Stability switch of tumor-immune system driven by delayed response in cell-to-cell contact

Considering the delayed response in cell-to-cell contact is essential for accurately predicting the progression of cancer. In this paper, the stability switch phenomenon of a tumor-immune interaction system induced by the delayed response is presented. Firstly, the qualitative properties of the system including non-negativity and boundedness of the solutions are discussed. Subsequently, a complete discriminant system for polynomials method is employed to scrutinize the local stability switch conditions, which circumvents the complicated calculation procedures and enhances the analytical accuracy for high-dimensional nonlinear time-delay systems. The delayed response in cell-to-cell contact is observed to exert a suppressive influence on system stability. Specifically, the system with stable initial state will switch to instability when the delayed responding time of cell-to-cell contact exceeds a critical threshold, indicating a weakened resistance of the immune system against tumor cells. While the system with unstable initial state will maintain and even exhibit chaotic behavior, implying uncontrolled tumor proliferation. Finally, the major findings related to cancer therapy are summarized in the conclusion. These results help to understand the complex mechanism of the tumor-immune system and provide ideas for the clinical treatment of tumor.