A mathematical model is proposed to consider the effects of saturated diagnosis and vaccination on HIV/AIDS infection. By employing center manifold theory, we prove that there exists a backward bifurcation which suggests that the disease cannot be eradicated even if the basic reproduction number is less than unity. Global stability of the diseasefree equilibrium is investigated for appropriate conditions. When the basic reproduction number is greater than unity, the system is uniformly persistent. The proposed model is applied to describe HIV infection among injecting drug users (IDUs) in Yunnan province, China. Numerical studies indicate that new cases and prevalence are sensitive to transmission rate, vaccination rate, and vaccine efficacy. The findings suggest that increasing vaccination rate and vaccine efficacy and enhancing interventions like reducing share injectors can greatly reduce the transmission of HIV among IDUs in Yunnan province, China.
Acquired immunodeficiency syndrome (AIDS) is spreading rapidly in the world ever since it was firstly detected in 1981 and continues to threaten the health of human seriously, especially among sex workers and injecting drug users. Furthermore, AIDS also influences the economy of many countries which has attracted great attention of governments. For such a severe scenario, the governments have taken intervention measures to reduce HIV transmission.
Mathematical models play a vital role in gaining a quantitative insight into HIV transmission dynamics and suggesting the effective control strategies. In order to study the effect of various intervention strategies on HIV transmission, extensive mathematical models have been formulated. Traditionally, models of HIV/AIDS dynamics often incorporate staged progression (see, e.g., [
The majority of mathematical models consider only one control strategy, vaccination or diagnosis, for instance [
The paper is organized as follows. The model is formulated in Section
The model describes the spread of HIV/AIDS in a high risk population. The total high risk population size at time
The equations of the model are
Since the model monitors change in the human population, the variables and parameters are assumed to be nonnegative for all
Model (
Denote by
Employing the center manifold theory as described in [
If
The local bifurcation analysis near
From [
If
Due to existence of backward bifurcation we know that, for positive
We now analyze the endemic equilibrium of model (
(a) Plot of the function
First, we have the following result on the local stability of
The diseasefree equilibrium
By checking the Jacobian matrix of system (
If the real parts of the roots of the equation
Then, using Lyapunov function we can get global stability of
The diseasefree equilibrium
We note that no endemic equilibrium exists for
Note that
In this section, we will prove that system (
Model (
If
Define
Note that
Note that
We initially investigate variation in
Parameter description and values.
Parameters  Description  Estimated values  Source 


Recruitment rate  4348  [ 

Transmission coefficient  0.304  [ 

Per capita waning rate of vaccine 

[ 

Per capita vaccination rate  0.4  Variable 

Vaccine efficacy  0.4  Variable 

Natural death rate  0.0246  [ 

Diseaseinduced death rate  0.7114  [ 

Progression rate to AIDS stage for the infection stage  0.0413  [ 

Progression rate to AIDS stage for the diagnosed stage  0.116  [ 

Diagnosis rate  0.304  [ 

Modification factor in transmission coefficient of diagnosed HIVpositive individuals  0.491  [ 

Modification factor in transmission coefficient of AIDS patients  0.1  Variable 
Contour plots of
Next, we consider the effect of different transmission rate, vaccination rate, vaccine efficacy, and recruitment rate on transmission of HIV/AIDS. We take the year 2004 as starting time; since then the policy of diagnosis is consistent. In [
Variation in the number of HIVpositive individuals with different (a) transmission coefficient, (b) vaccination rate, (c) vaccine efficacy, and (d) recruitment rate.
In this section, we use sensitivity analysis method [
Figures
Sensitivity coefficients of new cases (a) and prevalence (b) on
It follows from Figure
In this paper, we established an epidemic model to investigate effects of saturated diagnosis and vaccination on HIV transmission. It proved that backward bifurcation occurs by employing center manifold theory, which causes the diseasefree equilibrium to be locally asymptotically stable instead of globally asymptotically stable for
It is interesting to note that if the diagnosis is described linearly backward bifurcation does not happen. This implies that nonlinear diagnosis due to limited medical resources leads to backward bifurcation, and consequently complete elimination of HIV infection becomes difficult. That is, HIV infection might be extinct only by improving integrated interventions, which ensures that
Since several candidate HIV vaccines are in development, it is useful to study the effectiveness. Moreover, the detection of HIVpositive individuals is limited due to medical resources. We then applied the proposed model with nonlinear diagnosis and vaccination to examine HIV infection among IDUs in Yunnan province, China. Sensitivity analysis shows that new cases and prevalence are sensitive to transmission rate, vaccine efficacy, and vaccination rate, whereas diagnosis rate and recruitment rate slightly affect both of them. Therefore, enlarging vaccination rate, improving vaccine efficacy, and lowering transmission rate by reducing sterile injecting equipment are beneficial to reduce transmission of HIV infection. In order to efficiently reduce HIV transmission, combined intervention strategies are suggested to be implemented simultaneously.
Effective antiretroviral therapy (ART) is an important strategy to slow down the progression to AIDS due to great reduction in viral loads and is not included in our model. Note that when HIV infected individuals are diagnosed and CD4 T cell counts decrease to 350 copies/
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are supported by the National Megaproject of Science Research no. 2012ZX10001001, the National Natural Science Foundation of China (NSFC, 11171268 (YX)), the Fundamental Research Funds for the Central Universities (GK 08143042 (YX)), and the International Development Research Center, Ottawa, Canada (104519010).