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DCDS-B

Investigating limit cycle oscillator with extended delay feedback is an efficient way to understand the dynamics of a global coupled
ensemble or a large
system with periodic oscillation. The stability and bifurcation of the arisen neutral
equation are obtained. Stability
switches and Hopf bifurcations appear when
delay passes through a sequence of critical values. Global continuation of Hopf
bifurcating periodic solutions and double--Hopf bifurcation are studied. With the help of
the unfolding system near double--Hopf bifurcation obtained
by using method of normal forms, quasiperiodic oscillations are found.
The number of the
coexisted periodic solutions is estimated. Finally, some numerical
simulations are carried out.

MBE

In this paper, we incorporate an extra logistic growth term for uninfected CD4$^+$ T-cells into an HIV-1 infection model with both
intracellular delay and immune response delay which was studied by Pawelek et al. in [26]. First, we proved that if the basic
reproduction number $R_0<1$, then the infection-free steady state is globally asymptotically stable. Second, when $R_0>1$, then the system is uniformly persistent, suggesting that the clearance or the uniform persistence of the
virus is completely determined by $R_0 $. Furthermore, given both the two delays are zero, then the infected steady
state is asymptotically stable when the intrinsic growth rate of the extra logistic term is sufficiently small.
When the two delays are not zero, we showed that both the immune response delay and the intracellular delay may destabilize the infected steady
state by leading to Hopf bifurcation and stable periodic oscillations, on which we analyzed the direction of the Hopf bifurcation
as well as the stability of the bifurcating periodic orbits by normal form and center manifold theory introduced by Hassard et al
[15]. Third, we engaged numerical simulations to explore the rich dynamics like chaotic oscillations, complicated bifurcation
diagram of viral load due to the logistic term of target cells and the two time delays.

keywords:
uniform persistence.
,
logistic growth
,
HIV-1 model
,
intracellular delay
,
Hopf bifurcation

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