Demography

For each of the seasons of the study, the number W_i of individuals in the population with age i is taken from the Office for National Statistics (http://www.ons.gov.uk). Seven age groups were considered (1, 1–4, 5–14, 15–24, 25–44, 45–64, 65+ y). Each of these age groups is divided into individuals at low or high risk of complications associated with influenza, later simply referred to as low or high risk. Individuals are considered at high risk if they have one of the following conditions: chronic respiratory, heart, or renal disease; diabetes; or immunosuppression due to disease or treatment. The proportion of people in a risk group in a particular age group is assumed to be constant over the period of the study. We derived the proportion of individuals in a risk group for each age group by analysing data from the Royal College of General Practitioners (RCGP) Weekly Returns Service over a period of 5 y (2003–2008). The numbers of people in the different age groups and the percent classified as high risk are given in Table 1. On the last line of the table, the actual numbers resulting from the RCGP analysis can be seen.

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Table 1. Sizes of the modelled population compartments by age and risk group averaged over the period of the study.

https://doi.org/10.1371/journal.pmed.1001527.t001

Contact Survey

In 2006, a pan-European survey was conducted to measure and compare the structure of contacts in eight different European countries (Belgium, Finland, Germany, United Kingdom, Italy, Luxembourg, the Netherlands, and Poland). Attention was paid to recruiting participants representative in terms of geography, age, and sex [14]. Recruitment methods differed between countries, but were based in the United Kingdom on face-to-face interviews. Participants were asked to complete diaries recording with whom they had contacts during a day. Diaries also recorded different information relative to those contacts. Among them, the age of the contact was recorded, the nature of the contact (conversational or physical contact) and the nature of the day (weekday, weekend, or holiday). Details about the project can be found in Mossong et al. [15]. We use data from the United Kingdom arm of the study: the final data consisted of 1,012 participants aged 0 to 79 y who recorded 11,876 contacts in total.

Consultations in General Practices

Since January 1967, the Weekly Returns Service of the RCGP has monitored the activity of acute respiratory infections in general practices. As part of this scheme, the weekly number of persons consulting for influenza-like-illness (ILI) is recorded. For each week of the studied period (from the influenza season 1995/1996 until 2008/2009), we obtained the size of the monitored population included in the RCGP Weekly Return Service and the number of individuals in that monitored population consulting for ILI stratified in five age groups (0–4, 5–14, 15–44, 45–64, 65+ y). For a given season, we note by the size of the monitored population and by m_ij the number of people recorded as consulting for ILI in age group i at week j in that population.

In 2008, roughly 1.7% of the total population in England and Wales was included in the RCGP network of practices. Although this number might seem small in comparison with other existing surveillance networks, this sample is considered to be reasonably representative of the whole population in terms of demography and geography [16].

Respiratory Virus RCGP Surveillance

In order to complement the syndromic surveillance, a new surveillance activity of virological confirmation of cases was set up starting from the 1995/1996 season. During weeks of potential influenza activity, samples are taken from people presenting with ILI and sent for testing to the Respiratory Virus Unit of the Health Protection Agency. In the period considered (i.e., from the start of the virological surveillance activity until the season 2008/2009), the dataset includes 12,575 virological samples with recorded age group tested, which corresponds to an average of 900 samples per season. In total, however, during the last season, because of the novel H1N1 pandemic, 3,395 samples were tested. If this season is excluded, the dataset has an average of 700 samples per season.

The predominant strain in the period investigated was A/H3N2. In every season except 2000/2001, A/H3N2 was identified, while in several seasons either A/H1N1 or B was absent. In any given week, the number of confirmed samples by age group was relatively small. In addition, no samples were taken in the age group 1 y; most of the samples (47.7%) were taken in the 15–44-y age group, while the younger age groups—though higher transmitters—are less represented (9.6% and 13.7% for the age groups 1–4 y and 5–14 y, respectively). Elderly adults (65+ y) are under-represented (7.8%), which is a problem as the incidence appears to be smaller in this age group. Positivity rates are difficult to accurately quantify in this age group.

Among these tested samples, 17.3% were positive for influenza (H3N2: 53.8%, H1N1: 11.9%, H1N2: 0.9%, H1N1pdm: 10%, B: 18.9%). In the 2002/2003 season, only seven samples were taken, among which five were positive. However, three out of the five samples were taken in one age group (1–4 y) at a time when the level of ILI was extremely low (middle of May 2003).

In the rest of the manuscript, we note by the number of positive samples from a given subtype (A/H1N1, A/H3N2, or B) taken among the n_ij samples tested at week j in the age group i.

Vaccination Uptake and Match

Coverage by age and risk group by week was taken from Joseph et al. [17] for the seasons before 2003/2004. Figures for the 65+-y age group from 2004/2005 onwards were taken from the Health Protection Agency/Department of Health (HPA/DH) annual reports on the influenza programme [18].

In order to be able to derive estimates of the vaccine efficacy for each strain and season, data from the Health Protection Agency were used to establish the match between the circulating and vaccine strains (Table 2).

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Table 2. Match of the vaccine strains to the circulating seasonal strains during the period 1995/1996 to 2008/2009.

https://doi.org/10.1371/journal.pmed.1001527.t002

Serology for the Season 2003/2004 in A/H3N2

During the winter of 2003/2004, the United Kingdom experienced an unusual level of influenza activity following the emergence of A/Fujian/411/02-like antigenic variant strains. In order to investigate the severity of these new drifted strains, an age-stratified serological survey was conducted by the Health Protection Agency using sera sampled before and after the season [19]. Resulting data thus contain information on haemaglutination inhibition titres and RCGP age group for 875 sera collected pre- and post-season. The haemaglutination inhibition assays were performed using the A/Wyoming/3/03 strain, antigenically equivalent to the A/Fujian/411/02 strain that was circulating in the United Kingdom during the 2003/2004 season.

Methods Overview

Based on these data sources, an inference and modelling framework embedding social, immunisation, epidemiological, and surveillance components was developed. A Bayesian approach to statistical inference was adopted, as it provides the most suitable framework for synthesising diverse sources of evidence in a coherent manner [20]. Specifically, we utilised adaptive Markov chain Monte Carlo (MCMC) techniques [21], which represent a generic tool for inference from complex stochastic systems. An attractive aspect of the Bayesian approach when coupled to the MCMC methodology is its particular suitability for the natural propagation of uncertainty. This is especially crucial in the present analysis since the quantities of interest, like the final numbers of individuals infected under different scenarios, are non-linear functionals of the basic model parameters. Additionally, MCMC is highly modular, thus accommodating the inclusion of a novel algorithm for exploring the space of contact matrices (i.e., rates of epidemiologically relevant contacts).

We use a directed acyclic graph [22] to represent the model structure and the way the data streams are integrated in the inference scheme (Figure 1). In what follows we refer to the notation adopted in Figure 1 in order to identify the different processes operating at the different levels (transmission, vaccination, epidemiology, and observational processes) of the model.

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Figure 1. Directed acyclic graph showing the link between the different modelling components, data, and parameters.

Double and simple arrows indicate, respectively, deterministic and stochastic relationships, ellipses indicate variables, and rectangles indicate data. The circled letters indicate which relationship connects the variables or data involved. These relationships are as follows: process a, drawing with replacement; process b, calculation of the transmission matrix by rescaling the mixing matrix obtained by reweighting for age and weekday and weekend days; process c, derivation of the immunisation rate from uptake data and match of vaccine; process d, integration of the SEIR model of transmission; process e, ascertainment of cases through ILI recording at GPs; process f, virological testing scheme following a hypergeometric distribution.

https://doi.org/10.1371/journal.pmed.1001527.g001

Transmission Model

A series of studies have linked routes of transmission for infectious diseases with the structure of contacts in the community [15],[23]. The assumption is that when in contact with a susceptible individual, an infectious individual will have a probability q of transmitting the disease. This transmission probability q will depend on the type of contact involved (e.g., in the case of a respiratory virus, a physical contact might be more effective in transmitting than a conversational contact of the same duration) and the type of pathogen.

Sociological surveys can thus replace expert opinion [24] to characterise transmission matrices. However, though well established for sexually transmitted diseases, where the implication and interpretation of “contacts” is more obvious, the notion of “contact” is problematic for other diseases such as respiratory infections. Additionally, because of sample size or biases, the results may need some reworking in order to become directly interpretable. For example, we demonstrated previously that the transmission matrix of the A/H1N1/2009 virus directly inferred from the mean contact matrix derived from the POLYMOD study could not explain the change of dynamics during the summer holidays [25]. However, using matrices produced from resampling (with replacement) the original data did allow the dynamics to be accurately captured.

We thus assume here that there exists among the possible sets a set of resampled participants (with some participants sampled several times) that represents an appropriate structure of contacts in the population in terms of disease transmission. The mixing in our model is thus described by a resampled subset of participants from the original UK POLYMOD dataset. From this list of participants, we uniquely derive a mixing matrix using the methodology developed below (similar to the one described in Wallinga et al. [23] and Hens et al. [14]).

For a set of entries resampled from the original POLYMOD dataset, let T_j be the number of participants in class j, A_k and be, respectively, the age and number of contacts made in age group i for and by participant k. We can then associate to each participant k a weight w_k depending on participant age and the day recorded in the diary:(1)with N_wd and N_we the total number of contacts recorded, respectively, during week days and weekends, and the number of individuals of age A_k.

The re-normalised average number of contacts per day d_ij made by participants from age group j with persons in age group i standardised for age and weekdays is(2)As contact making is symmetric, the number of contacts in the population resulting from people from age group i meeting with people from age group j is the same as the number of contacts made by people from age group j meeting with people from age group i. If we call c_ij the probability that two randomly selected individuals in group i and j get in contact, we get by symmetry c_ij = c_ji. By using the direct formula c_ij = d_ij/T_i, symmetry will not usually be achieved because of reporting or participation biases. To achieve symmetry of the contact matrix {c_ij}, we thus set(3)To obtain the transmission matrix, we finally multiply this mixing matrix by q, describing the transmissibility of the virus, which is the probability that a contact between an infectious person and a susceptible person leads to transmission. The transmission matrix is thus broken down into its biological and social components.

In this study, we consider an average contact matrix over the epidemic season, thus not considering holiday periods or weekends. When the epidemic is simultaneous with long periods of holidays (as during the A/H1N1/2009 pandemic in the United Kingdom), it is necessary to make the distinction between term and holiday contact matrices [25].

At each step in the MCMC chain the matrix is updated by resampling from the POLYMOD data with replacement (Figure 1, process a) and is thus inferred within the MCMC algorithm via a novel random-walk type of proposal on the matrix space where the number of contacts between distinct age groups is re-normalised, reducing potential reporting or participation biases via symmetry. The contact matrix is combined with the demographic data and scaled by the transmissibility of the virus to give the transmission matrix of the virus (Figure 1, process b).

Vaccination Model

The different epidemiological compartments of the model are split into two types of compartments based on vaccine history (indexed by N for naive or V for vaccinated; see Figure S1 in Text S1). As the vaccine is not 100% effective, a proportion α_i of the vaccinees become protected (we assume full protection), while the rest 1−α_i remain fully susceptible. The vaccine efficacy α_i depends on age group and the degree of match between the strain in the vaccine and the circulating strain in that year.

As in [25], we assume a 2-wk delay between infection and development of protective antibodies. This is based on an analysis of the dynamics of seroconversion following cases of A/H1N1pdm infection confirmed by PCR [26]. We do not model the dynamics of antibody production as in [12] as these measurements are done for cases following infection rather than vaccination, and vaccine trials usually measure level of antibodies after a longer period. The period of 2 wk to confer protection appears to be a reasonable assumption for our purpose.

Combining the vaccination uptake and the vaccination match results in a time varying immunisation rate v_ik in age group i and risk group k (process c). The rate of immunisation v_ik is assumed to be constant over a monthly period. The total number of persons immunised in each age group by month is taken from Joseph et al. [17] and HPA/DH reports. For the purpose of the transmission fitting, we assume that the change of status (whether as S^V or R^V) occurs two weeks after vaccination.

A recent Cochrane review [27] suggested that vaccine efficacy was 73% in years in which the vaccine was well matched, and 44% in years when there was a poor match between the vaccine and circulating strains. In addition, a recent analysis by Fleming et al. [28] on seasonal influenza vaccine efficacy suggested that efficacy was lower in elderly (46%) compared to younger adults (70%). Since all of the studies included in the Cochrane review were performed on healthy young adults, we assume that efficacy was 70% and 46% in those under 65 y and 65 y or older, respectively, in a well-matched year, which was reduced to 42% and 28% in poorly matching years. We assume that children were immunised with a live attenuated influenza vaccine and that this type of vaccine produced protection similar to that of the current trivalent inactivated vaccine in adults.

When the age grouping of the coverage data differed from that used in our model, we reweighted the coverage values proportionally to the population sizes. Figures for the 65+-y age group from 2004/2005 onwards are taken from the HPA/DH annual report on the influenza programme [29]. For the coverage in high-risk individuals under 65 y for the season 2007/2008, we used the HPA/DH report reweighted depending on the risk group. For the seasons 2004/2005, 2005/2006, and 2006/2007, the coverage in each age group is taken as the figure from 2007/2008 rescaled by the ratio of coverage in the total non-risk under-65-y population in each year and in 2007/2008. S1 in Text S1 gives the final coverage by age/risk group and season assumed in the analysis.

Only infants over 6 mo received vaccination. Infants under 6 mo were assumed to be fully susceptible. Possible protection by remaining maternal antibodies was not considered.

Epidemiological Model

The transmission matrix, the immunisation rate, the susceptibility profile, and the initial number of infections in the different age groups feed into the epidemiological model (Figure 1, process d). The susceptibility profiles and initial number of infections are derived from the inference procedure. The epidemiological model uses an age- and risk-specific SEIR (susceptible–exposed–infected–recovered) epidemic model [25] with gamma-distributed latent and infectious periods. We assumed that the background immunity in the population (inferred by the susceptibility profiles) results in a reduced probability of infection rather than full protection.

The epidemiological model describing the transmission dynamics of the virus is similar to that used during the 2009 pandemic [25]. The model has a modified SEIR structure. We assume random mixing (within an age group) between the clinical groups. The size of these groups is given for each season by demographic figures from the Office for National Statistics (average figures over the period of the study are presented in Table 1). To allow the latent and infectious periods to be gamma distributed (rather than exponential), we assume that each of the E and I compartments are defined by two classes (hence SEEIIR structure), with the same rate of loss of latency (γ₁) and infectiousness (γ₂) in both groups. Hence, the average latent period is 2/γ₁, and the average infectious period is 2/γ₂. Following Ferguson et al. [30], we chose γ₁ = 2.5 and γ₂ = 1.1, corresponding to a latent period of 0.8 d and an infectious period of 1.8 d.

We assume that at the outset of the influenza epidemic a small fraction of individuals in each age class is infectious, and the remainder are susceptible. This fraction is obtained by scaling the initial infectious population by a factor l. Preseasonal susceptibility is not necessarily assumed to be absolute, and may vary by age. An age-dependent susceptibility profile {σ_i} is assumed, the parameters of which are estimated from the model-fitting process for all strains and years, except for H3N2 in 2003/2004, for which a pre-epidemic serological profile was available [19] (more detail about the derivation of susceptibility priors for this year is given in Section 2.2.3 of Text S1). As susceptibility has to be inferred, we chose to limit the model to three age bands to avoid overfitting. We considered an average susceptibility for children (0–14 y old), younger adults (15–64 y old), and elderly adults (65+ y old).

Uncertainty in estimates of these quantities reflects the joint uncertainty in the parameters from which they are derived. Therefore, any correlation structures that may be present are appropriately propagated. We start our model at week 35 of the epidemiological season (rather than week 40, which is usually considered as the starting date of the flu season) to match the start of the epidemic with the reopening of schools, the period at which we considered that ILI starts to increase.

The equations of the SEIR epidemiological model are given by Equation 1 in Text S1. The models age-group-specific force of infection is given by(4)where q is the transmissibility parameter, c_ij is the rate at which individuals in age group i make contact with those in age group j, and σ_i is the susceptibility of age group i.

The incidence Z_ik(n) of new infections in age group i and risk group k at week n is(5)Note that the general practitioner (GP) consultation and swabbing data are only available in five age groups. The seven age groups in the epidemic model are thus collapsed into these five age groups for the purpose of the observation model, by combining the 1 and 1–4 y age groups and the 15–24 and 25–44 y age groups. (For simplicity of notation, we do not explicitly write the step that consists of collapsing the seven age groups into five. This leads us to a slight abuse of notation. The i index of Z_ik(n) in the epidemic model is varying between 1 and 7 and is thus different from the i index of Z_ik(n) in the observation model, which is varying between 1 and 5. Passing from one to the other is simply done by adding the sizes of the populations that are being grouped.)

Observation Model

The last step of the model is to link syndromic surveillance data with the number of infections due to circulating influenza viruses in the population. Although GP consultations for ILI are often used as a proxy to monitor transmission of influenza in a population, individuals consulting for ILI and individuals infected with one of the circulating strains from the influenza family are typically two different sets. A large proportion of individuals recorded as having ILI by their GP consult for symptoms resulting from infection by pathogens other than influenza (e.g., respiratory syncytial virus). Also, during an influenza epidemic, only a fraction of cases present symptoms [31], among whom only a fraction will consult in general practice. Of them, another fraction will be recorded as ILI (others being recorded as having other respiratory symptoms), and if PCR-tested, not all of them will end up positive because of the sensitivity of the test. We derive hereafter a statistical model to rigorously link syndromic surveillance and influenza infections in the population.

The reappearance of influenza in temperate countries from one year to another is determined by the introduction by people travelling of some of the new variants from strains circulating globally [32]. Patterns of circulation between the two global hemispheres and persistence in some regions appear to be governing the global circulation of influenza. In a country such as the United Kingdom, this translates into exposure throughout the year to importation from travellers. The H1N1/2009 pandemic, with its intense testing and tracking of travellers acquiring influenza abroad, shed light on the pattern of introduction of the virus. In-country epidemics of influenza are characterised by a balance of constant re-introduction from outside and local in-country transmission until a widespread epidemic emerges (for a comparative description of the initial phase of the epidemic during the 2009 spring in 12 European countries, see [33]).

We thus considered that in addition to the main epidemic modelled by the (deterministic) equations described in the previous section, each individual in the monitored population is associated with a weekly risk of being infected outside of the national (deterministic) epidemic that we are modelling. This risk can be associated with travelling abroad (e.g., in the southern hemisphere during the influenza season in the northern summer) or with a local outbreak independent of the national one. This risk is modelled by a probability ψ that we assume, for simplicity, is independent of time and age.

If, for ease of exposition, we describe by θ the parameters of the epidemic model, the incidence among the monitored population of age group i at week j of new cases arising from the main deterministic epidemic (assumed to be homogeneous across the country) is(6)with θ the epidemiological parameters (q, σ_i, l, and A), and the size of, respectively, the monitored and the total population in age group i and week j, and the function rounding to the nearest integer.

At the same time, the incidence of infection acquired from outside of the main epidemic follows a binomial law of probability ψ:(7)The ascertainment of a case by the surveillance system is dependent on two steps. First, the infected case needs to go to the GP, and the GP needs to record the case as ILI. Second, a swab sample has to be taken and confirmed as containing influenza viruses. Because of these two steps, the number of ascertained cases is usually much smaller than the number of ascertainable cases: not all persons presenting with ILI are tested for influenza. The number of ascertained cases depends on the testing scheme, while the number of ascertainable cases depends on the number of individuals with true influenza infections in the population that present to their GP. We thus are interested in the number of ascertainable cases and assume that each individual in age group i infected by the strain of influenza studied has a probability i of being ascertainable, i.e., going to the GP, being recorded as ILI, and having a detectable viral load (the number of ascertainable cases is thus the number of positives that we would get if all the patients with ILI were virologically tested for influenza in a given week).

The number of ascertainable cases that get infected outside of the main epidemic can then be modelled as binomial of probability as the number of cases with infection from outside is a binomial of probability ψ, and the number of ascertained cases, conditional on the number of cases, is also binomial with probability :(8)As ψε_i is small and is big, can less cumbersomely be defined by a Poisson distribution of rate :(9)This approximation is accurate for all the values considered in this paper.

Among the m_ij individuals reported with ILI at week j among age group i, we are interested in the who have a detectable viral infection for a certain subtype of influenza. We assume that they are a binomial sample from the total number of infected (equal to ) with ascertainment probability , assumed to be constant over time (Figure 1, process e). The virological testing scheme can be seen as randomly drawing n_ij samples without replacement from the population of the m_ij individuals consulting for ILI, in which have a virologically detectable infection. The number of positive samples that are expected can thus be represented using a hypergeometric distribution (Figure 1, process f). The consultation and positivity model is then described by the following system of equations:(10)A representation of the surveillance system in terms of sets is given in Figure 2. Following this set representation, we can express as a product of elementary epidemiological quantities (derivation in Text S1):(11)where P(E|A ∩ C ∩ D) is the GP recognition ability, P(D|A ∩ C) the propensity to consult among symptomatic influenza cases, P(C|A) the proportion with symptoms among the infected, and P(B|A ∩ C) the sensitivity of the test against clinical influenza cases.

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Figure 2. Venn diagram giving a schematic representation of the different surveillance schemes and clinical statuses.

The relative proportions of the different sets vary from one week to another and are different for each age group.

https://doi.org/10.1371/journal.pmed.1001527.g002

Using this, we can derive an order of magnitude for . In Table 3, we summarise the range of values that could take the elementary epidemiological parameters forming and obtain values between 0.006 and 0.05.

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Table 3. Order of magnitude of the ascertainment probability ε derived from the elementary probabilities of measurable underlying phenomena.

https://doi.org/10.1371/journal.pmed.1001527.t003

Reproduction Numbers and Mixing between Groups

The transmission potential of a pathogen is traditionally summarised by way of its basic reproduction number R₀, the average number of secondary cases following the introduction of an infectious individual in a totally susceptible large population [34]. In a heterogeneous population, R₀ will be a function of each of the reproduction numbers within population sub-groups and of the degree of assortativity of these sub-populations. As a result, similar R₀ values can arise from very different epidemiological situations (see Section 2.1.4 of Text S1). We thus measured, in addition to the basic reproduction number for the overall population, the specific reproductive numbers for children under 15 y and adults (R_C and R_A, respectively) and the degree of assortativity d_R of the two subpopulations.

Generating Alternative Vaccination Scenarios

To estimate the impact of potential changes to the current vaccination strategy, we ran for each of the 14 seasons and each of the three strains, 1,000 simulated epidemics with parameters sampled from the posteriors obtained using MCMC techniques. In addition to the actual strategy (used in the fitting process), we analysed an additional ten strategies. The ten strategies are three “basic” strategies and incremental extensions of them (at 15%, 30%, 50%, and 70% coverage):

• No vaccination: Nobody receives any vaccine doses.
• Pre-2000: The pre-2000 scheme is kept throughout the 14 y, i.e., the risk groups are targeted as they were during these years; the change to targeting the low-risk 65+-y group that occurred in 2000 is not implemented. For the seasons 2000/2001 to 2008/2009, the average coverage from the pre-2000 seasons is kept for this group (29.34% coverage in the low-risk 65+-y group).
• Post-2000: The post-2000 scheme is applied throughout the 14 y, i.e., the risk groups are targeted as they were during these years; the low-risk 65+-y group is targeted in all seasons. For the seasons 1995/1996 to 1999/2000, the coverage of 1999/2000 is applied for this group (70% coverage in the low-risk 65+-y group).
• S1: Low-risk 0.5–4-y-olds are vaccinated at 15%, 30%, 50%, and 70% coverage, incremental on the post-2000 scenario.
• S2: Low-risk 50–64-y-olds are vaccinated at 15%, 30%, 50%, and 70% coverage, incremental on the post-2000 scenario.
• S3: Low-risk 5–16-y-olds are vaccinated at 15%, 30%, 50%, and 70% coverage, incremental on the post-2000 scenario.
• S4: S1+S2, i.e., combination of scenarios 1 and 2.
• S5: S1+S3, i.e., combination of scenarios 1 and 3.
• S6: S1+S2+S3, i.e., combination of scenarios 1, 2 and 3.
• S7: Universal, i.e., everybody among the low-risk population is vaccinated at 15%, 30%, 50%, and 70% coverage, incremental on the post-2000 scenario.

To quantify the efficiency of the assessed programmes, we measured the reduction in infections and deaths induced by one dose of vaccine for each of the different strategies. For this, we used the estimated total reduction in number of infections and deaths during the period (removing the season 2002/2003, where virological samples are too scarce) and divided it by the numbers of doses of vaccines given by the programme over the same period. For the extension strategies, we took the median over the four considered coverages (15%, 30%, 50%, and 70%).

Estimation of the Number of Deaths due to Influenza

Our model provides estimates of the number of influenza infections (N^infec) under different vaccination scenarios for the period 1995–2009. In order to compare these scenarios on the basis of the predicted number of influenza deaths (N^death), we need the case fatality ratio (CFR = N^death/N^infec) of influenza. Ideally, because our model provides estimates of the N^infec by strain (s), age (a), and risk (r) group we would like to use CFRs of influenza as a function of these three variables: CFR_s,a,r. However, such estimates are currently lacking in the literature, mainly because quantifying the magnitude of both infection and mortality attributable to influenza is not straightforward [1].

In theory, it would be possible to extend our current framework to include some influenza mortality data in order to estimate N^death and then deduce CFRs. Unfortunately, in contrast to the weekly age- and strain-specific data available for influenza morbidity, the best information on influenza mortality in England consists only in annual, age-specific estimates for the restricted period 1999–2009 [1]. In this context (i.e., only one data point per year for all three strains and only two-thirds of the period considered), it would be illusory to try to estimate annual CFR_s,a,r with our dynamical model approach. Instead, we propose to combine available estimates of annual N^death by age group in England—obtained from a recently published regression analysis by Hardelid et al. [1] (see Table S2 in Text S1)—with our estimates of the annual N^infec in order to construct and fit generalised linear models whose coefficients equal the desired CFRs.

More precisely, because there was evidence of overdispersion in the mortality time series of Hardelid et al. [1], we start with the following age-specific negative binomial regression model with identity link and no intercept:(12)where the negative binomial distribution is parametrized by its mean (μ_a) and dispersion (ϕ_a), the index a denotes the age group considered (we fit each age-specific model independently), and t runs over the restricted period 1999–2009. Given that (Hardelid et al. data) and (our model outputs) are “known” (with uncertainty in both sources), the objective is to estimate the coefficients β_s,a,r that, given the identity link and the absence of intercept, correspond to the desired CFR_s,a,r values.

Note that the age groups in the study of Hardelid et al. [1] (15, 15–44, 45–74, 75+ y) do not exactly match those of our model (1, 1–4, 5–14, 15–24, 25–44, 45–64, 65+ y). We tackle this issue by (1) aggregating the three youngest age groups in our model and (2) refitting the Hardelid et al. regression model for the age groups 45–64 and 65+ y (H. Green, Health Protection Agency, personal communication).

In principle, because the severity of a strain is expected to change from one year to another, CFRs should also depend on the season t, thus greatly increasing the number of parameters. Here, we instead assume that the severity effect can be captured by the overdispersion parameter φ_a. Such a simplification is required for two reasons: first, we have a single (uncertain) estimate of per year, and, second, these estimates are not available before the 1999/2000 season [1]. Put another way, we assume that the CFR_s,a,r values are constant from 1999 to 2009 and can be applied to the 1995–1999 period.

In addition, one can see that each age-specific model (Equation 12) suffers from overparametrization as it intends to estimate six parameters (three strains times two risk groups) from only nine data points (of the ten seasons provided by the study of Hardelid et al. [1], only nine are used since our estimates of N^infec for the 2002/2003 season are not reliable because of a lack of virological data during that particular season). To tackle this issue, we assumed that the risk ratio CFR_s,a,2/CFR_s,a,1 depends only on the age group a and is equal to R_a: the age-specific risk ratio of the CFRs for acute respiratory infection among patients not in a high-risk group versus patients in a high-risk group. The R_a values were obtained by analysing extracts from the Hospital Episode Statistics database (Health Social Care Information Centre) for England, for the period April 2000–March 2009. Patients were included when they had an acute respiratory ICD-10 diagnostic code (J0*, J1*, J2*, J3*, J40*, J41*, J42*, J43*, J44*, J47*) in any diagnostic fields and were divided into risk group/non-risk group based on a risk-group-related ICD-10 code. (We used the following ICD-10 codes for risk groups [where a two-digit code is given, it includes all ICD-10 codes beginning with that code]: D73, J4, J6, J7, J8, Q30, Q31, Q32, Q33, Q34, Q35, Q36, Q37, I05, I06, I07, I08, I09, I11, I12, I13, I20, I21, I22, I25, I27, I28, I3, I40, I41, I42, I43, I44, I45, I47, I48, I49, I5, I6, Q2, N0, N11, N12, N14, N15, N16, N18, N19, N25, Q60, Q61, K7, P788, Q44, E10, E11, E12, E13, E14, E24, G1, G2, G3, G4, G5, G6, G7, G8, G9, N083, O24 P700, P701, P702, C, D37, D38, D39, D4, B20, B21, B22, B23, B24, Z94, Z85, D73, Z9621, G960, D561, D578, D571, D570, K900, D70, D71, D72, D76, D80, D81, D82, D83, D84.) Mortality was identified by discharge method 4, and only mortality within 30 d of admission date was considered attributable to the admission (results of the analysis are given in Table S3 in Text S1).

Following this further simplification we obtain a new death model, with only three coefficients to estimate:(13)Finally, because the values of , , and R_a come with confidence intervals (CIs), we need to account for this uncertainty when computing CFRs. For each age group a, we proceed as follows. (1) We generate K = 1,000 sample datasets , where and are sampled from a normal distribution with the same mean and 95% CI as in Tables S2 and S3 of Text S1, whereas values are obtained by running our model parameters sampled from the joint posterior distribution of our MCMC analysis. (2) We fit the death model for each replicate k by maximum likelihood using the function glm.nb of R 2.15.1 [35].

We obtain the distribution of the K maximum likelihood estimates of Figure S2 in Text S1. These distributions are used to account for uncertainty when computing the distribution of the cumulated number of influenza deaths over the 1995–2009 period predicted under each vaccination scenario:(14)where t = 7 corresponds to the 2002/2003 season and is therefore excluded. The 95% credible intervals of the cumulated number of influenza deaths are calculated for each scenario using the 2.5% and 97.5% quantiles of the posterior distributions of the quantities of interest.

Statistical Inference

The multiple sources of observation give rise to an intractable likelihood, which we explore via data augmentation [36] techniques. In order to manage to integrate numerically the likelihood, we rewrite the likelihood by marginalising some of the augmented variables. Then, we derive a recursive definition of each of the elementary components of the likelihood in order to minimise the number of computational steps involved in the estimation of the likelihood of the model. Finally, we modify the summation surface of the likelihood to derive an accurate approximation of this likelihood by truncating some of the terms with very low probability (see Text S1 for more details of the statistical inference). This allows us to accelerate the computation of the likelihood in order to run 11-million-length chains for each strain and season (1 million for the burn-in). Samples of size 1,000 are obtained by thinning the chain by 10,000.

Serology data were available for the H3N2-dominated 2003/2004 season and facilitated inference for the transmissibility and ascertainment probabilities during this season. The transmissibility between an infectious individual and a susceptible contact is assumed to be constant irrespective of age.

For the remaining 41 subtype-seasons, the estimates of the transmissibility and ascertainment probabilities derived from the 2004/04 H3N2 season with serology were used as priors, allowing us to infer the age-specific susceptibility profiles at the beginning of each of the 41 subtype-seasons.