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

Qingfeng Tang; Guohong Zhang; Qingfeng Tang; Guohong Zhang

doi:10.3934/mbe.2021101

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 3: 1941-1965. doi: 10.3934/mbe.2021101

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

Qingfeng Tang ,
Guohong Zhang ^,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received: 21 December 2020 Accepted: 01 February 2021 Published: 24 February 2021

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.
- bistable phenomenon,
- tumor relapse,
- intrinsic recruitment delay,
- chemotherapy,
- Hopf bifurcation,
- numerical simulations
Citation: Qingfeng Tang, Guohong Zhang. Stability and Hopf bifurcations in a competitive tumour-immune system with intrinsic recruitment delay and chemotherapy[J]. Mathematical Biosciences and Engineering, 2021, 18(3): 1941-1965. doi: 10.3934/mbe.2021101

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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.

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