^{1}

^{1,2}

^{1}

^{2}

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive number

The human immunodeficiency virus (HIV) mainly targets a host’s

In recent years, there is much work done on HIV infection from different points of view, such as pathology [

Equation (

There is a discussion about the process of the HIV RNA transcribing into DNA: when an HIV enters a resting

Recently, some mathematical models of HIV infection have been proposed based on the assumption that a fraction of infected

In (

On biological grounds, during primary HIV infection, the rate of virus infection should be approximately proportionate to the virus load

Based on the argument above, this paper describes an amended model. In this model, we use a saturated infection rate

The rest of this paper is organized as follows. Section

Based on (

The infection-free steady state

represents the virus infection free.

The endemic infected steady state

represents persistent virus infection.

Here,

According to (

It is easy to show that the solutions of (

According to the first two equations of (

According to (

In this section, we discuss locally asymptotical stability and globally asymptotical stability of the infection-free equilibrium point

If

The Jacobi matrix of (

Substituting the equilibrium point

Solving equation (

If

If

Define a global Lyapunov function by

If

Define a global Lyapunov function by

Since the arithmetic mean is greater than or equal to the geometric mean, we obtain

Therefore,

Hence if

In this section, we analyze local asymptotical stability and global asymptotical stability of the endemic infection equilibrium point

If

Put the equilibrium point

Solving the eigenequation of the matrix above, here is

Hence all inequalities of the Routh-Hurwitz criterion are satisfied. Therefore, the endemic infection equilibrium point

In this subsection, we firstly introduce a lemma outlined by Li and Wang [

The lemma is briefly summarized as follows.

Let

there exists a compact absorbing set

equation (

Li and Wang (see Theorem 2.5 in [

Assume that

assumptions (

equation (

for each periodic solution

is asymptotically stable, where

Then the unique equilibrium

Now one uses Lemma

If

Based on Lemma

Meanwhile,

The results above verify the condition

If

According to [

And then the second additive compound matrix of the Jacobian matrix of (

Since

Along a solution

Hence, if

In the first subsection, we determine some parameter values of an anti-HIV infection treatment model based on (

Baxter et al. [

In the following subsections, we select, from [

Based on (

The infection-free equilibrium point

The endemic infection equilibrium point