Bifurcation Analysis in a Tumor-Immune System Interaction Model with Dendritic Cell Therapy and Immune Response Delay
Abstract.
In this paper, we study a tumor-immune system interaction model with dendritic cell therapy and immune response delay. First, it is shown that the ODE version of the model has a Bogdanov–Takens (BT) singularity or a weak focus with multiplicity at most 1 for different parameter values. As the parameters vary, the ODE model undergoes supercritical Hopf bifurcation and supercritical BT bifurcation. Our analysis indicates that there exists a threshold value of the activation rate of T cells, below which tumor immune escape occurs, above or at which T cells and tumor cells coexist in the form of a stable periodic oscillation or steady state. Second, we study how the immune response delay affects the dynamics of the model. Our results reveal that the delay can destabilize the stable positive equilibrium through Hopf bifurcation. Furthermore, the direction and stability of Hopf bifurcation are derived. When there is a cusp, we show that it is a BT singularity for any delay and the delay model also undergoes BT bifurcation. Finally, numerical simulations are presented to illustrate the theoretical results.
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Acknowledgments.
We thank the four anonymous reviewers for their helpful comments and suggestions which helped us to improve the manuscript significantly.
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Published In
SIAM Journal on Applied Mathematics
Pages: 1892 - 1914
DOI: 10.1137/22M1533979
ISSN (online): 1095-712X
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 10 November 2022
Accepted: 28 June 2023
Published online: 21 September 2023
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Funding Information
National Natural Science Foundation of China (NSF): 11871235, 12231008, 11901225
Funding: This research was partially supported by the National Natural Science Foundation of China grants 11871235, 12231008, and 11901225 and by the Japan Society for the Promotion of Science “Grand-in-Aid 20K03755”.
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