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 Reff=R0s. Because Reff needs to be <1, s needs to be <1/R0, or the proportion vaccinated needs to be >1-1/R0. Measles has an R0 of 10-12; therefore, the proportion required to be vaccinated to attain herd immunity is approximately 90%-95%. Unfortunately, these conventional modeling equations may be too optimistic and are based on assumptions that the population makes contacts randomly. If there is an uneven distribution of susceptible individuals [1, 2], a more complex model is needed to anticipate the effect of vaccination on herd immunity. Opinion clustering can cause concentrations of susceptible individuals that have an impact on disease outbreaks, as demonstrated in a network model [3]. I describe a stochastic simulation that demonstrates the impact of the clustering of contact networks on the effective vaccination threshold. The simulation begins with 1000 individuals, with a proportion of nonvaccinated individuals and an additional 2% of the population who are not immune, typically infants and individuals in whom the vaccine failed. Individuals have a mean of 20 contacts each, formed stochastically; the OR of contacts of nonvaccinated individuals being another nonvaccinated individual is set at r. An epidemic simulation is seeded with 1 case of measles and spreads in generations until extinction. The probability of an infectious case contact developing measles is R0;c if they are susceptible, where c is the mean number of contacts. An individual's infectiousness lasts 1 generation and that individual is thereafter immune. Figure 1 demonstrates the impact of changing the proportion of individuals vaccinated and the cohorting ratio, r, on outbreak probability. If r⩽1, the conventional vaccination thresholds lead to adequate herd immunity, but as r increases, outbreaks occur, even with very high vaccination rates. Outbreak occurrence as a function of vaccination coverage and tendency of non-vaccinees to cohort. The black dots on the figure represent simulations in which greater than 50 of 1000 individuals were involved in the outbreak. This simulation study demonstrates that more sophisticated models are required to examine vaccine effects in the settings described by Dominguez et al [1]. It also quantifies the problem of cohorting of susceptible individuals. Although this simulation avoids the assumption of homogenous population mixing from conventional models, it introduces other simplifying assumptions. Social network analysis is a means to derive data-driven models to investigate this phenomenon further. Financial support. Centre for Clinical Research Excellence in Infectious Diseases and the National Health and Medical Research Council Capacity Building Grant in Population Health (358424). Potential conflicts of interest. E.S.M.: no conflicts.
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Emma S. McBryde (2009) studied this question.