The right-hand side of equation (16) is interpreted as follows. Figure 3A shows a transmission tree, i.e., a representation of who infected whom, where each primary case on average generates two secondary cases. A transmission tree of this kind is usually unobserved unless rigorous contact-tracing with microbiological examination is implemented. Thus, a likelihood-based approach to reconstructing the tree is considered ( Figure 3B ) [29] [30] [31] . Given the total number of effective contacts that have equal potential for resulting in secondary transmission, the probability of a single effective contact resulting in a secondary transmission (or the probability that a secondary case is linked to an effective contact made by a single primary case in year t) is given by 1/R 0 . This is a simple binomial sampling process. In other words, the likelihood function for estimating R 0 is: (17) where T is the most recent time point of observation within an early (linear) epidemic growth stage. The maximum likelihood estimate of R 0 is obtained by minimizing the negative logarithm of (17) , and the 95% CI are derived from the profile likelihood. Table 1 shows the estimates of r and R 0 for HIV in France, Western Germany and the UK. The maximum likelihood estimates of r ranged from 1.15 to 2.15 per year with the smallest estimate in France and the highest in Western Germany. The 95% CI for Western Germany did not overlap with those of the other two countries, reflecting the steep rise in incidence in this region in Figure 2B . Based on the exponential growth assumption in (5), the estimate of R 0 ranged from 3.65-4.08. Again, Western Germany yielded the highest estimate without an overlap of the uncertainty bound with the other two countries. The maximum likelihood estimate of R 0 based on the proposed new method ranged from 3.59 to 3.74. Not only were the qualitative patterns for the expected values of R 0 consistent with those based on an exponential growth assumption, but the 95% CI also broadly overlapped with those based on the other method. In particular, although R 0 in Western Germany using the proposed method is smaller than that based on an exponential growth assumption, the 95% CIs of the two methods overlapped. The 95% CI based on the proposed method appeared to be wider than those based on exponential growth assumption. Since HIV is mainly transmitted via sexual contact, the above mentioned estimate may vary with the mixing pattern and contact frequency (thus, there is no general disease-specific R 0 , especially for HIV/AIDS). At least, compared with a previous estimate of R 0 as ranging from 13.9 to 54.5 in the USA, based on an exponential growth assumption that adopted a mean infectious period of 10 years [32] , R 0 in the present study appeared to be much smaller using a precise estimate of the generation time distribution.