Network structure can play a role in vaccination thresholds and herd immunity: a simulation using a network mathematical model.

To the Editor—Dominguez et al. [1] described outbreaks of measles occurring in Catalonia despite good vaccination rates. The way in which heterogeneity in the distribution of nonimmune individuals diminishes herd immunity can be explained using network simulation models. Herd immunity is the indirect impact of vaccination on an epidemic. This effect is caused by having a sufficient number of immune individuals to prevent chains of transmission, thus protecting unvaccinated individuals. Mathematical models have derived vaccination thresholds that are predictive of herd immunity on the basis of the average number of secondary cases that directly result from a primary case of contagious disease, assuming all contacts are susceptible (this is the “basic reproduction ratio,” R0). To attain herd immunity, the number of susceptible people in the population needs to be reduced to a new “effective reproduction ratio” (Reff) that is !1, by diminishing the proportion of susceptible individuals, s, through vaccination. In conventional modeling, the relationship between the effective and basic reproduction ratios is . Because Reff R p R s eff 0