Akad. Nauk SSSR, 28 (1964), 1297. 
[28] 
H. Poincaré, Oeuvres,, Paris, (1928). 
[29] 
E. Shochat, D. Hart and Z. Agur, Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols,, Math. Models&Methods in Applied Sciences, 9 (1999), 599. doi: 10.1142/S0218202599000312. 
[30] 
K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord, Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion,, J. Neurol. Sci., 216 (2003), 1. doi: 10.1016/j.jns.2003.06.001. 
[31] 
J. Wigginton and D. Kirschner, A model to predict cellmediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis,, J. Immunol., 166 (2001), 1951. 
[32] 
Q. S. Zhang, A. Y. T. Leung and J. E. Cooper, Computation of normal forms for higher dimensional semisimple systems,, Dynamics of Continuous, 8 (2001), 559. 
show all references
References:
[1] 
A. Algaba, E. Friere and E. Gamero, Characterizing and computing normal forms using Lie transforms: A survey,, Dyn. Continuous, 8 (2001), 449. 
[2] 
V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations,", Springer Verlag, (1982). 
[3] 
J. Archuleta, P. Mullens and T. P. Primm, The relationship of temperature to desiccation and starvation tolerance of the Mycobacterium avium complex,, Arch. Microbiol., 178 (2002), 311. doi: 10.1007/s002030020455x. 
[4] 
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics,, in, (1993). 
[5] 
R. F. M. Bevers, K. H. Kurth and D. J. H. Schamhart, Role of urothelial cells in BCG immunotherapy for superficial bladder cancer,, Brit. J. Cancer, 91 (2004), 607. 
[6] 
Y. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,", Lecture Notes in Mathematics, 702 (1979). 
[7] 
G. D. Birkhoff, "Dynamical Systems,", New York, (1927). 
[8] 
A. Bohle and S. Brandau, Immune mechanisms in bacillus CalmetteGuerin immunotherapy for superficial bladder cancer,, J. Urol., 170 (2003). doi: 10.1097/01.ju.0000073852.24341.4a. 
[9] 
A. D. Bruno, "Local Methods in Nonlinear Diff. Equations,", Springer Verlag, (1989). 
[10] 
S. BunimovichMendrazitsky, E. Shochat and L. Stone, Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer,, Bull. Math. Biol., 69 (2007), 1847. doi: 10.1007/s115380079195z. 
[11] 
S. BunimovichMendrazitsky, H. M. Byrne and L. Stone, Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer,, Bull. Math. Biol., 70 (2008), 2055. doi: 10.1007/s115380089344z. 
[12] 
C. W. Cheng, M. T. Ng, S. Y. Chan and W. H. Sun, Low dose BCG as adjuvant therapy for superficial bladder cancer and literature review,, Anz Journal of Surgery, 74 (2004), 569. doi: 10.1111/j.14452197.2004.02941.x. 
[13] 
S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory,", New York: Springer, (1982). 
[14] 
S. N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcations of Planar Vector Fields,", Cambridge University Press, (1994). doi: 10.1017/CBO9780511665639. 
[15] 
Y. M. Goltser, On the strong stability of resonance systems with parametrical perturbations,, Applied Mathematics and Mechanics (PMM), 41 (1977), 251. 
[16] 
Y. M. Goltser, Bifurcation and stability of neutral systems in the neighborhood of third order resonance,, Applied Mathematics and Mechanics (PMM), 43 (1979), 429. 
[17] 
Y. M. Goltser, On the extent of proximity of neutral systems to internal resonance,, Applied Mathematics and Mechanics (PMM), 50 (1986), 945. 
[18] 
Y. M. Goltser, Some bifurcation problems of stability,, Nonlinear Analysis, 30 (1997), 1461. doi: 10.1016/S0362546X(97)000448. 
[19] 
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Applied Mathematical Sciences, (1983). 
[20] 
G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications,", World Scientific, (1992). 
[21] 
A. Jemal, T. Murray, E. Ward, A. Samuels, R. C. Tiwari, A. Ghafoor, E. J. Feuer and M. J. Thun, Cancer Statistics,, CA Cancer. J. Clin., 55 (2005), 10. doi: 10.3322/canjclin.55.1.10. 
[22] 
F. A. Kelley, The stable, center stable, center, center unstable, and unstable manifolds,, J. Diff. Eqns, 3 (1967), 546. doi: 10.1016/00220396(67)900162. 
[23] 
V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumours: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol., 56 (1994), 295. 
[24] 
A. M. Liapunov, "Probléme géné ral de la stabilité du Mouvement,", Princeton University Press, (1947). 
[25] 
I. G. Malkin, "Theory of Stability of Motion,", Translated by Atomic Energy Commission, (1952), 92. 
[26] 
J. J Patard, F. Saint, F. Velotti, C. C. Abbou and D. K. Chopin, Immune response following intravesical bacillus CalmetteGuerin instillations in superficial bladder cancer: A review,, Urol. Res., 26 (1998), 155. doi: 10.1007/s002400050039. 
[27] 
V. A Pliss, The reduction principle in the theory of the stability motion,, Izv.Akad. Nauk SSSR, 28 (1964), 1297. 
[28] 
H. Poincaré, Oeuvres,, Paris, (1928). 
[29] 
E. Shochat, D. Hart and Z. Agur, Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols,, Math. Models&Methods in Applied Sciences, 9 (1999), 599. doi: 10.1142/S0218202599000312. 
[30] 
K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord, Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion,, J. Neurol. Sci., 216 (2003), 1. doi: 10.1016/j.jns.2003.06.001. 
[31] 
J. Wigginton and D. Kirschner, A model to predict cellmediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis,, J. Immunol., 166 (2001), 1951. 
[32] 
Q. S. Zhang, A. Y. T. Leung and J. E. Cooper, Computation of normal forms for higher dimensional semisimple systems,, Dynamics of Continuous, 8 (2001), 559. 
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