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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

Gladis Torres-Espino , and  Claudio Vidal

Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, Chile

* Corresponding author: Gladis Torres-Espino

Received  February 2020 Revised  August 2020 Published  October 2020

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.

Keywords: Tumor-immune system, immunotherapy, periodic solutions, averaging theory, stability.
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 & Continuous Dynamical Systems - B, 2021, 26 (8) : 4523-4547. doi: 10.3934/dcdsb.2020301
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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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 $
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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 $
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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 $.
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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} $.
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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} $
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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} $
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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} $
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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} $
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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} $
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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} $
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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} $
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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} $
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Table 1.  Definition of the parameters in model (2)
Functions Biological meaning
$ \xi(x) $ Growth rate of the tumor
$ \phi(x)y $ Functional response
$ g(x) $ External inflow of effector cells
$ \beta(x) $ Tumor-stimulated proliferation rate of effector cells
$ \mu(x) $ Tumor-induced loss of effector cells
$ \sigma g(x) $ Influx of effector cells
$ \theta(\omega t) $ Immunotherapy
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Functions Biological meaning
$ \xi(x) $ Growth rate of the tumor
$ \phi(x)y $ Functional response
$ g(x) $ External inflow of effector cells
$ \beta(x) $ Tumor-stimulated proliferation rate of effector cells
$ \mu(x) $ Tumor-induced loss of effector cells
$ \sigma g(x) $ Influx of effector cells
$ \theta(\omega t) $ Immunotherapy
Table 2.  Definition of the parameters in model (3)
Parameter Biological meaning
$ a $ Intrinsic growth rate of the tumor
$ b $ Death malignant cells rate due to interaction with lymphocyte cells
$ d $ Increased lymphocyte rate due to interaction with malignant cells
$ f $ Death rate of the lymphocytes
$ \kappa $ Immunosuppression coefficient
$ \sigma g(x) $ Influx external of effector cells
$ \omega $ Immunotherapy dosage frequency
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Parameter Biological meaning
$ a $ Intrinsic growth rate of the tumor
$ b $ Death malignant cells rate due to interaction with lymphocyte cells
$ d $ Increased lymphocyte rate due to interaction with malignant cells
$ f $ Death rate of the lymphocytes
$ \kappa $ Immunosuppression coefficient
$ \sigma g(x) $ Influx external of effector cells
$ \omega $ Immunotherapy dosage frequency
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