An HIV/AIDS treatment model with different stages is proposed in this paper. The stage of the HIV infection is divided into two stages, that is, HIVpositive in the asymptomatic stage of HIV infection and HIVpositive individuals in the preAIDS stage. The fact that some individuals with HIVpositive individuals after the treatment can be transformed into the compartment of HIVpositive individuals in the asymptomatic stage of HIV infection, the compartment of HIVpositive individuals in the preAIDS stage, or the compartment of individuals with fullblown AIDS is also considered. Mathematical analyses establish the idea that the global dynamics of the HIV/AIDS model are determined by the basic reproduction number
Human immunodeficiency virus (HIV) is one of the major life threatening viruses that are spreading worldwide. AIDS is caused by the human immunodeficiency virus (HIV), which has developed into a global pandemic since the first patient was identified in 1981, making it one of the most destructive epidemics in history [
Because of the fact that there is no vaccine, there are many obstacles in the AIDS treatment. Recently, the most prevalent treatment strategy for HIV infected patients is highly active antiretroviral therapies (HAART), which can prolong the lifespans and improve their life quality of infected individuals [
It is well known that mathematical modeling is a very important approach to understand the dynamics of any epidemic and to further develop various control and prevention policies [
Huo and Chen [
Motivated by the above, in this paper, we will divide the stage of the HIV infection into two stages, that is, HIVpositive individuals in the asymptomatic stage of HIV infection and HIVpositive individuals in the preAIDS stage. We also consider the fact that some individuals with HIVpositive after the treatment can be transformed into the compartment of HIVpositive individuals in the asymptomatic stage of HIV infection, the compartment of HIVpositive individuals in the preAIDS stage, or the compartment of individuals with fullblown AIDS.
The organization of this paper is as follows. In the next section, we propose an HIV/AIDS model. In Section
The total population is divided into six compartments:
Description and estimation of parameters.
Parameter  Description  Estimated value  Date source 


Recruitment rate of the population  0.55 year^{−1}  Estimate 

Transmission coefficient for contact with the H class  0.3 year^{−1}  Estimate 

Transmission coefficient for contact with the P class  0.2 year^{−1}  Estimate 

Natural death rate  0.0196 year^{−1}  [ 

Progression rate from the H class into the P class  0.498 year^{−1}  Estimate 

Progression rate from the P class into the A class  0.08 year^{−1}  Estimate 

Proportion of the H class receiving treatment  Variable  Estimate 

Proportion of the P class receiving treatment  Variable  Estimate 

The proportion of the H class successful treatment  0.2 year^{−1}  Estimate 

The proportion of the P class treatment failure  0.05 year^{−1}  Estimate 

The proportion of the P class successful treatment  0.15 year^{−1}  Estimate 

The proportion of the P class treatment failure  0.03 year^{−1}  Estimate 

The proportion of the P class treatment failure and seriousness  0.01 year^{−1}  Estimate 

The diseaserelated death rate of the AIDS  0.0909 year^{−1}  [ 

The diseaserelated death rate of being treated  0.0667 year^{−1}  [ 

The rate of susceptible individuals who changed their habits  0.03 year^{−1}  [ 
Transfer diagram of the model
It is necessary to prove that all solutions of system
The feasible region
Adding the equations of system
If
Under the given initial conditions, we need to prove that the solutions of system
Thus, the solutions
There exist one diseasefree equilibrium
The model has a diseasefree equilibrium given by
Letting
The diseasefree equilibrium
According to [
While
If
If
To study the globally asymptotic stability of the endemic equilibrium, motivated by [
Biologically speaking,
In this section, we present some numerical simulations of system
First, when
When
Second, we choose
When
When
Third, we choose
Finally, we show the relation between
The relationship among
A mathematical model is proposed and analyzed to study the spread of HIV/AIDS with treatment. In this paper we have analyzed a stage structured model for HIV and the effect of treatment has also been studied. In the presented model, we get two equilibria: the diseasefree equilibrium and the endemic equilibrium. We further consider global asymptotic stability of the diseasefree equilibrium by using the wellknown LyapunovLaSalle invariance principal. It is found that the diseasefree equilibrium is globally asymptotically stable when the basic reproduction number is less than one. When the basic reproduction number is greater than one and
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the NNSF of China (11461041), the NSF of Gansu Province of China (2013GS09485, 1107RJZA088), the NSF for Distinguished Young Scholars of Gansu Province of China (1111RJDA003), the Special Fund for the Basic Requirements in the Research of University of Gansu Province of China, and the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology.