Tanzania to be only 20% (P crit ¼ 1 -1/R 0 ), and even in areas where R 0 is higher, P crit rises to just 40% (Table 2 ). Our observations and simulations ( Figure 4 ) demonstrate that small outbreaks occur by chance even when coverage exceeds P crit and should be expected more frequently when there is individual variation in transmission (Text S2). Herd immunity declines rapidly in the interval between vaccination campaigns because of births and deaths in the domestic dog population (Figure 4 , inset). To maintain herd immunity above P crit between campaigns, therefore, requires a larger proportion of the dog population, P target , to be vaccinated (P target ¼ e (mþdþr)T P crit , where r is the rate of dog population growth, d is the death rate, 1/m is the duration of vaccineinduced immunity, and T is the interval between campaigns (see Materials and Methods)). By incorporating demographic parameters (Table 3) , we estimate that annual campaigns should therefore aim to vaccinate 60% of the dog population to avoid coverage falling below P crit .