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A delayed
*HI*
*V/A*
*IDS* epidemic model with treatment and vertical transmission is investigated. The model allows some infected individuals to move from the symptomatic phase to the asymptomatic phase; next generation of infected individuals may be infected and it will take them some time to get maturity and infect others. Mathematical analysis shows that the global dynamics of the spread of the
*HIV/AIDS* are completely determined by the basic reproduction number
*R _{0}* for our model. If

*R*< 1 then disease free equilibrium is globally asymptotically stable, whereas the unique infected equilibrium is globally asymptotically stable if

_{0}*R*> 1.

_{0}Mathematical models play an important role in the study of the transmission dynamics of HIV/AIDS, and in some sense, delay models give better compatibility with reality, as they capture the dynamics from the time of infection to the infectiousness. Some HIV/AIDS models are introduced in [

In [

In model (1), it is assumed that some individuals with the symptomatic phases J can be transformed into asymptomatic individuals I after treatment and they get the result that when

In [

Here, the authors assume that a fraction of newborns, who sustain treatment, join the infective class while others, who do not sustain treatment, join the AIDS class after getting sexual maturity. The infectives through vertical transmission at any time t are given by

The organization of the paper is as follows. In the next section we present the model with delay. Section 3 presents the basic properties of the model. In Section 4, we analyze local and global stability of equilibrium points. In the last section, we present a brief conclusion.

We propose an HIV/AIDS model which incorporates time delay during which a newly born infected child attains sexual maturity and becomes infectious. In this model, the sexually mature population is divided into four subclasses: the susceptibles (S), the asymptomatic infectives (I), the symptomatic infectives (J) and full-blown AIDS group (A). The number of total population is denoted by

With the above considerations and assumptions, the spread of the disease is assumed to be governed by the following model:

where

For model (2), let the initial condition be

Let

which gives,

Define

This implies that if

It is reasonable to assume that the general death rate

Since the variable A of model (3) does not appear in the first three equation, in the subsequent analysis, we only consider the submodel:

Model (4) always has a disease-free equilibrium

By straightforward computation, when

First we will study the local and global stability of disease free equilibrium

The variational matrix of model (4) is given by

Theorem 4.1. If

Proof. The Jacobian matrix corresponding to model (4) about

where

The characteristic equation of this matrix is given by

where

Clearly, one root of this equation is

If

Since

when

If

Separating the real and imaginary parts, we have

Eliminating

Let

Through simple computation, we can found that all the coefficients of this equation is positive, so Equation (7) have no solution, it implies that Equation (6) have not the root like

We are now in a position to investigate the global stability of the disease-free equilibrium

Theorem 4.2. If

Proof. Consider the following Lyapunov functional.

Calculating the derivative of L along with the solution of model (4), we have

This implies that

invariant set of

Now, when

Theorem 4.3. If

Proof. For this purpose, we obtain the Jacobian matrix corresponding to model (4) about

where

The characteristic equation of this matrix is

where

Notice that

Hence

when

where

Obviously,

Now we study the stability behavior of

We assume that

Separating the real and imaginary parts, we have

Eliminating

where

Substituting

when

Next, we consider the global stability of

Theorem 4.4. If

Proof. Firstly, we define a function,

Next calculating the derivative of V along with the solution of model (4), we have

Since

Next,we consider the following variables substitutions by letting,

Then,

Further let

Then, through a straight computation, we have

Since the arithmetic mean is greater than or equal to the geometric mean and function g is a positive function, we have

Thus,

In this paper, we have considered an HIV/AIDS model with treatment, vertical transmission and time delay. Under the assumption that asymptomatic infectives (J) have the symptoms of AIDS, AIDS patients (A) are isolated; hence their probability of producing children is small; and it is neglected. From the local stability of disease free equilibrium, we calculated the basic reproduction number

This work was supported by the National Natural Science Foundation of China (Grants nos. 11261056, 11261058 and 11271312).

ZohragulOsman,XamxinurAbdurahman, (2015) Stability Analysis of a Delayed HIV/AIDS Epidemic Model with Treatment and Vertical Transmission. Applied Mathematics,06,1781-1789. doi: 10.4236/am.2015.610158