Approximate Formula for Number of Mcs Needed Per Infection Averted

Approximate formula for number of male circumcisions needed per infection averted

In a stable epidemic with prevalence and incidence rate among both males and females ( is the duration of infection), the number of HIV infections over a time horizon in absence of male circumcision (MC) is where is the total population size. Meanwhile in presence of MC, the number of MCs is given by where is the fraction of males that are circumcised, and the number of HIV infections over the time horizon is given by

Here, is the male circumcision efficacy against HIV acquisition and is the male circumcision efficacy against HIV transmission.

The number of MCs needed per each infection averted can be calculated as the number of MCs divided by the difference between the number of HIV infections with and without MC. This leads to the expression

This approximation is derived using a crude model for HIV transmission at the population level. It does not capture non-linear effects in HIV transmission dynamics such as heterogeneities in disease spread, indirect effects, and the epidemic threshold singularity. Accurate and precise estimates of this measure are best done using models that can account for many of the complexities of HIV transmission such as the deterministic transmission models discussed in the main text. Nevertheless, the formula provides a reasonable approximation for settings at hyperendemic HIV prevalence if the male circumcision intervention coverage is not very high (<75%) (Figure A).

In other settings, this formula may not provide quantitatively a good approximation, but still it can provide a very rough estimate of this quantity. It bears notice that this formula is more likely to overestimate the number of MCs needed per infection averted in the long term rather than underestimate it. In the short term however, this formula is more likely to underestimate the number of MCs needed per infection averted.

Figure A: Number of circumcision operations per infection averted at equilibrium at different levels of baseline HIV prevalence (model 2).