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Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy
Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, Chile |
In this paper, we consider a mathematical model of a tumor-immune system interaction when a periodic immunotherapy treatment is applied. We give sufficient conditions, using averaging theory, for the existence and stability of periodic solutions in such system as a function of the six parameters associated to this problem. Finally, we provide examples where our results are applied.
References:
[1] |
P. Amster, L. Berezansky and L. Idels,
Periodic solutions of angiogenesis models with time lags, Nonlinear Analysis: Real World Applications, 13 (2012), 299-311.
doi: 10.1016/j.nonrwa.2011.07.035. |
[2] |
A. d'Onofrio,
A general framework for modeling tumor-inmune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena, 208 (2005), 220-235.
doi: 10.1016/j.physd.2005.06.032. |
[3] |
A. d'Onofrio,
Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637.
doi: 10.1016/j.mcm.2007.02.032. |
[4] |
D. I. Gabrilovich,
Combination of chemotherapy and immunotherapy for cancer: A paradigm revisited, Lancet Oncology, 8 (2007), 2-3.
doi: 10.1016/S1470-2045(06)70985-8. |
[5] |
V. A. Kuznetsov, I. A. Makalkin, M. Taylor and A. Perelson,
Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
doi: 10.1007/BF02460644. |
[6] |
Z. Liu and C. Yang,
A mathematical model of cancer treatment by radiotherapy, Comput. Math. Meth. Med., 124 (2014), 1-12.
doi: 10.1155/2014/172923. |
[7] |
O. Sotolongo-Costa, L. Morales-Molina, D. Rodríguez-Pérez, J. C. Antonraz and M. Chacón-Reyes,
Behaviour of tumors under nonstationary therapy, Physica D: Nonlinear Phenomena, 178 (2003), 242-253.
doi: 10.1016/S0167-2789(03)00005-8. |
[8] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2$^{nd}$ edition, Universitext, Springer-Verlag, Berlin Heidelberg, 1996.
doi: 10.1007/978-3-642-61453-8. |
show all references
References:
[1] |
P. Amster, L. Berezansky and L. Idels,
Periodic solutions of angiogenesis models with time lags, Nonlinear Analysis: Real World Applications, 13 (2012), 299-311.
doi: 10.1016/j.nonrwa.2011.07.035. |
[2] |
A. d'Onofrio,
A general framework for modeling tumor-inmune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D: Nonlinear Phenomena, 208 (2005), 220-235.
doi: 10.1016/j.physd.2005.06.032. |
[3] |
A. d'Onofrio,
Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614-637.
doi: 10.1016/j.mcm.2007.02.032. |
[4] |
D. I. Gabrilovich,
Combination of chemotherapy and immunotherapy for cancer: A paradigm revisited, Lancet Oncology, 8 (2007), 2-3.
doi: 10.1016/S1470-2045(06)70985-8. |
[5] |
V. A. Kuznetsov, I. A. Makalkin, M. Taylor and A. Perelson,
Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
doi: 10.1007/BF02460644. |
[6] |
Z. Liu and C. Yang,
A mathematical model of cancer treatment by radiotherapy, Comput. Math. Meth. Med., 124 (2014), 1-12.
doi: 10.1155/2014/172923. |
[7] |
O. Sotolongo-Costa, L. Morales-Molina, D. Rodríguez-Pérez, J. C. Antonraz and M. Chacón-Reyes,
Behaviour of tumors under nonstationary therapy, Physica D: Nonlinear Phenomena, 178 (2003), 242-253.
doi: 10.1016/S0167-2789(03)00005-8. |
[8] |
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2$^{nd}$ edition, Universitext, Springer-Verlag, Berlin Heidelberg, 1996.
doi: 10.1007/978-3-642-61453-8. |













Functions | Biological meaning |
Growth rate of the tumor | |
Functional response | |
External inflow of effector cells | |
Tumor-stimulated proliferation rate of effector cells | |
Tumor-induced loss of effector cells | |
Influx of effector cells | |
Immunotherapy |
Functions | Biological meaning |
Growth rate of the tumor | |
Functional response | |
External inflow of effector cells | |
Tumor-stimulated proliferation rate of effector cells | |
Tumor-induced loss of effector cells | |
Influx of effector cells | |
Immunotherapy |
Parameter | Biological meaning |
Intrinsic growth rate of the tumor | |
Death malignant cells rate due to interaction with lymphocyte cells | |
Increased lymphocyte rate due to interaction with malignant cells | |
Death rate of the lymphocytes | |
Immunosuppression coefficient | |
Influx external of effector cells | |
Immunotherapy dosage frequency |
Parameter | Biological meaning |
Intrinsic growth rate of the tumor | |
Death malignant cells rate due to interaction with lymphocyte cells | |
Increased lymphocyte rate due to interaction with malignant cells | |
Death rate of the lymphocytes | |
Immunosuppression coefficient | |
Influx external of effector cells | |
Immunotherapy dosage frequency |
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