# Minimal impact of circumcision on HIV acquisition in men who have sex with men

Gregory J. Londish^{A}, David J. Templeton

^{B}

^{C}, David G. Regan

^{B}, John M. Kaldor

^{B}and John M. Murray

^{A}

^{B}

^{D}

^{A} School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia.

^{B} National Centre in HIV Epidemiology and Clinical Research, University of New South Wales, Sydney, NSW 2052, Australia.

^{C} Royal Prince Alfred Sexual Health, Royal Prince Alfred Hospital, Sydney, NSW 2050, Australia.

^{D} Corresponding author. Email: J.Murray@unsw.edu.au

*Sexual Health* 7(4) 463-470 https://doi.org/10.1071/SH09080

Submitted: 11 August 2009 Accepted: 20 April 2010 Published: 10 November 2010

## Abstract

**
Background:
** Men who have sex with men (MSM) are disproportionately affected by HIV. The proven efficacy of circumcision in reducing the risk of HIV acquisition among African heterosexual males has raised the question of whether this protective effect may extend to MSM populations. We examined the potential impact of circumcision on an HIV epidemic within a population of MSM.

**A mathematical model was developed to simulate HIV transmission in an MSM population. The model incorporated both circumcision and seropositioning, and was used to predict the reduction in HIV prevalence and incidence as a result of the two interventions. Estimates for the time required to achieve these gains were also calculated.**

*Methods:***We derive simple formulae for the decrease in HIV prevalence with increased circumcision. Our model predicts that if an initially uncircumcised MSM population in a developed country with a baseline HIV prevalence of 10% underwent universal circumcision, HIV incidence would only be reduced to 95% of pre-intervention levels and HIV prevalence to 9.6% after 20 years. In the longer term, our model predicts that prevalence would only decrease from 10% to 6%, but this would take several generations to achieve. The effectiveness of circumcision increases marginally with higher degrees of seropositioning.**

*Results:***The results of these calculations suggest that circumcision as a public health intervention will not produce a substantial decrease in HIV prevalence or incidence among MSM in the near future, and only modest reductions are achievable in the long-term.**

*Conclusions:***Additional keywords:** mathematical model, MSM, seropositioning.

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## Appendix

## Derivation of equations

The mathematical model for the number of uninfected *X* and infected *Y* MSM is described by the following system of differential equations:

where *β* = 1 – (1 – *p*
_{
R
})^{
n
R
} (1 – *p*
_{
I
} (1 – γ*m*))^{
n
I
} with the following parameters:

Population sizes are scaled so that the rate of entry of new MSM to the uninfected population is 1 per year. The infection term (*β*
*Y*/(*X* + *Y*))*X* is the harmonic mean of the uninfected and infected populations scaled by an infection transmission parameter *β*. The harmonic mean has the useful property of reducing to the smaller population if there is a large discrepancy between these populations, since the number of infections should depend more on the number of infected men, who are the minority in the population. Also the harmonic mean increases linearly as the populations increase and so the number of partnerships per man increases linearly as more men enter the population. Since the probability of HIV transmission per UAI act is small,^{12}
*β* simplifies to

From these equations, the stable equilibrium solution can be found:

with the HIV prevalence *Y*/(*X* + *Y*) given by:

when *β* > δ_{
Y
}. For this parameter region, the uninfected state with *Y* = 0 is unstable. If *β* ≤ δ_{
Y
} then the uninfected state *Y* = 0 and *X* = 1/δ_{
X
} is the only stable equilibrium (i.e. the epidemic cannot be sustained). This prevalence being non-negative indicates that the infected state is the stable solution, whereas a negative value indicates that the uninfected state is the stable equilibrium solution and prevalence reverts to zero.

## Circumcision in a MSM population without seropositioning

Since *p*
_{I} is very much less than *p*
_{R} (the probability of infection for insertive UAI is small compared to the probability of being infected when receptive) and with no seropositioning (*n*
_{
R
} = *n*
_{
I
} = *n*), we have for a fraction *m* of the population being circumcised:

so that:

The HIV prevalence when there is no circumcision (*m* = 0) is given by

so that this last equation can be rewritten as:

where *r* = *p*
_{
R
}/*p*
_{
I
} expresses the relative risk of receptive to insertive acts. This is the approximation used in estimates of the protective effect of circumcision in an MSM population.

## Circumcision in a MSM population practicing seropositioning

We now consider how prevalence changes with circumcision where seropositioning is practised. With seropositioning, the number of acts between infected and uninfected individuals is assumed to remain constant, but the ratio of receptive to insertive acts between these individuals changes. We assume here that the number of UAI acts, *N*, is constant regardless of the level of seropositioning.

The level of seropositioning is described by the variable σ = 1 – *n*
_{
R
}/*n*
_{
I
} such that σ = 0 represents no seropositioning so that the number of receptive and insertive acts are equal between HIV-infected and uninfected individuals, and σ = 1 represents full seropositioning so that there are no receptive acts by an uninfected MSM with an infected MSM. HIV prevalence with circumcision under different levels of seropositioning is given by:

Since we assume that the total number of UAI is unchanged by seropositioning,

then:

and the number of insertive UAI acts is given by *n*
_{
I
} = *N*/(2 – σ). Therefore:

and:

Prevalence in the absence of circumcision and seropositioning is given by

so that: