Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy

Gladis Torres-Espino; Claudio Vidal

doi:10.3934/dcdsb.2020301

Article Contents

2021, Volume 26, Issue 8: 4523-4547. Doi: 10.3934/dcdsb.2020301

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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
^* Corresponding author: Gladis Torres-Espino

Received: January 31, 2020

Revised: July 31, 2020

Early access: October 2020

Published: August 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34C29, 37N25.

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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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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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