where the superscript "IDU" represents injecting drug users and "sex" represents sex workers. They estimate R IDU 0 = 32 and R sex 0 = 1.7. Abu-Raddad et al. [50] examined the population-level implications of imperfect HIV vaccines. They divided the population into two groups: unvaccinated and vaccinated, both of whom could become infected (albeit at different rates, depending on the efficacy of the vaccine). By forming two independent next-generation matrices, they derived R 0 and R 0V , the basic reproductive ratios for the unvaccinated and vaccinated populations, respectively. They note that the condition R 0V < 1 may not be sufficient for disease eradication, due to the possibility of a backward bifurcation. The two formulas are equivalent in the absence of the vaccine, assuming that risky behaviour does not change. They estimate R 0V = 1.4 in the absence of vaccine protection.