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

• * Corresponding author: Gladis Torres-Espino
• 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: • • Figure 1.  Intersection between the graph of the function $g(X)$ and the line $l_2:\, Y = \frac{KA}{BS}X-\frac{\overline{\beta}}{S}$.

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 $\xi(x)$ 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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