Throughout this manuscript, the following definitions apply:
Effective coverage in children
vaccination coverage in children
vaccine efficacy in children
Change in effective coverage in the entire population (induced by effective coverage in children)
effective coverage in children
proportion of children in the total population
Linear approximation of herd effect
Bauch et al. (2009)
] describe a pseudo-dynamic approximation to allow incorporation of herd effect in a focal cohort (vaccinated in year X), induced by vaccinating subsequent cohorts not included in the focal cohort model (vaccination in years X
3, etc.). Equations 2 and 3 in this publication estimate an adjustment factor ω by which the incidence in susceptible individuals should be multiplied to capture partial herd effect benefits. Both equation functions are linear. Assuming ω is a good approximation of the relative risk (RR) of infection induced by herd effect, the linear relationship between RR and effective coverage estimated from Equations 2 and 3 is presented in a figure in the publication by Bauch et al. (2009)
] (the second figure in Bauch et al. (2009)
Derived from Equation 2 (known R0)
basic reproduction number (average number of secondary infectious persons resulting from the introduction of an infectious person into a totally susceptible population).
The RR of infection can be described as decreasing linearly with increasing effective coverage. The slope of the line (or the value of effective coverage at which a RR of zero is achieved, i.e. the elimination threshold) is dependent on the value of R0
: the lower the R0
, the steeper the decrease in RR, i.e. the higher the impact of herd effect (the second figure in Bauch et al. (2009)
]). Detailed information on the relationship between R0
and the magnitude of herd effect can be found in Bauch et al. (2009)
Derived from Equation 3 (unknown R0)
This equation is only dependent on effective coverage, and is the most conservative approach for estimating the relationship between effective coverage and the RR of infection, since it does not account for any incremental herd immunity induced by R0
approaching 1 (the second figure in Bauch et al. (2009)
Although Bauch et al. (2009)
] suggest a linear approximation of herd effect, their settings and assumptions deviate substantially from those generally accepted for seasonal influenza. Further confirmation was required on whether a linear approximation could also be considered valid for annual vaccination against seasonal influenza and therefore this was the rationale for conducting a literature review to identify published evidence to test this hypothesis.
Structured literature review with a focus on seasonal influenza
A structured literature review was performed with a specific focus on herd effect induced by vaccination against seasonal influenza. The objectives of this review were: to validate whether a linear relationship between effective coverage in a subpopulation and RR of symptomatic influenza infection in the non-vaccinated population forms a valid approximation for herd effect; and to identify point estimates of this relationship, expressed as RR as a function of effective coverage in children. Methods of analysis, i.e. keywords, limitations, inclusion criteria, as well as the data extraction sheet, were defined a priori.
Free-text PubMed searches were conducted using the following search terms, limited to English-language publications in humans with abstracts available:
2. herd immunity OR herd protection OR herd effect
3. population protection OR community protection
4. community vaccination OR community disease transmission
5. 1 AND (2 OR 3 OR 4)
No time limits were applied. The last search was run on 3 August 2011.
Relevant references cited in articles identified through the database search, as well as literature identified from other sources, were included. Literature identified through other sources was clearly stated as such, as these may be subject to search bias.
Articles were included if they met the following pre-defined criteria:
1. Clinical study or observational study or review or modelling or health economic study;
2. Inclusion of a subpopulation for mass vaccination;
3. Reporting of one of the following outcomes (either directly reported, or reported outcomes allowing a recalculation to obtain these data):
a. A relationship (mathematical function) between varying degrees of vaccine coverage and efficacy in subgroup populations (not restricted to children) and the reduction of influenza transmission (i.e. reduction in probability of infection) in a larger unvaccinated population;
b. Point estimates of the reduction of influenza infection in the unvaccinated population after vaccination of children, which allow for a fitting of the mathematical function to published data (as defined under (a)).
Titles and abstracts were scanned, and the full text of publications meeting the eligibility criteria or requiring further evaluation was reviewed. Publications meeting the eligibility criteria after evaluation of the full text were included in the full data extraction process.
The data extraction sheet was pre-defined and only minor changes, mainly to improve clarity, were applied after the start of review. Data extraction was conducted by one reviewer and reassessed by an independent reviewer (included studies only). Any discrepancies, which were only minor and non-substantial, were resolved by discussion between the two reviewers.
Outcomes considered and additional analyses
The main outcomes and additional analyses from the publications included in the literature review were as follows:
• Vaccination coverage and direct effectiveness of vaccination in subgroup population;
Additional analysis (if not reported): calculation of effective coverage in subpopulation, based on vaccination coverage in subpopulation and effectiveness expressed as a reduction in the probability of infection in vaccinated individuals;
• Indirect effectiveness in unvaccinated individuals after vaccination of subpopulations;
Additional analysis (if not reported): calculation of the reduction in probability of infection in the unvaccinated population, based on the probability of infection in the absence (or baseline level) of effective coverage in subpopulations, and the probability of infection in the presence of increased effective coverage in subpopulations;
• Relationship (mathematical function and point estimates) between different levels of effective coverage in subpopulation and indirect effectiveness in unvaccinated individuals after vaccination of subpopulation;
Additional analysis (if not reported): calculated relationship (mathematical function) between different levels of effective coverage in subpopulation and changes in RR in unvaccinated population.
Function fitting process
The linear function calculated from Equation 3 in Bauch et al. (2009)
] did not contain a fitting parameter and hence was not fitted to the point estimates. This function accounts only for the reduction in the number of susceptible individuals due to vaccination, and can therefore be applied to estimate on a yearly basis the RR for seasonal influenza infection.
In a second approach, a linear function was fitted to the point estimates identified through the structured literature review as best predictors of the functional relationship between effective coverage in children and RR of infection in the unvaccinated remainder of the population. Theoretically, the linear function calculated from Equation 2 in Bauch et al. (2009)
] could have been used for this purpose; R0
would then be the fitting parameter. However, Equation 2 in Bauch et al. (2009)
] was developed for a particular situation, in which – amongst others – natural and vaccine-derived immunity are lifelong. Since this is not the case for seasonal influenza, any value attributed to R0
as a result of the fitting process would be of no epidemiological meaning.
Therefore, a simple linear function of the form y
was fitted to the point estimates identified through the structured literature review as best predictors, by minimizing the sum of squared residuals (SSR) using the methodology described by Kemmer and Keller (2010)
]. In this function, y
effective coverage, a
1 (ensuring RR
1 at zero per cent effective coverage), and b
fitting parameter. The slope and intercept of the resulting linear function obtained by this fitting process are identical to those that would have been obtained by applying Equation 2 in Bauch et al. (2009)
], but there is no epidemiological meaning attributable to the fitting parameter b.