Figure 1.
Intersection between the graph of the function $ g(X) $ and the line $ l_2:\, Y = \frac{KA}{BS}X-\frac{\overline{\beta}}{S} $ .
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August
2021, 26(8): 4523-4547.
doi: 10.3934/dcdsb.2020301
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.
Mathematics Subject Classification:
Primary: 34C25; Secondary: 34C29, 37N25.
Citation:
Gladis Torres-Espino, Claudio Vidal. Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy. Discrete and Continuous Dynamical Systems - B,
2021, 26
(8)
: 4523-4547.
doi: 10.3934/dcdsb.2020301
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References:
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
Figure 2.
Intersection between the graph of the function $ g(X) $ and the line $ l_2:\, Y = \frac{KA}{BS}X+\frac{FA}{BS}-\frac{\overline{\beta}}{S} $ , when $ 0<\frac{FA}{BS}-\frac{\overline{\beta}}{S}<1 $ and $ \frac{FA}{BS}-\frac{\overline{\beta}}{S}<0 $
Figure 3.
Intersection between the graph of the function $ g(X) $ and the line $ l_2:\, Y = (K-D)\frac{A}{BS}X-\frac{\overline{\beta}}{S} $ , when $ K-D>0 $
Figure 4.
Intersection between the graph of the function $ g(X) $ and the line $ l_2:\, Y = (K-D)\frac{A}{BS}X+\frac{FA}{BS}-\frac{\overline{\beta}}{S} $ , when $ 0<\frac{FA}{BS}-\frac{\overline{\beta}}{S}<1 $ and $ \frac{FA}{BS}-\frac{\overline{\beta}}{S}<0 $ .
Figure 5.
Intersection between the graph of the function $ g(X) $ and the line $ l_2:\, Y = (K-D)\frac{A}{BS}X+\frac{FA}{BS}-\frac{\overline{\beta}}{S} $ .
Figure 6.
Intersection between the graph of the function $ g(X) $ and the line $ l_2:\, Y = -\frac{DA}{BS}X+\frac{FA}{BS}-\frac{\overline{\beta}}{S} $
Figure 7.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.1, with initial conditions $ x_0 = 1/19+10^{-40} $ and $ y_0 = 100000+10^{-40} $
Figure 8.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.2, with initial conditions $ x_0 = 5/32(-3+\sqrt{73})+10^{-40} $ and $ y_0 = 80000+10^{-40} $
Figure 9.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.3, with initial conditions $ x_0 = 1/60 (-27 + \sqrt{1009})+10^{-40} $ and $ y_0 = 50000+10^{-40} $
Figure 10.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.4, with initial conditions $ x_0 = 1.0099381+10^{-40} $ and $ y_0 = 25000+10^{-40} $
Figure 11.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.5, with initial conditions $ x_0 = 1.15201+10^{-40} $ and $ y_0 = 25000+10^{-40} $
Figure 12.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.5, with initial conditions $ x_0 = 9.99873+10^{-40} $ and $ y_0 = 25000+10^{-40} $
Figure 13.
Malignant cells $ x(t) $ and Lymphocyte Cells $ y(t) $ for the periodic solution of Theorem 2.6, with initial conditions $ x_0 = 0.0123435+10^{-40} $ and $ y_0 = 25000+10^{-40} $
Table 1.
Definition of the parameters in model (2)
| 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 |
Table 2.
Definition of the parameters in model (3)
| 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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