A delayed HIV-1 infection model with CTL immune response is investigated. By using suitable Lyapunov functionals, it is proved that the infection-free equilibrium is globally asymptotically stable if the basic reproduction ratio for viral infection is less than or equal to unity; if the basic reproduction ratio for CTL immune response is less than or equal to unity and the basic reproduction ratio for viral infection is greater than unity, the CTL-inactivated infection equilibrium is globally asymptotically stable; if the basic reproduction ratio for CTL immune response is greater than unity, the CTL-activated infection equilibrium is globally asymptotically stable.

Recently, many mathematical models have been developed to describe the infection with HIV-1 (human immunodeficiency virus 1). By investigating these models, researchers have gained much important knowledge about the HIV-1 pathogenesis and have enhanced progress in the understanding of HIV-1 infection (see, e.g., [

Moreover, infection rate plays an important role in the modelling of epidemic dynamics. Holling type-II functional response seems more reasonable than the bilinear incidence rate (see, [

In [

Motivated by the works of Nowak and Bangham [

The initial conditions for system (

It is well known by the fundamental theory of functional differential equations [

This paper is organized as follows. In Section

In this section, we discuss the existence of three equilibria and prove that all the solutions are positive and bounded.

Clearly, system (

Denote

If

Supposing that

Let

In this section, we study the global stability of each equilibrium of system (

Define the following function:

If

Let

Noting that

If

Let

For clarity, we will calculate the derivatives of

Since

Noting that

If

Let

Next, we will calculate the derivatives of

Similar to (

Hence, from (

In the following, we give three examples to illustrate the main theoretical results above. All the parameters were obtained from [

In system (

The infection-free equilibrium

In system (

The CTL-inactivated infection equilibrium

In system (

The CTL-activated infection equilibrium

In this paper, we have studied the global dynamics of a delayed HIV-1 infection model with CTL immune response. By constructing suitable Lyapunov functionals, sufficient conditions have been derived for the global stability of three equilibria. It is easy to show that if the basic reproduction ratio for viral infection

From Theorems

This work was supported by the National Natural Science Foundation of China (no. 11071254).