# American Institute of Mathematical Sciences

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

Received  February 2020 Revised  August 2020 Published  August 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (8) : 4523-4547. doi: 10.3934/dcdsb.2020301
##### 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.

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.
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
 [1] Giulio Caravagna, Alex Graudenzi, Alberto d’Onofrio. Distributed delays in a hybrid model of tumor-Immune system interplay. Mathematical Biosciences & Engineering, 2013, 10 (1) : 37-57. doi: 10.3934/mbe.2013.10.37 [2] Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282 [3] Sophia R-J Jang, Hsiu-Chuan Wei. On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3261-3295. doi: 10.3934/dcdsb.2021184 [4] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5227-5249. doi: 10.3934/dcdsb.2020341 [5] Jianquan Li, Xiangxiang Ma, Yuming Chen, Dian Zhang. Complex dynamic behaviors of a tumor-immune system with two delays in tumor actions. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022033 [6] Martina Conte, Maria Groppi, Giampiero Spiga. Qualitative analysis of kinetic-based models for tumor-immune system interaction. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2393-2414. doi: 10.3934/dcdsb.2018060 [7] Min Yu, Gang Huang, Yueping Dong, Yasuhiro Takeuchi. Complicated dynamics of tumor-immune system interaction model with distributed time delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2391-2406. doi: 10.3934/dcdsb.2020015 [8] Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787 [9] Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157 [10] Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971 [11] J.C. Arciero, T.L. Jackson, D.E. Kirschner. A mathematical model of tumor-immune evasion and siRNA treatment. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 39-58. doi: 10.3934/dcdsb.2004.4.39 [12] Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925 [13] Lifeng Han, Changhan He, Yang Kuang. Dynamics of a model of tumor-immune interaction with time delay and noise. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2347-2363. doi: 10.3934/dcdss.2020140 [14] Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 [15] Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Mathematical Biosciences & Engineering, 2012, 9 (2) : 347-368. doi: 10.3934/mbe.2012.9.347 [16] Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151 [17] Denise E. Kirschner, Alexei Tsygvintsev. On the global dynamics of a model for tumor immunotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 573-583. doi: 10.3934/mbe.2009.6.573 [18] Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi. Mathematical modeling on helper T cells in a tumor immune system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 55-72. doi: 10.3934/dcdsb.2014.19.55 [19] K. Renee Fister, Jennifer Hughes Donnelly. Immunotherapy: An Optimal Control Theory Approach. Mathematical Biosciences & Engineering, 2005, 2 (3) : 499-510. doi: 10.3934/mbe.2005.2.499 [20] Daniel Vasiliu, Jianjun Paul Tian. Periodic solutions of a model for tumor virotherapy. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1587-1597. doi: 10.3934/dcdss.2011.4.1587

2020 Impact Factor: 1.327