Existence of periodic solution for a tumor growth model with vaccine interaction

Canan Çelik, Nigar Özarslan Tuncer


This paper is devoted to the study of existence of positive periodic solutions of a tumor-immune competition model with vaccine interaction. By using the continuation theorem of coincidence degree theory developed by Gains and Mahwin, we establish the sufficient conditions for the existence of periodic solutions.


Coincidence degree theory; tumor-immune system; periodic solution

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J.A. Adam and N. Bellomo, Survey of Models for Tumor-Immune System Dynamics, Birkhäuser, Boston, (1997).

J.C. Arciero, T.L. Jackson and D.E. Kirschner, Mathematical model of tumorimmune evasion and siRNA treatment, Discrete Contin. Dyn. Syst. Ser. B 4 (2004), 39–58.

N. Bellomo and I. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comp. Model 32 (2000), no. 3–4, 413–452.

R.E. Gaines and J.L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematic, vol. 568, Springer, 1977.

M. Galach, Dynamics of the tumor-immune system competition – the effect of time delay, Int. J. Appl. Math. Comput. Sci. 13, (2003), 395–406.

D. Kirschner and J.C. Panetta, Modeling immunotherapy of the tumor – immune interaction, J. Math. Biol. 37 (1998), 235—252

V. Kuznetsov, I. Makalkin, M. Taylor and A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol. 56 (1994), no. 2, 295–321.

V. Kuznetsov and G. Knott, Modeling tumor regrowth and immunotherapy, Math. Comp. Model 33 (2001), 1275–1287.

Y. Li and Y. Kuang, Periodic solutions in periodic state-dependent delay equations and population models, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1345–1353.

J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, 1979.

L.G. de Pillis, W. Gu and A. Radunskaya, Mixed immunotherapy and chemotherapy of tumors; modeling, applications and biological interpretations, J. Theor. Biol. 238 (2006), no. 4, 841–862.

L.G. de Pillis, A. Radunskaya and C.L. Wiseman, A validated mathematical model of cellmediated immune response to tumor growth, Cancer Res. 65 (2005), no. 17, 7950–7958.

S. Wilson and D. Levy, A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy, Bull. Math. Biol. 74 (2012), no. 7, 1485–1500.

M. Terabe, E. Ambrosino, S. Takaku, J. O’Konek, D. Venzon, S. Lonning, J.P. McPherson and J.A. Berzofsky, Synergistic enhancement of CD8+ T cell-mediated tumor vaccine efficacy by an anti-transforming growth factor-U3b2 monoclonal antibody, Clin. Cancer Res. 15 (2009), no. 21, 6560–6569.

R. Thomlinson, Measurement and management of carcinoma of the breast, Clinical Radiology 33 (1982), no. 5, 481–493.

S. Wilson and D. Levy, A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy, Bull. Math. Bio. 74 (2012), 1485–1500.

R. Yafia, The effect of time delay and Hopf bifurcation in a tumor-immune system competition model with negative immune response, Appl. Math. (Warsaw) 36 (2009), no. 3, 349–364.

Z. Yang and J. Cao, Existence of periodic solutions in neutral state-dependent delays equations and models, J. Comput. Appl. Math. 174 (2005), 179–199.


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