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Bautin bifurcation in a minimal model of immunoediting
1. | Departamento de Matemáticas, UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México City, C.P. 09340, México |
2. | Posgrado en Ciencias Naturales e Ingeniería, UAM-Cuajimalpa, Av. Vasco de Quiroga 4871, Col. Santa Fé Cuajimalpa, México City, 05348, México |
3. | Universidad Autónoma de la Ciudad de México, Plantel San Lorenzo Tezonco, Calle Prolongación San Isidro 151, Col. San Lorenzo Tezonco, México City, 09790, México |
One of the simplest model of immune surveillance and neoplasia was proposed by Delisi and Resigno [
References:
[1] |
J. A. Adam,
Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features, Math. Comput. Modelling, 23 (1996), 1-10.
|
[2] |
B. Al-Hdaibat, W. Govaerts, Y. A. Kuznetsov and H. G. E. Meijer,
Initialization of homoclinic solutions near Bogdanov-Takens points: Lindstedt-Poincaré compared with regular perturbation method, SIAM J. Applied Dynamical Systems, 15 (2016), 952-980.
doi: 10.1137/15M1017491. |
[3] |
G. I. Bell,
Predator-prey simulating an immune response, Mathematical Biosciences, 16 (1973), 291-314.
|
[4] |
W.-J. Beyn,
Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point, IMA Journal of Numerical Analysis, 14 (1994), 381-410.
doi: 10.1093/imanum/14.3.381. |
[5] |
A. R. Champneys and Y. A. Kuznetzov,
Numerical detection and continuation of codimension-two homoclinic bifurcation, International Journal of Bifurcation and Chaos, 4 (1994), 785-822.
doi: 10.1142/S0218127494000587. |
[6] |
C. DeLisi and A. Rescigno,
Immune surveillance and neoplasia-I. A minimal mathematical model, Bull. Math. Bio., 39 (1977), 201-221.
doi: 10.1007/bf02462859. |
[7] |
G. P. Dunn, L. J. Old and R. D. Schreiber,
The three ES of Cancer Immunoediting, Annual Review of Immunology, 22 (2004), 329-360.
|
[8] |
R. Kim, M. Emi and K. Tanabe,
Cancer immunoediting from immune surveillance to immune escape, Immunology, 21 (2007), 11-14.
|
[9] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. |
[10] |
D. Liu, S. G. Ruan and D. M. Zhu,
Bifurcation analysis in models of tumor and immune system interactions, Discrete and Continuous Dynamical Systems series B, 12 (2009), 151-168.
doi: 10.3934/dcdsb.2009.12.151. |
show all references
References:
[1] |
J. A. Adam,
Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features, Math. Comput. Modelling, 23 (1996), 1-10.
|
[2] |
B. Al-Hdaibat, W. Govaerts, Y. A. Kuznetsov and H. G. E. Meijer,
Initialization of homoclinic solutions near Bogdanov-Takens points: Lindstedt-Poincaré compared with regular perturbation method, SIAM J. Applied Dynamical Systems, 15 (2016), 952-980.
doi: 10.1137/15M1017491. |
[3] |
G. I. Bell,
Predator-prey simulating an immune response, Mathematical Biosciences, 16 (1973), 291-314.
|
[4] |
W.-J. Beyn,
Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point, IMA Journal of Numerical Analysis, 14 (1994), 381-410.
doi: 10.1093/imanum/14.3.381. |
[5] |
A. R. Champneys and Y. A. Kuznetzov,
Numerical detection and continuation of codimension-two homoclinic bifurcation, International Journal of Bifurcation and Chaos, 4 (1994), 785-822.
doi: 10.1142/S0218127494000587. |
[6] |
C. DeLisi and A. Rescigno,
Immune surveillance and neoplasia-I. A minimal mathematical model, Bull. Math. Bio., 39 (1977), 201-221.
doi: 10.1007/bf02462859. |
[7] |
G. P. Dunn, L. J. Old and R. D. Schreiber,
The three ES of Cancer Immunoediting, Annual Review of Immunology, 22 (2004), 329-360.
|
[8] |
R. Kim, M. Emi and K. Tanabe,
Cancer immunoediting from immune surveillance to immune escape, Immunology, 21 (2007), 11-14.
|
[9] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. |
[10] |
D. Liu, S. G. Ruan and D. M. Zhu,
Bifurcation analysis in models of tumor and immune system interactions, Discrete and Continuous Dynamical Systems series B, 12 (2009), 151-168.
doi: 10.3934/dcdsb.2009.12.151. |






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