# American Institute of Mathematical Sciences

## 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

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.

Citation: Gladis Torres-Espino, Claudio Vidal. Periodic solutions of a tumor-immune system interaction under a periodic immunotherapy. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020301
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##### 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
Intersection between the graph of the function $g(X)$ and the line $l_2:\, Y = \frac{KA}{BS}X-\frac{\overline{\beta}}{S}$.
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 $\frac{FA}{BS}-\frac{\overline{\beta}}{S}<0$
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 $K-D>0$
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 $\frac{FA}{BS}-\frac{\overline{\beta}}{S}<0$.
Intersection between the graph of the function $g(X)$ and the line $l_2:\, Y = (K-D)\frac{A}{BS}X+\frac{FA}{BS}-\frac{\overline{\beta}}{S}$.
Intersection between the graph of the function $g(X)$ and the line $l_2:\, Y = -\frac{DA}{BS}X+\frac{FA}{BS}-\frac{\overline{\beta}}{S}$
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 $y_0 = 100000+10^{-40}$
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 $y_0 = 80000+10^{-40}$
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 $y_0 = 50000+10^{-40}$
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 $y_0 = 25000+10^{-40}$
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 $y_0 = 25000+10^{-40}$
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 $y_0 = 25000+10^{-40}$
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 $y_0 = 25000+10^{-40}$
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
 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
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
 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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