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August  2017, 22(6): 2365-2387. doi: 10.3934/dcdsb.2017121

## Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays

 a. College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, China b. The Basic Science Department, Xinjiang Institute of Engineering, Urumqi, Xinjiang 830091, China

Received  April 2015 Revised  March 2017 Published  March 2017

Fund Project: This research was supported by the Natural Science Foundation of Xinjiang (Grant No. 2016D03022) and the Doctorial Subjects Foundation of the Ministry of Education of China (Grant No. 2013651110001), the National Natural Science Foundation of China (Grant No. 11661076)

In this paper, the dynamical behaviors of a viral infection model with cytotoxic T-lymphocyte (CTL) immune response, immune response delay and production delay are investigated. The threshold values for virus infection and immune response are established. By means of Lyapunov functionals methods and LaSalle's invariance principle, sufficient conditions for the global stability of the infection-free and CTL-absent equilibria are established. Global stability of the CTL-present infection equilibrium is also studied when there is no immune delay in the model. Furthermore, to deal with the local stability of the CTL-present infection equilibrium in a general case with two delays being positive, we extend an existing geometric method to treat the associated characteristic equation. When the two delays are positive, we show some conditions for Hopf bifurcation at the CTL-present infection equilibrium by using the immune delay as a bifurcation parameter. Numerical simulations are performed in order to illustrate the dynamical behaviors of the model.

Citation: Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
##### References:

show all references

##### References:
The time series of model (2) before Hopf bifurcation occurs for $\tau_2=0.0516$
The time series and the phase trajectories of model (2) when Hopf bifurcation occurs for $\tau_2=1.3215$
The time series of model (2) after Hopf bifurcation occurs for $\tau_2=4.3215$
The time series and the phase trajectories of model (2) when Hopf bifurcation occurs again for $\tau_2=6.8000$
List of parameters
 Parameter Interpretation Value Source $s$ production rate of uninfected cells 10 $\mu l^{-1}day^{-1}$ [22,26] $d$ death rate of uninfected cells 0.01 $day^{-1}$ [22,26] $a$ death rate of infected cells 0.5 $day^{-1}$ [17,26] $p$ CTL effectiveness 1 $\mu l day^{-1}$ [17,26] $\beta$ the infection rate 0.45 $\mu l day^{-1}$ [26] $\alpha$ Saturation coefficient 0.01 Assumed $k$ production rate of free virus 0.4 $cell^{-1}day^{-1}$ [17,26] $u$ clearance rate of free virus 3 $day^{-1}$ [27,17] $c$ proliferation rate of CTL response 0.1 $\mu l day^{-1}$ [27,17] $b$ death rate of CTL 0.15 $day^{-1}$ [27,17] $m$ death rate for infected cells during $[t-\tau_1, t]$ 0.01 Assumed
 Parameter Interpretation Value Source $s$ production rate of uninfected cells 10 $\mu l^{-1}day^{-1}$ [22,26] $d$ death rate of uninfected cells 0.01 $day^{-1}$ [22,26] $a$ death rate of infected cells 0.5 $day^{-1}$ [17,26] $p$ CTL effectiveness 1 $\mu l day^{-1}$ [17,26] $\beta$ the infection rate 0.45 $\mu l day^{-1}$ [26] $\alpha$ Saturation coefficient 0.01 Assumed $k$ production rate of free virus 0.4 $cell^{-1}day^{-1}$ [17,26] $u$ clearance rate of free virus 3 $day^{-1}$ [27,17] $c$ proliferation rate of CTL response 0.1 $\mu l day^{-1}$ [27,17] $b$ death rate of CTL 0.15 $day^{-1}$ [27,17] $m$ death rate for infected cells during $[t-\tau_1, t]$ 0.01 Assumed
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