Studies have shown that the potential benefit of vaccinating children against influenza extends to other members of their families, which supports the recommendation to make wider use of influenza vaccine in healthy children of any age in order to reduce the burden of infection on the community. The vaccination of otherwise healthy day-care and school-aged children may significantly reduce indirect influenza-related costs, thus supporting earlier economic modelling analyses of immunization programs
[36]. The methods described in the present study allow an approximate assessment of this herd effect in traditional static models used in economic evaluation of annual vaccination of children against seasonal influenza. The estimation of herd effect is expressed as a function of effective coverage in children, a notion which combines both vaccine efficacy and coverage. As such, these approximations inherently incorporate the flexibility of estimating changes in magnitude of herd effect associated with varying levels of vaccination coverage in children, as well as for the impact of varying vaccine efficacy, which is – amongst others – dependent on the degree of strain matching and type of vaccine. A plausible range for the magnitude of this indirect effect can be estimated from the two methods of approximation identified in this research: (1) a general approach, irrespective of disease area, provides more conservative estimates, and (2) a data-driven approach, fitted to published data specific for influenza vaccination in children, provides less conservative estimates.

The structured literature review provided evidence to support the hypothesis of a linear relationship between effective coverage and RR within an effective coverage range (vaccine efficacy combined with coverage) of 20% to 80% of the subgroup targeted for vaccination. Point estimates identified from the literature review allowed the fitting of a linear equation of the form y = a + bx for each of two broad age groups, the age group targeted by a childhood vaccination strategy (i.e. children) and the group not targeted by the vaccination strategy (i.e. adults and/or elderly people). In children, the fitted equation was not very different from the slightly more conservative function derived from Equation 3 in Bauch et al. (2009)
[12]. In the other age group, there was a large difference between the fitted linear equation and the linear function derived from Equation 3 of Bauch et al. (2009)
[12], with the latter being much more conservative. Thus, using the linear approximations derived from Equation 3 in Bauch et al. (2009)
[12] for both age groups would provide a conservative estimate of herd effect, while using the linear functions fitted to data from this structured literature review would provide a less conservative estimate of herd effect. Both approximations require only simple adjustments to the annual baseline risk of influenza for the two age groups, and can therefore be easily incorporated into static models to provide an approximate estimate of the likely range of possible herd effects.

### Limitations

A non-dynamic approximation such as those presented here cannot replace a fully dynamic modelling approach, and should only be intended for a preliminary assessment of herd effect
[12]. However, the linear approximations derived from Equation 3 in Bauch et al. (2009)
[12] are considered by the respective authors to be more conservative than a full dynamic assessment. Our second linear approximation was fitted to point estimates that included estimates derived from dynamic models, and can therefore be considered as more closely mimicking a full dynamic assessment of herd effect (which is the ultimate objective of a non-dynamic approximation). This second approximation offers a method for making a less conservative estimate of herd effect, and should thus help to allow a fuller exploration of the potential impact of herd effect within a static model.

Our second linear approximation is only intended for exploratory purposes, since it implicitly assumes a constant basic reproduction number (R_{0}) for seasonal influenza. The potential bias induced by this assumption is likely to be marginal for seasonal influenza, since R_{0} estimates for these epidemics are low and fairly constant
[37]. However, as a consequence of this assumption, our second linear approximation cannot as such be applied to provide a preliminary assessment of potential herd effect in pandemic situations.

Although the literature review conducted was not systematic, it was structured in a transparent and reproducible manner, with search terms, eligibility criteria and data extraction defined in advance. An independent reviewer checked all included studies and data extracted, in an effort to minimise selection bias. However, the initial screening process included studies that could not be ruled out with certainty, and reasons for exclusion were documented for all studies rejected after full text review. In addition, the inclusion of studies from sources other than the database search (in this review, mainly from reference lists) also bears a risk of selection bias. Most of the studies identified as useful for the main aim of the project were derived from the database search, and the two which came from other sources
[21, 23] reported outcomes that did not differ from the other studies.

The literature review did not reveal a mathematical function for the relationship between the relative risk in unvaccinated and very low (<20%) or very high (>81.1%) effective coverage levels in a subpopulation. However, findings have indicated that herd effect is relevant even with very low levels of coverage and can be even greater than direct effect
[13]. This finding is supported by other authors, who reported that the extent to which the elderly benefit from indirect effects depends (among other factors) on disease transmissibility
[38]. Below a certain transition point, the elderly were protected more by the indirect effects of the morbidity-based strategy than by direct effects of the mortality-based strategy
[38]. Accordingly, in epidemics a relevant indirect effect can also be assumed for very low levels of effective coverage, and can even be higher than the direct effects
[13]. However, this is highly dependent on the transmissibility, which is linearly related to R_{0}[38].

For very high levels of effective coverage, i.e. very high coverage and vaccine efficacy, a linear function might overestimate the impact of herd effect and a flattening of the curve, i.e. a more exponential function with exponent <1 in age groups others than those considered for mass vaccination might be expected. However, this is a more intuitive conclusion, rather than based on evidence from literature search.

Depending on the study, the RR of infection was calculated from either the probability of infection (modelling studies) or the probability of symptomatic influenza (observational studies). Thus, we implicitly assumed that both probabilities are linearly related, so that the RR is identical irrespective of which outcome is considered. It is however important to note that the RR obtained with our approximations refer to the baseline risk of true influenza infections (whether or not symptomatic), and do not reflect the reduction in risk of influenza-like-illness (ILI). Seasonal influenza vaccination is not efficacious against ILI other than true influenza, and hence will only partially reduce transmission of all ILI. As such, the effective coverage estimates to be applied in our linear approximations should be based on vaccine efficacy against true influenza, and not vaccine effectiveness against ILI. And consequently, our approximations can only be applied in cohort models operating on the basis of true influenza and its health and economic consequences.

Our second linear equation fitted to the point estimates in this literature review assumes that individuals in a population mix randomly within and between all age groups, and do not take account of the variety of mixing and contact patterns apparent in real life. The wide range between the minimum and maximum point estimates derived from Vynnycky et al. (2008)
[29] clearly demonstrates the impact of different mixing contact patterns on the size of the indirect effects of vaccination. Empirical data such as the POLYMOD contact survey
[39] indicate that mixing between age groups is often highly assortative, i.e. people have contacts primarily with people of the same age group as themselves. Thus, this assumption of random mixing, inherent to our second linear approximation, is likely to overestimate the importance of herd effect on age groups other than those targeted by vaccination in communities with a relatively low inter-age mixing (e.g. communities with low frequency of multi-generational households).

A further limitation is that the approximation of herd effect in age groups not targeted for vaccination does not account for any effective vaccine coverage already present in those age groups. If effective coverage is already substantial in these age groups, a modest increase in effective coverage in the total population induced by vaccinating children might result in a situation where the elimination threshold is exceeded and RR falls to zero. As such, the magnitude of the herd effect reported by the studies identified in this review is dependent on the pre-existing vaccine coverage in the age groups not targeted for vaccination. This could explain why the point estimate derived from the study by Loeb et al. (2010)
[11] was less favourable than the other studies shown in Figure
4B. In the study by Loeb et al. (2010)
[11] vaccination coverage in the remainder of the population was quite low (<13%), whereas in Halloran et al. (2002)
[24] 22.9% of adults aged 19–64 years and 68.1% of the elderly were vaccinated. Consequently, the less conservative linear function, derived by fitting to these point estimates, is likely to overestimate herd effect in groups that have little or no vaccine coverage.

For the purpose of this study, point estimates of effective coverage were derived or calculated from vaccine efficacy data reported in the various publications. There is a risk of bias when using data from observational studies since the vaccinated population might also potentially benefit from a reduction in the baseline risk of influenza (indirect effect), where observed vaccine efficacy is in fact the sum of both direct and indirect effects of vaccination. However, the linear fitting in our study was performed against data extracted from three publications in which this risk of bias is not present or negligible: the two modelling studies compared the post-vaccination population against a pre-vaccination population
[24, 29], and the vaccine efficacy reported in the one observational study statistically corrected for this bias
[11]. However, this aspect needs to be considered thoroughly, in case future studies are included in the fitting process in further research.

As a result of these limitations, we would recommend using the more conservative approach (the linear function derived from Equation 3 of Bauch et al. (2009)
[12]) as the base case for cost-effectiveness analyses using a static model. We would recommend using the less conservative approach, using the linear functions fitted to the point estimates in this literature review, in sensitivity analyses. The less conservative approach may overestimate the effects of herd effect induced by childhood vaccination, particularly for age groups with a low likelihood of mixing with children and/or with little or no pre-existing vaccination coverage. However, it allows a fuller exploration of the potential impact of herd effect than the conservative approach alone. Both approximations require only simple adjustments to be made to the annual baseline risk of influenza for the two age groups, and can therefore be incorporated into static models. They can be used together to explore the likely range of herd effects in static models, without requiring dynamic modelling processes.