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Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

Ben Niu

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, P. R. China

E-mail Address: niu@hit.edu.cn

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Yuxiao Guo

http://orcid.org/0000-0001-9293-4238

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, P. R. China

E-mail Address: gyx@hit.edu.cn

Corresponding author.

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, and

Yanfei Du

Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, P. R. China

E-mail Address: duyanfei@sust.edu.cn

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https://doi.org/10.1142/S0218127418501365Cited by:10 (Source: Crossref)

Abstract

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay $τ$ of immune system where the immune cell has a probability $p$ in killing tumor cells. We find increasing time delay $τ$ destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability $p$ on Hopf bifurcation values is also discussed.

Keywords:

References

Adam, J. A. [1996] “ Effects of vascularization on lymphocyte/tumor cell dynamics: Qualitative features,” Math. Comput. Model. 23, 1–10. Crossref, Web of Science, Google Scholar
Adam, J. & Bellomo, N. [2012] A Survey of Models for Tumor-Immune System Dynamics (Springer Science & Business Media, USA). Google Scholar
Bi, P. & Ruan, S. [2013] “ Bifurcations in delay differential equations and applications to tumor and immune system interaction models,” SIAM J. Appl. Dyn. Syst. 12, 1847–1888. Crossref, Web of Science, Google Scholar
Galach, M. [2017] “ Dynamics of the tumor-immune system competition — The effect of time delay,” Int. J. Appl. Math. Comput. Sci. 13, 395–406. Google Scholar
Hale, J. & Verduyn Lunel, S. [1993] Introduction to Functional Differential Equations (Springer, NY). Crossref, Google Scholar
Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. [1981] Theory and Applications of Hopf Bifurcation (Cambridge University Press). Google Scholar
Karaoğlu, E., Yılmaz, E. & Merdan, H. [2016] “ Hopf bifurcation analysis of coupled two-neuron system with discrete and distributed delays,” Nonlin. Dyn. 85, 1039–1051. Crossref, Web of Science, Google Scholar
Kayan, S. & Merdan, H. [2017] “ An algorithm for Hopf bifurcation analysis of a delayed reaction–diffusion model,” Nonlin. Dyn. 89, 345–366. Crossref, Web of Science, Google Scholar
Kayan, S., Merdan, H., Yafia, R. et al. [2017] “ Bifurcation analysis of a modified tumor-immune system interaction model involving time delay,” Math. Model. Nat. Phenom. 12, 120–145. Crossref, Web of Science, Google Scholar
Kirschner, D. & Panetta, J. C. [1998] “ Modeling immunotherapy of the tumor-immune interaction,” J. Math. Biol. 37, 235–252. Crossref, Web of Science, Google Scholar
Ku-Carrillo, R. A., Delgadillo, S. E. & Chen-Charpentier, B. M. [2016] “ A mathematical model for the effect of obesity on cancer growth and on the immune system response,” Appl. Math. Model. 40, 4908–4920. Crossref, Web of Science, Google Scholar
Kuznetsov, V. A., Makalkin, I. A., Taylor, M. A. et al. [1994] “ Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,” Bull. Math. Biol. 56, 295–321. Crossref, Web of Science, Google Scholar
Kuznetsov, V. A. & Knott, G. D. [2001] “ Modeling tumor regrowth and immunotherapy,” Math. Comput. Model. 33, 1275–1287. Crossref, Web of Science, Google Scholar
López, Á. G., Seoane, J. M. & Sanjuán, M. A. F. [2017a] “ Bifurcation analysis and nonlinear decay of a tumor in the presence of an immune response,” Int. J. Bifurcation and Chaos 27, 1750223-1–7. Link, Web of Science, Google Scholar
López, Á. G., Seoane, J. M. & Sanjuán, M. A. F. [2017b] “ Dynamics of the cell-mediated immune response to tumour growth,” Phil. Trans. Roy. Soc. A 375, 20160291. Crossref, Web of Science, Google Scholar
Pardoll, D. [2003] “ Does the immune system see tumors as foreign or self?” Ann. Rev. Immunol. 21, 807–839. Crossref, Web of Science, Google Scholar
Protter, M. H. & Weinberger, H. F. [1984] Maximum Principles in Differential Equations (Springer). Crossref, Google Scholar
Rihan, F. A., Hashish, A., Al-Maskari, F. et al. [2016] “ Dynamics of tumor-immune system with fractional-order,” J. Tumor Res. 2, 109. Crossref, Google Scholar
Ruan, S. & Wei, J. [2003] “ On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dyn. Contin. Discr. Impuls. Syst. Ser. A 10, 863–874. Web of Science, Google Scholar
Schreiber, R. D., Old, L. J. & Smyth, M. J. [2011] “ Cancer immunoediting: Integrating immunity’s roles in cancer suppression and promotion,” Science 331, 1565–1570. Crossref, Web of Science, Google Scholar
Song, Y., Peng, Y. & Zou, X. [2014] “ Persistence, stability and Hopf bifurcation in a diffusive ratio-dependent predator–prey model with delay,” Int. J. Bifurcation and Chaos 24, 1450093-1–18. Link, Web of Science, Google Scholar
Vesely, M. D., Kershaw, M. H., Schreiber, R. D. et al. [2011] “ Natural innate and adaptive immunity to cancer,” Ann. Rev. Immunol. 29, 235–271. Crossref, Web of Science, Google Scholar
Wiggins, S. [1980] Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, NY). Google Scholar
Wu, J. [1996] Theory and Applications of Partial Functional Differential Equations (Springer Science & Business Media, USA). Crossref, Google Scholar
Yu, M., Dong, Y. & Takeuchi, Y. [2017] “ Dual role of delay effects in a tumour-immune system,” J. Biol. Dyn. 11, 334–347. Crossref, Web of Science, Google Scholar