Stability and Hopf bifurcations in a competitive tumour-immune system with intrinsic recruitment delay and chemotherapy

Abstract

In this paper, a three-dimensional nonlinear delay differential system including Tumour cells, cytotoxic-T lymphocytes, T-helper cells is constructed to investigate the effects of intrinsic recruitment delay and chemotherapy. It is found that the introduction of chemotherapy and time delay can generate richer dynamics in tumor-immune system. In particular, there exists bistable phenomenon and the tumour cells would be cleared if the effect of chemotherapy on depletion of the tumour cells is strong enough or the side effect of chemotherapy on the hunting predator cells is under a threshold. It is also shown that a branch of stable periodic solutions bifurcates from the coexistence equilibrium when the intrinsic recruitment delay of tumor crosses the threshold which is new mechanism, which can help understand the short-term oscillations in tumour sizes as well as long-term tumour relapse. Numerical simulations are presented to illustrate that larger intrinsic recruitment delay of tumor leads to larger amplitude and longer period of the bifurcated periodic solution, which indicates that there exists longer relapse time and then contributes to the control of tumour growth.

Keywords

• bistable phenomenon
• tumor relapse
• intrinsic recruitment delay
• chemotherapy
• hopf bifurcation
• numerical simulations