A sexrolepreference model for HIV transmission among men who have sex with men in China
 Jie Lou^{1},
 Jianhong Wu^{2},
 Li Chen^{1},
 Yuhua Ruan^{3} and
 Yiming Shao^{3}Email author
https://doi.org/10.1186/147124589S1S10
© Lou et al; licensee BioMed Central Ltd. 2009
Published: 18 November 2009
Abstract
Background
Men who have sex with men (MSM) are much more likely to be infected with HIV than the general population. China has a sizable population of MSM, including gay, bisexual men, money boys and some rural workers. So reducing HIV infection in this population is an important component of the national HIV/AIDS prevention and control program.
Methods
We develop a mathematical model using a sexrolepreference framework to predict HIV infection in the MSM population and to evaluate different intervention strategies.
Results
An analytic formula for the basic reproduction ratio R_{0} was obtained; this yields R_{0} = 3.9296 in the current situation, so HIV will spread very fast in the MSM population if no intervention measure is implemented in a timely fashion. The persistence of HIV infection and the existence of disease equilibrium (or equilibria) are also shown. We utilized our model to simulate possible outcomes of antiretroviral therapy and vaccination for the MSM population. We compared the effects of these intervention measures under different assumptions about MSM behaviour. We also found that R_{0} is a decreasing function of the death rate of HIVinfected individuals, following a power law at least asymptotically.
Conclusion
HIV will spread very fast in the MSM population unless intervention measures are implemented urgently. Antiretroviral therapy can have substantial impact on the reduction of HIV among the MSM population, even if disinhibition is considered. The effect of protected sexual behaviour on controlling the epidemic in the MSM population largely depends on the sexratio preference of different subpopulations.
Background
The report from the American Foundation for AIDS Research [1] suggests that the group originally at the most risk of HIV  gay and bisexual men  still remains at the highest risk. This is largely due to anal sex which, when unprotected, carries a high risk of HIV transmission, especially for the receptive partner. Men who have sex with men (MSM) are 19 times more likely to be infected with HIV than the general population. Gay and bisexual men are only a part of the total MSM population, since MSM is a description of a behavioural phenomenon, not an identity.
China's first, and most recent, official figure on male homosexuality was released in 2004, putting the total of gay men in the country at between five and ten million [2]. But this is only a conservative estimation [3]. The HIV infection rate among gay men in China is climbing at an alarming rate, largely due to neglecting this subpopulation. Recent studies suggest unprotected risk behaviour or sexually transmitted diseases (STDs) among MSM have been found in several cities in China. Disturbing HIV prevalence rates from 1.0 to 5.0% among MSM have been reported in several urban cities [4]; higher than the overall prevalence (0.05%) for China. Without timely action, MSM could become the second most risky group for HIV infection following injection drug users in China.
In China, sociologists and public health workers have long been aware of the commercial sex workers serving MSM, who are called money boys. Beijing, for example, has thousands of male sex workers, working in bathhouses, bars and clubs or finding their own clients on the streets or via the internet [5, 6]. It is shown in [5] that, even if money boys are normally managed by a socalled "Mommy", it is not uncommon for some of them to suffer physical violence and rape from clients [7]. In such circumstances, it is hardly realistic to hope that male sex workers will always use condoms. Migrant rural workers now also become a source of MSM [5, 6]. China's fastgrowing economy creates a lot of new jobs for migrant rural workers, who move frequently between big cities and their home town. A sizable proportion of these migrant workers have sex with men. Some of them also act as money boys to some old or not so popular gay men; in this situation, migrant workers normally only prefer insertive anal intercourse (AI) [6]. Thus, the population of money boys includes both professional money boys, and also a small number of rural workers. Some deterministic models have been proposed to understand the HIV epidemic in homosexual populations [8, 9]. In [8], Valle et al looked at the impact of education, temporarily effective vaccines and therapies on the dynamics of HIV in homosexually active populations. Their study assumed that some individuals possess one or two mutant alleles (like D32 of CCR5) that prevent the successful invasion or replication of HIV, and the study examined separate or combined effects of therapies, education, vaccines and genetic resistance. Breban et al. [9] evaluated the potential impact of rectal microbicides for reducing HIV transmission in bathhouses. In addition, Tan & Kiang [10] proposed a state space model (Kalman filter model) for the HIV epidemic in homosexual populations stratified into subpopulations by their sexual activity levels.
In our study here, we assume that HIV transmission takes place exclusively through AI (both receptive and insertive acts occur). We develop a mathematical model, based on the above categorization and the assumption that the viral transmission probability per anal sex act is different when transmission happens through receptive acts or insertive acts. Therefore, we divide MSM into three subgroups:

Only Bottom, including gay and bisexual men who prefer receptive AI, and some money boys;

Versatile, including all gay and bisexual men who are versatile in sex role, and some migrant rural workers;

Only Top, including gay and bisexual men who prefer insertive AI, and some money boys (such as some migrant rural workers who earn subsidy income by having sex with gay men).
We use this model to examine the effect of highly active antiretroviral therapy on controlling the HIV spread in the MSM population. HAART has led to dramatic decrease in morbidity and mortality among individuals infected with HIV. However, HAART coverage remains suboptimal, even in the resourcerich areas of the world. Our modelbased simulations therefore assume a small portion of MSM in China with HIV1 will start to take HAART. Since HAART predictably decreases plasma HIV1 RNA levels to below the levels of detection of currently available assays [12], we assume that individuals taking HAART are no longer infectious. An increase in adverse behaviour can result from the availability of interventions, the socalled disinhibition. Several early mathematical modeling studies raised the concern that any possible benefit of HAART on the spread of HIV could be readily offset by even modest increases in HIV risk behaviour [13]. However, our model shows that antiretroviral therapy for MSM in China will have both individual and publichealth benefits even if risk behaviour increases. We also discuss the effect of vaccination for general MSM in order to compare different strategies.
Methods
Mathematical model
where N_{ T }= S_{ T }+ I_{ T }, N_{ V }= S_{ V }+ I_{ V }and N_{ B }= S_{ B }+ I_{ B }denote the total population of the Only Top, the Versatile and the Only Bottom categories, respectively. Note that the Only Top category can have sex with the Only Bottom and the Versatile population. The Only Bottom category can have sex with the Only Top and the Versatile population. The Versatile category can have sex with all categories. In this model, we assume that susceptible and infected MSM can die at rates d_{ M }and d_{ I }respectively. Also, we assume new MSM are recruited into the appropriate susceptible compartment at rates r_{ T }, r_{ B }and r_{ V }respectively.
The HIV transmission rate (β_{ yx })
The HIV transmission rate through anal sex in MSM depends on six quantities:

the number of different AI sex partners per year, n_{ x }, for individuals from compartment x;

the number of AI with each sex partner per year, c_{ x }, for individuals from compartment x;

the viral transmission probability per anal sex act, h_{ yx };

the level of protection against HIV infection due to condom usage (if condoms are used, HIV transmission is decreased by a factor of (1  η^{ c }ρ^{ c }), where η^{ c }is the condom efficacy and ρ^{ c }is the proportion of condom use);

the proportion of infected MSM who know that they are infected, α_{ y }. This term denotes the effect of the 2008 HIV census in MSM population, where many men discovered they were HIV positive; ν_{ y }denotes the proportion of these infected MSM who begin to control their behaviour (such as condom use) to avoid the spreading of HIV, if they did not use condoms before they knew that they have been infected by HIV.

other STIs increase both the rate of transmission and acquisition of HIV (the proportion with other STIs is assumed to be ψ^{ s }, with μ^{ s }being the multiplication factor for HIV);
HAART and HIV vaccination in MSM
We assume that only 20% of MSM with HIV1 start to take HAART each year in China, although some of the simulations below permit variable rates of HARRT treatment. To model the effects of HAART, we add one additional compartment to each of the infected groups. We assume that individuals taking HAART extend their lifespan by 5 years, so their annual death rate is 0.069. Since HAART predictably decreases plasma HIV1 RNA levels to below the level of detection of currently available assays [12], we also assume that individuals taking HAART are no longer infectious. This reduction of plasma HIV1 leads to an increase in adverse behaviour (disinhibition), and we model this behaviour change by reducing condom use between MSM from ρ^{ c }to zero (again, some of the simulations below allow for variable condom use rates). We also consider the effect of a potential vaccine, by adding one compartment for each of the uninfected groups. This vaccine has the property that vaccinated individuals may become infected, if the efficacy of the vaccine is less than 100%. In our baseline simulations, we assume that uninfected individuals are vaccinated at a rate of 20%, (the same as the HAART rate), and we explore vaccine efficacies of 30% and 70%. Equations for both the HAART model and the vaccination model can be found in Additional File 1.
Parameters and initial values
Parameters.
Para  Description  Value  Source 

r _{ T }  source rate of OnlyTop MSM  96667  estimate 
r _{ V }  source rate of Versatile MSM  603950  estimate 
r _{ B }  source rate of OnlyBottom MSM  70817  estimate 
d _{ M }  death rate of susceptible MSM  0.022  estimate 
d _{ I }  death rate of infected MSM  0.105  estimate 
n _{ T }  number of AI sex partners per year of OnlyTop MSM  11.5  [16] 
n _{ V }  number of AI sex partners per year of Versatile MSM  12.4  [16] 
n _{ B }  number of AI sex partner per year of OnlyBottom MSM  13.6  [16] 
c _{ T }  number of AI acts with each sex partners per year of OnlyTop MSM  4.4  [16] 
c _{ V }  number of AI acts with each sex partners per year of Versatile MSM  4.5  [16] 
c _{ B }  number of AI acts with each sex partners per year of OnlyBottom MSM  4.2  [16] 
h _{ TB }  transmissibility of HIV from Top to Bottom  0.01  
h _{ VB }  transmissibility of HIV from Versatile to Bottom  0.01  
h _{ TV }  transmissibility of HIV from Top to Versatile  0.01  
h _{ BT }  transmissibility of HIV from Bottom to Top  0.005  
h _{ VT }  transmissibility of HIV from Versatile to Top  0.005  
h _{ BV }  transmissibility of HIV from Bottom to Versatile  0.005  
h _{ VV }  transmissibility of HIV from Versatile to Versatile  0.0075  estimate 
α_{ y }  proportion of infected MSM who know they are infected  0.15  [16] 
η ^{ c }  condom efficacy  0.9  [16] 
ρ ^{ c }  rate of condom use  0.3  [16] 
ψ ^{ s }  proportion with STI  0.158  [16] 
μ ^{ s }  multiplication factor of STI for HIV  2.89  [21] 
ν _{ y }  proportion of infected MSM who begin to control their behaviour  0.5  [16] 
Parameters and R_{0}.
Parameters  Case 1  Case 2  Case 3 

h_{ TB }, h_{ VB }, h_{ TV }  0.01  0.001  0.001 
h_{ BT }, h_{ VT }, h_{ BV }  0.005  0.0005  0.0005 
h _{ VV }  0.0075  0.00075  0.00075 
ρ ^{ c }  0.3  0.6  0.6 
d _{ I }  0.105  0.105  0.026 
R _{0}  3.9296  0.2528  1.0212 
Infection rates of Table 1.
Compartment  Description  Value 

β _{ TB }  rate of infection by I_{ T }of S_{ B }  0.5786 
β _{ TV }  rate of infection by I_{ T }of S_{ V }  0.5653 
β _{ BT }  rate of infection by I_{ B }of S_{ T }  0.2563 
β _{ VB }  rate of infection by I_{ V }of S_{ B }  0.5786 
β _{ VT }  rate of infection by I_{ V }of S_{ T }  0.2563 
β _{ BV }  rate of infection by I_{ B }of S_{ V }  0.2826 
β _{ VV }  rate of infection by I_{ V }of S_{ V }  0.4239 
Initial Conditions.
Compartment  Description  Initial Value 

S _{ T }  susceptible OnlyTop MSM  2.4251 × 10^{6} 
S _{ V }  susceptible Versatile MSM  1.5418 × 10^{7} 
S _{ B }  susceptible OnlyBottom MSM  2.5908 × 10^{6} 
I _{ T }  infected OnlyTop MSM  3.6931 × 10^{4} 
I _{ V }  infected Versatile MSM  2.3480 × 10^{5} 
I _{ B }  infected OnlyBottom MSM  3.9455 × 10^{4} 
Results and discussion
Model analysis: the basic reproductive ratio, R_{0}
Following the nextgeneration operator method of [15], we linearize the second, the fourth and the sixth equations of our model around the diseasefree state and look for conditions that guarantee the growth of the three infected classes, I_{ T }, I_{ V }and I_{ B }.
R_{0} is the spectral radius of the nextgeneration matrix. Therefore, to find R_{0} we must find the largest eigenvalue of .
The reproductive number, R_{0}, is the number of secondary cases produced by a typical infected individual during his entire period of infectiousness in a demographically steady susceptible population. Calculating this number for our model is critical to determine whether HIV can invade the MSM population and/or stabilize in the population. We can show that when R_{0} < 1, HIV will not be sustained in this MSM population; otherwise, the infection will approach an endemic equilibrium of constant incidence and prevalence.
Persistence of HIV infection
By Thieme's persistence theory, we prove that the system is persistent of HIV infection when R_{0} > 1; i.e., when R_{0} > 1, HIV will spread in the MSM population so long as one infected MSM is introduced in this population, regardless of whether he is an OnlyTop, a Versatile or an OnlyBottom. We also get the existence of disease equilibrium (or equilibria) from the persistence of HIV infection. The proof is in Additional File 1.
Numerical simulations
Outcomes without any intervention
Outcomes of HAART and vaccination
Suppose the vaccination rate in MSM is also 20%. Then the effects of a potential vaccine with efficacy of 30% and 70% are shown as the solidring curve and the emptyring curve, respectively. R_{0} = 2.8676 and R_{0} = 1.4514 for each situation. It shows that the effect of a potential vaccine appears worse than that of HAART, even if the vaccine efficacy is as high as 70%. Simulation shows that even if disinhibition occurs, the effect of HAART is still much better than that of no intervention. This result is remarkably different from those in [13], but agrees with those in [17].
Reproduction number and lifespan: power law
Conclusion
We developed a mathematical model using a sexrolepreference framework to predict HIV infection in the MSM population. An analytic expression of the basic reproduction ratio R_{0} was obtained using model parameters, and we estimated the current R_{0} as 3.9296.
Our simulations suggest that both antiretroviral therapy and a potential vacce are powerful interventions, even if disinhibition is considered. Our simulations also suggest that having protected sexual behaviour has limited effect on controlling an epidemic in the MSM population, and medicine which can reduce the transmission and extend the lifespan of the infected has a complex impact.
There are three points that we should pay attention to. First, we suppose that most of these professional money boys are being in the Bottom Only category. This is from the investigation of China's current situation. Maybe there are some of them being in the other two categories. But since the proportions are small, the effect should be also very limit to the final outcomes. Second, considering the large variation of the data of different sexual partners for each MSM, which is possible to obey a powerlaw distribution, maybe the complex network (such as the ScaleFree network) is a more suitable method to model the spreading of HIV in MSM. This is also what we want to try in our next work. Third, many MSM in China, whether occasionally or frequently having sex with men, do not necessarily regard themselves as homosexual or bisexual. They are very often married. Even if they are not, they may have sex with women as well. This applies particularly to those societies wherein marriage is strongly promoted by the society and the family. This is largely true for rural workers, most of whom are married. Thus, as a bridging population, infected MSM transmit the infection to their heterosexual partners and thereafter to the general community.
Declarations
Acknowledgements
JL's work was supported in part by the Natural Science Foundations of China (Grant No.10701053 and No.10531030), by the China National Grand Program on Key Infectious Disease Control (2008ZX10001002 project 2), and by the Shanghai Leading Academic Discipline Project (S30104). JW's work was supported by the Canada Research Chairs Program, by the Natural Sciences and Engineering Research Council of Canada, and by the Mathematics for Information Technology and Complex Systems. This work was also supported by a CRCIDRC International Research Chair Program: CanadaChina Program on Disease Modeling and Management (104519018) and the Ministry of Science and Technology of China (2007DFC30230).
This article has been published as part of BMC Public Health Volume 9 Supplement 1, 2009: The OptAIDS project: towards global halting of HIV/AIDS. The full contents of the supplement are available online at http://www.biomedcentral.com/14712458/9?issue=S1.
Authors’ Affiliations
References
 MSM 19 times more likely to be infected with HIV, says AIDS group. Fridae: Empowering Gay Asia. 2008, [http://www.fridae.com/newsfeatures/2008/08/05/2099.msm19timesmorelikelytobeinfectedwithhivsaysaidsgroup]
 HIV infection rate climbing among gay men. [http://www.chinadaily.com.cn/china/200610/22/content_713884.htm]
 Li Y: Selected Works of Li Yinhe. 2006, Neimenggu University PressGoogle Scholar
 Qu SQ, Zhang DP, Wu YH, Zhu H, Zheng XW: Seroprevalence of HIV and risk behaviours among men who have sex with men in a northeast city of China. Chin J STD/AIDS Prev Cont. 2002, 8: 145147.Google Scholar
 Tong G: An Inquiry into Commercial Sex in the Community of Men Who Have Sex with Men in China. 2007, Beijing Gender Health Education InstituteGoogle Scholar
 Tong G: MSM Volunteers Participation in HIV/AIDS Prevention Activities. 2001, Public PublicationGoogle Scholar
 Tong G: Research on the HIV/AIDS behavioural Intervention among MSM. 2004, Beijing Gender Health Education InstituteGoogle Scholar
 Valle SD, Evangelista AM, Velasco MC, KribsZaleta CM, Schmitz SFH: Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity. Mathematical Biosciences. 2000, 187: 111133. 10.1016/j.mbs.2003.11.004.View ArticleGoogle Scholar
 Breban R, McGowan I, Topaz C, Schwartz EJ, Anton P, Blower S: Modeling the potential impact of rectal microbicides to reduce HIV transmission in bathhouses. Mathematical Biosciences and Engineering. 2006, 3: 450466.Google Scholar
 Tan WY, Xiang ZH: A state space model for the HIV epidemic in homosexual populations and some applications. Mathematical Biosciences. 1998, 152: 2961. 10.1016/S00255564(98)100135.View ArticlePubMedGoogle Scholar
 Yee N: 2002, [http://www.nickyee.com/ponder/topbottom.html]
 Hogg RS, Rhone SA, Yip B, Sherlock C, Conway B, Schechter MT, O'Shaughnessy MV, Montaner JS: Antiviral effect of double and triple drug combinations amongst HIVinfected adults: lessons from the implementation of viral loaddriven antiretroviral therapy. AIDS. 1998, 12: 27984. 10.1097/0000203019980300000005.View ArticlePubMedGoogle Scholar
 Blower SM, Gershengorn HB, Grant RM: A tale of two futures: HIV and antiretroviral therapy in San Francisco. Science. 2000, 287: 650654. 10.1126/science.287.5453.650.View ArticlePubMedGoogle Scholar
 The total number of rural workers at the end of 2008 is 225,420,000. [http://www.cpirc.org.cn/tjsj/tjsj_cy_detail.asp?id=10471]
 Diekmann O, Heesterbeek JAP, Metz JAJ: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990, 28: s36510.1007/BF00178324.View ArticleGoogle Scholar
 Highrisk behaviours and HIV/syphilis prevalence among men who have sex with men in Beijing. 2008, China CDCGoogle Scholar
 Abbas UL, Anderson RM, Mellors JW: Potential impact of antiretroviral therapy on HIV1 transmission and AIDS mortality in resourcelimited settings. J Acquir Immune Defic Syndr. 2006, 41: 63241. 10.1097/01.qai.0000194234.31078.bf.View ArticlePubMedGoogle Scholar
 Traditional Chinese medicine gives hope to HIV infected people. [http://www.people.com.cn/GB/paper3024/13507/1209794.html]
 An antiHIV traditional Chinese medicine has won state invention patent. [http://zyb.gzst.gov.cn/dongtai.asp?id=506]
 Porco TC, Martin JN, PageShafer KA, Cheng A, Charlebois E, Grant RM, Osmond DH: Decline in HIV infectivity following the introduction of highly active antiretroviral therapy. AIDS. 2004, 18: 8188. 10.1097/0000203020040102000010.PubMed CentralView ArticlePubMedGoogle Scholar
 Deschamps MM, Pape JW, Hafner A, Johnson WD: Heterosexual transmission of HIV in Haiti. Annals of Internal Medicine. 1996, 125 (4): 324330.View ArticlePubMedGoogle Scholar
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