Global stability of an HIV infection model with saturated CTL immune response and intracellular delay

Jian Ren; Rui Xu; Liangchen Li; Jian Ren; Rui Xu; Liangchen Li

doi:10.3934/mbe.2021003

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 57-68. doi: 10.3934/mbe.2021003

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Global stability of an HIV infection model with saturated CTL immune response and intracellular delay

1.
Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, P. R. China
2.
Institute of Applied Mathematics, Army Engineering University, Shijiazhuang 050003, Hebei, P. R. China

Received: 13 September 2020 Accepted: 10 November 2020 Published: 19 November 2020

In this paper, we consider an HIV infection model with saturated infection rate, intracellular delay and saturated cytotoxic T lymphocyte (CTL) immune response. By calculation, we obtain immunity-inactivated reproduction number $\mathscr{R}_0$ and immunity-activated reproduction number $\mathscr{R}_1$. By analyzing the distribution of roots of the corresponding characteristic equations, we study the local stability of an infection-free equilibrium, an immunity-inactivated equilibrium and an immunity-activated equilibrium of the model. By constructing suitable Lyapunov functionals and using LaSalle's invariance principle, we show that if $\mathscr{R}_0 < 1$, the infection-free equilibrium is globally asymptotically stable; If $\mathscr{R}_1 < 1 < \mathscr{R}_0$, the immunity-inactivated equilibrium is globally asymptotically stable; If $\mathscr{R}_1>1$, the immunity-activated equilibrium is globally asymptotically stable. Sensitivity analyses are carried out to show the effects of parameters on the immunity-activated reproduction number $\mathscr{R}_{1}$ and the viral load.
- saturated infection rate,
- intracellular delay,
- saturated CTL immune response,
- global stability,
- Lyapunov functional
Citation: Jian Ren, Rui Xu, Liangchen Li. Global stability of an HIV infection model with saturated CTL immune response and intracellular delay[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 57-68. doi: 10.3934/mbe.2021003

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Abstract

In this paper, we consider an HIV infection model with saturated infection rate, intracellular delay and saturated cytotoxic T lymphocyte (CTL) immune response. By calculation, we obtain immunity-inactivated reproduction number $\mathscr{R}_0$ and immunity-activated reproduction number $\mathscr{R}_1$. By analyzing the distribution of roots of the corresponding characteristic equations, we study the local stability of an infection-free equilibrium, an immunity-inactivated equilibrium and an immunity-activated equilibrium of the model. By constructing suitable Lyapunov functionals and using LaSalle's invariance principle, we show that if $\mathscr{R}_0 < 1$, the infection-free equilibrium is globally asymptotically stable; If $\mathscr{R}_1 < 1 < \mathscr{R}_0$, the immunity-inactivated equilibrium is globally asymptotically stable; If $\mathscr{R}_1>1$, the immunity-activated equilibrium is globally asymptotically stable. Sensitivity analyses are carried out to show the effects of parameters on the immunity-activated reproduction number $\mathscr{R}_{1}$ and the viral load.

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