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A mathematical model of HIV/AIDS transmission incorporating treatment and drug resistance was built in this study. We firstly calculated the threshold value of the basic reproductive number (

It was reported that 2.7 million people were newly infected by HIV/AIDS virus and 1.8 million patients died of AIDS-related causes in 2010 worldwide. By the end of 2010, about 34 million people were living with HIV/AIDS in the world [

Since the initial infectious diseases model was presented by Anderson et al. in 1986 [

In this work, we established a model by adding effect of drug resistance into the similar models in the literatures [

According to the progression of disease, the total populations were separated into five groups: susceptible population, early-stage HIV population, symptomatic population, AIDS patients, and those who are accepting ART; we marked them with

We made some assumption as below.

Infection occurred when susceptible and infected contact with each other took place.

Only people in the period of AIDS may die of AIDS disease-related, then use

Although AIDS patients have the higher viral load, we assume they will not infect others because they have the obvious clinical symptoms and were accepting ART.

When in the situation (2) of treatment, we assume these people transformed into asymptomatic individuals.

According to the assumptions, the flow diagram of the six subpopulations was shown in Figure

Relationship between different populations.

The model was collected as the following differential equations:

The model is established in practice; thus we assume all parameters are nonnegative.

Since the

Let the initial data be

From the first equation of (

When

Equation (

When

Equation (

Similarly for the other equations of system (

Next, add all the the equations of system (

Then,

In a similar fashion we have

Thus, the feasible solution of system remains in the region

It is easy to see that the model has a disease-free equilibrium (DFE),

Let

Hence the reproduction number, denoted by

The disease-free equilibrium

(1) The Jacobian matrices of system (

Obviously,

Define the positive vector subsequently:

Then begin to show that all the eigenvalues of

In conclusion, if

Through the structure of the Jacobian matrix

(2) Let

(3) If

Equating each equation in system (

Endemic equilibrium

Equating each equation in system (

Setting

Computing the time derivative of

In order to determine the coefficient

Let

In conclusion, the limit sets of solutions in

This paper is an extended model about the works in [

For public health view, to bring HIV/AIDS into control, the prerequisite is reducing the threshold value of basic reproductive number

Limitations in our model exist. We ignored the changing drug when treatment failed in practice. A refinement of the model can be done in future. Additionally, we did do simulation with actual data, which are ongoing under further study.

This study was supported by Chinese government grants administered under the Twelfth Five-Year Plan (2012ZX10001-002), the Beijing Municipal Commission of Education (KM201010025010), and National Nature Science Foundation (30973981).