A compartmental model with antiviral therapy was proposed to identify the important factors that influence HIV infection among gay men in China and suggest some effective control strategies. We proved that the disease will be eradicated if the reproduction number is less than one. Based on the number of annual reported HIV/AIDS among MSM we used the MarkovChain MonteCarlo (MCMC) simulation to estimate the unknown parameters. We estimated a mean reproduction number of 3.88 (95% CI: 3.69–4.07). The estimation results showed that there were a higher transmission rate and a lower diagnose rate among MSM than those for another highrisk population. We compared the current treatment policy and immediate therapy once people are diagnosed with HIV, and numerical studies indicated that immediate antiviral therapy would lead to few HIV new infections conditional upon relatively low infectiousness; otherwise the current treatment policy would result in low HIV new infection. Further, increasing treatment coverage rate may lead to decline in HIV new infections and be beneficial to disease control, depending on the infectiousness of the infected individuals with antiviral therapy. The finding suggested that treatment efficacy (directly affecting infectiousness), behavior changes, and interventions greatly affect HIV new infection; strengthening intensity will contribute to the disease control.
After the detection of the first acquired immunodeficiency syndrome (AIDS) patient, human immunodeficiency virus (HIV) spreads at an alarming rate worldwide. In recent years, though the number of people newly infected with HIV is decreasing, the prevalence of HIV among men who have sex with men (MSM) has increased significantly in China since 2005. In China, the first AIDS patient among MSM was found in 1989 and the proportion of reported cases resulting from homosexual contact in year 2005 to 2011 is 0.4%, 2.5%, 3.4%, 5.9%, 8.6%, 10.8%, and 13.0%, respectively [
Discovered in 1981, HIV is one of the few things that draw attention from both mathematicians, medical scientists and behavioral scientists. Many models have been proposed in order to predict and control the spread of HIV effectively. A basic SIR model was formulated by Gran et al. [
In mainland China, the government initiated a large scale of antiviral therapy since 2003, and free treatment has been expanded to all HIVpositive individuals whose CD4+ T count is less than 350 cells per
This paper is divided into 6 sections. In Section
Our model is formulated based on the key epidemiological properties of HIV/AIDS and some implemented public heath interventions such as condom use and antiviral therapy. The underlying structure of the model comprises classes of individuals among MSM who are highrisk susceptibles
The flow diagram of model with antiviral therapy.
We assume that people enter into the susceptible class at a rate
Based on the current surveillance system we can get the number of annual reported HIVpositive cases and AIDS patients among MSM from year 1985 to 2009 in mainland China. The first HIV infected case among MSM in China was reported in 1989. However, the disease has spread very quickly among this population. The proportion of reported cases resulting from homosexual contact increases as follows: 2.5% (2006), 3.4% (2007), 5.9% (2008), and 8.6% (2009) [
The natural death rate was estimated to be
Recently, antiviral therapy is not provided to HIVpositive individuals with
In order to determine the relative infectiousness for each class we follow the principle employed by Gran et al. [
We utilize the MarkovChain MonteCarlo (MCMC) simulation to estimate the main parameters, and initial values of model (
The basic reproduction number
As the model shows, the primary infected individual will stay at the compartments
Meanwhile, system (
When
Besides the diseasefree equilibrium the system has an endemic equilibrium
Moreover, according to the persistence theorem developed by Smith and Zhao in [
System (
Based on the number of individuals living with HIV (not AIDS) or AIDS among MSM from year 2005 to year 2009, we estimate mean values of parameters and their standard deviations which are listed in Table
Plots of data fitted results. (a) The number of annual reported AIDS patients. (b) The number of annual reported HIVpositive individuals. Squares denote the real data. Areas from light to dark mean the 50%, 90%, 95%, and 99% predictive probability limits due to parameter uncertainties.
MCMC plots for
We suppose that the treatment coverage rate will not change and the current treatment policy will remain the same (i.e., antiviral therapy starts when CD4+ T cell counts are less than 350 per
Prediction of HIV/AIDS among MSM in China from 2005 to 2020 and the uncertainties of the model. Areas from light to dark mean the 50%, 90%, 95%, and 99% predictive probability limits due to parameter uncertainties. (a) Total HIV/AIDS cases. (b) Total HIVpositive cases. (c) Total AIDS cases. (d) Total HIV/AIDS cases with antiviral therapy. Parameters and initial values used are shown in Table
Antiviral therapy can effectively decrease the viral load in the blood of HIV/AIDS infected individuals, thereby reduceing their infectivity. However, the decrease in viral load consequently alleviates the symptom of HIV/AIDS infected individuals; then they may become active or increase their highrisk behaviors. There are some studies showed that HIV infected individuals may increase their highrisk behaviors since they believe that antiviral therapy can make them more healthy [
Using the parameter values listed in Table
It is interesting to note that the value of
From (
(a) Plots of
We can verify that
Thus, there exists a critical value
In recent years many intervention measures have been implemented to control the quick increase of HIV epidemic among MSM. The intervention measures include (1) strengthening education to the highrisk population, which decreases the constant recruitment rate
To address the impact of each intervention measure on HIV infection among MSM, we investigate variation in number of HIV/AIDS infected individuals with varying transmission coefficients, treatment uptake rate. Figure
Plots of estimated number of HIV/AIDS cases vary with transmission coefficient
Plots of estimated number of HIV/AIDS cases vary with constant recruitment
Plots of estimated number of HIV/AIDS cases vary with diagnose rate
In order to investigate the effect of antiviral therapy, we study the variation in the incidence of HIV with different antiviral therapy efficacy represented by infectiousness (
Plots of incidence against factor
Note that in our model some parameters are known with uncertainties or have large variances, which may greatly affect outcomes. It is then necessary to do the uncertainty and sensitivity analysis such that the sensitive parameters can be detected. To examine the sensitivity of our results to parameter variations, we use latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCCs) [
We initially examine the sensitivity of basic reproduction number
Parameter values and ranges.
Parameters  Ranges  Initial values  Parameters  Ranges  Initial values 



402450 


763240 


0.8578 


0.0816 


0.5879 


0.2939 


1/12 


1/6 


0.2 


0.2 


0.7224 


0.4177 


0.2495 


0.172 


0.4177 


0.06 


0.3507 


0.1364 
Parameters and initial values.
Parameters  Definition  Value  Std  Source 


Transmission probability of HIV per highrisk behavior  —  —  — 

Contact rate per year  —  —  — 

Protection rate by intervention measures (condom use)  —  —  — 

Transmission coefficient, 
0.8578  0.013  MCMC 

Modification factor for HIV infected individuals with 
0.7224  —  [ 

Modification factor for HIV infected individuals with 
0.4177  —  [ 

Modification factor for HIV infected individuals with 
0.3507  —  — 

Modification factor for HIV infected individuals with 
0.3507  0.028  MCMC 

Modification factor for AIDS patients with antiviral therapy  0.2495  0.028  MCMC 

Recruitment rate of susceptible 

20764  MCMC 

Natural death rate  0.0149  —  [ 

Exit rate of susceptible  0.025  —  — 

Diagnose rate  0.0799  0.020  MCMC 

Proportion of diagnosed HIVpositive individuals 
0.8820  0.006  MCMC 

Proportion of diagnosed HIVpositive individuals with 
0.5879  —  [ 

Proportion of diagnosed HIVpositive individuals with 
0.2939  —  [ 

Progression rate from 
1/6  —  [ 

Progression rate from 
1/3  —  [ 

Progression rate from 
1/12  —  [ 

Progression rate from 
1/6  —  [ 

Antiviral therapy coverage rate for HIVpositive individuals with 
0  —  — 

Antiviral therapy coverage rate for HIVpositive individuals with 
0.2  —  [ 

Diseaserelated death rate for HIV infected individuals without receiving antiviral therapy  0.172  —  [ 

Diseaserelated death rate for HIV infected individuals with antiviral therapy  0.06  —  [ 

Diseaserelated death for AIDS patients with antiviral therapy  0.136  —  [ 

Initial value of 

87285  MCMC 

Initial value of 
5988  1274  MCMC 

Initial value of 
99  —  Database 

Initial value of 
49  —  Database 

Initial value of 
53  —  Database 

Initial value of 
0  —  Database 

Initial value of 
0  —  Database 

Initial value of 
0  —  Database 

The basic reproduction number  3.8840  0.097  Calculated 
(a) Partial rank correlation coefficients (PRCC) results for the dependence of
We also investigate the sensitivity of the expected number of people living with HIV/AIDS to parameter variations. Given the current treatment policy, parameters described in Table
According to the CD4+ T cell counts in the blood, we divided the HIVpositive individuals to several stages and formulated a mathematical model with antiviral therapy. The unknown parameters involved in this model were estimated using the MCMC simulation basing on the real data (i.e., the number of annual reported HIV/AIDS among MSM). We defined the threshold value (the basic reproduction number
We studied the effect of antiviral therapy in two situations: antiviral therapy started immediately once people are diagnosed with HIV and antiviral therapy started when CD4+ T counts are less than 350 cells per
Using the data on the number of new reported HIV/AIDS infected individuals by year among MSM, we obtained estimates of the reproduction number, intervention parameter values, and the highrisk population size. Our estimated reproduction number is 3.88 (95% CI 3.69–4.07) which is in the ranges of estimates for Western Germany (3.43–4.08) and UK (3.38–3.96) [
Simulation results show that strengthening education to highrisk population and increasing surveillance and testing can slow down the spread of disease. Further, sensitivity analysis implies that the most influential parameters are infection rate
In this paper, we concluded that if the infectiousness for HIV/AIDS infected cases is relatively small, treatment started immediately once diagnosed is more beneficial to disease control. It should be mentioned that we have not considered costs of antiviral therapy. However, early antiviral therapy will increase the financial burden of the government of China and may increase high risk of occurrence of drug resistance. We will consider these factors in the future study.
Since the number of susceptible cases is large compared to the number of HIV/AIDS infected cases,
For ensuring good convergence of Markov Chain, some prior information is given. Xiao et al. [
By fitting this reduced model to the annual reported HIV/AIDS cases from 2005 to 2009 we get the estimates of parameters
The model has a diseasefree equilibrium
Since
Let
Define a map
Define
If
If
According to the above verification, we can get that for
Obviously,
Since
Now we get the comparison system:
Suppose
The authors are supported by the National Megaproject of Science Research no. 2012ZX10001001, by the National Natural Science Foundation of China (NSFC, 11171268 (YX)), by the Fundamental Research Funds for the Central Universities (GK 08143042 (YX)), and by the International Development Research Center, Ottawa, Canada (104519010).