Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays

Juan Wang; Chunyang Qin; Yuming Chen; Xia Wang; Juan Wang; Chunyang Qin; Yuming Chen; Xia Wang

doi:10.3934/mbe.2019130

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2587-2612. doi: 10.3934/mbe.2019130

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Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays

1.
School of Mathematics and Statistics, Xinyang Normal University, Xinyang, Henan, 464000, P. R. China
2.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada N2L 3C5
3.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453000, P. R. China

Received: 30 December 2018 Accepted: 12 March 2019 Published: 22 March 2019

In this paper, we investigate a delayed HIV-1 infection model with immune response. Though a logistic growth is incorporated in the growth of the target cells, our focus is on the effect of delays on the infection dynamics. We first study the existence of steady states, which depends on the basic reproduction number $R_0$. When $R_0\le 1$, there is only the infection-free steady state, which is globally asymptotically stable if $R_0 < 1$. When $R_0> 1$, besides the unstable infection-free steady state, there is a unique infected steady state. We then study the local stability of the infected steady state and local Hopf bifurcation at it. The theoretical analysis indicates that the dynamics scenario is complicated. For example, there can be three sequences of critical values, stability switches and double Hopf bifurcation can occur. Some of the theoretical results are also illustrated with numerical simulations.
- HIV-1 infection,
- Hopf bifurcation,
- time delay,
- CTL immune response,
- stability
Citation: Juan Wang, Chunyang Qin, Yuming Chen, Xia Wang. Hopf bifurcation in a CTL-inclusive HIV-1 infection model with two time delays[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2587-2612. doi: 10.3934/mbe.2019130

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Abstract

In this paper, we investigate a delayed HIV-1 infection model with immune response. Though a logistic growth is incorporated in the growth of the target cells, our focus is on the effect of delays on the infection dynamics. We first study the existence of steady states, which depends on the basic reproduction number $R_0$. When $R_0\le 1$, there is only the infection-free steady state, which is globally asymptotically stable if $R_0 < 1$. When $R_0> 1$, besides the unstable infection-free steady state, there is a unique infected steady state. We then study the local stability of the infected steady state and local Hopf bifurcation at it. The theoretical analysis indicates that the dynamics scenario is complicated. For example, there can be three sequences of critical values, stability switches and double Hopf bifurcation can occur. Some of the theoretical results are also illustrated with numerical simulations.

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