Liljeros et al. reply

Jones and Handcock have reanalysed two of our four data sets of the number of different sexual partners for Swedish men and women. Specifically, they analyse the probability-distribution function p(k), which yields less reliable information than the cumulative distribution,

that we used in our analysis, and they propose that the tails of p(k) decay with a power-law exponent ρ > ρc = 3, which is larger than the threshold value above which the variance of the distribution is finite. They also question whether public-health strategies need to be refocused in view of their inferred existence of epidemic thresholds.

We argue that these claims are misleading. Jones and Handcock's point that the tail exponent ρ > ρc = 3, the threshold for finite variance, is not a new one. Their estimates, in fact, agree with ours1, which indicate that the tails of the distributions of the number of sexual partners decay asymptotically as power laws with ρc > 3: the apparent discrepancy arises because Jones and Handcock use the probability-distribution function, p(k), whereas we studied the cumulative distribution, P(≥ k). The exponent ρ is related to the exponent α by the equation ρ = α + 1; hence, our values of α are the same as those that Jones and Handcock derive for ρ.

Jones and Handcock ignore two of our data sets1, in which we analyse the number of lifetime sexual partners. This is important in view of the duration of the infectious period of some sexually transmitted pathogens such as HIV. When we re-analyse the data sets for the number of lifetime partners using the Yule distribution, we obtain exactly the same result as for a power-law fit because the Yule distribution closely resembles a power law in the interval analysed (further details are available from the authors).

Jones and Handcock argue that if the variance of the distribution of sexual partners is finite, then interventions aimed at reducing disease transmissibility have the potential to eradicate sexually transmitted infections. The two data sets that they analysed pertain to the number of different sexual partners during the previous year, rendering any discussion about the existence of epidemic thresholds speculative (because other factors besides the distribution of the number of partners in the previous year influence the existence of an epidemic threshold for infinitely large networks2,3,4,5). For finite networks, the variance of these distributions must be finite, as we are analysing finite populations of individuals who are sexually active for finite periods of time6. We did not, therefore, raise the issue of infinitely low thresholds in our communication1.

We contend that Jones and Handcock's conclusion is premature, given that contagious processes differ fundamentally for scale-free and single-scale networks4,5 and that, if a threshold were to exist, it would be very low, as the variance of the distribution of the number of sexual partners is much larger than the mean5,7,8.

Jones and Handcock claim that there is no need for a radical refocusing of current public-health strategies9. We agree, as targeting high-risk sexual behaviour and carrying out contact-tracing routines — the strategies best suited to scale-free networks — are well established practices in many countries.