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We consider a class of viral infection dynamic models with inhibitory effect on the growth of uninfected T cells caused by infected T cells and logistic target cell growth. The basic reproduction number

The human immunodeficiency virus (HIV) is a lentivirus, which replicates by infecting and destroying primarily CD4

It is widely known that mathematical models have made considerable contributions to understanding the HIV infection dynamics. Nowak et al. have proposed a class of classic mathematical model to describe HIV infection dynamics (see, for example, [

Incorporating the life cycle of the virus in the cells, some researchers have considered that the HIV virus from HIV infection to produce new virus takes time. To make a better understanding for this phenomenon in mathematics, HIV models including time delay have been proposed (see, for example, [

In the above model, there are two factors that accelerate the reduction of uninfected cells: one is the natural death of uninfected cells and the other is that uninfected cells become infected cells. HIV gene expression products can be toxic and directly or indirectly induce apoptosis in uninfected cells. Some data show that viral proteins interact with uninfected cells and produce an apoptotic signals that accelerate the death of uninfected cells. Recently, Wang and Zhang proposed a spatial mathematical model to describe the predominance for driving

Based on model (

Motivated by the above models, in this paper, we will study a delay differential equation model of HIV infection with a full logistic term of uninfected cells,

The main purpose of this paper is to carry out a pretty theoretical analysis on the stability of the equilibria of the model (

According to biological meanings, we assume that the initial condition of the model (

The existence and uniqueness, nonnegativity, and boundedness of the solutions of the model (

The solution

In fact, by using standard theorems for existence and uniqueness of functional differential equations (see, for example, [

We can denote the basic reproduction number of the HIV virus for the model (

(i) The model (

(ii) If

If

We consider linear system of the model (

Define

Motivated by the methods in [

Next, let us study the stability of the infected equilibrium

Therefore, (

Now, let us investigate the stability of

Now, we will illustrate the following conclusions, and it has been proved in [

For the polynomial (

If

If

If

Assume that

If

If

If the conditions of (ii) are all satisfied and

In the above section, we have given the sufficient condition where the model (

Since

We choose

Suppose that the conditions in (iii) of Theorem

If

If

If

For the main results in Sections

Based on the numerical simulations in [

(a) The solution curves of the model (

Furthermore, we also simulate the occurrence of Hopf bifurcations as the time delay

(a), (b), and (c) The solution curves of the model (

(a), (b), and (c) The solution curves of the model (

In this paper, we have proposed a delay HIV infection model (

If

If

It can be found that the basic reproduction number

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work is partly supported by the National Natural Science Foundation of China for W. Ma (no. 11471034).

^{+}T cells

^{+}T-cells

^{+}T cells

^{+}T cells death: a nonlocal spatial mathematical model