describes the range and distribution of
household structure in Great Britain. Households consisting of just one or two individuals
dominate, with decreasing numbers of households with larger occupancy (
*a*). We note that the 2001 census does not contain precise information on
households containing eight or more occupants, and therefore assume in our model that values
of ‘eight or more’ are exactly eight, which makes little quantitative or
qualitative difference to our results compared to any other realistic approach. This
variability in household size is clearly important: large households have a greater chance
of being infected (as there are more members to potentially bring infection in from the
community) and a higher rate of internal transmission (due to the greater number of
contacts). However, quantifying the impact of these features requires the type of detailed
mathematical model developed in the next section (and online Supplementary information).

Given the importance of large households it is important to consider their composition in
more detail and to determine relationships with other demographic measures. Considering
dependent children (
*b*) we find, as expected, that larger households tend to have more dependent
children. As such less than 10% of households of five or more are solely occupied by adults,
whereas over 90% of households of size two have no dependent children. For this we can see
that numbers of dependent children and household sizes are closely linked. We observe, as
shown in
*c*, great geographic diversity in the proportion of dependent children in
each ward – while the average is around 23%, extremes as low as 5% and as high as
40% exist. Finally, we observe that the proportion of dependent children within a ward
closely correlates with the average household size in that ward (*d*), although we note that there are several
points this figure exhibiting large mean household size but with few dependent children; the
demographic features (such as student houseshares) that can lead to this are discussed more
fully in the Supplementary information.

It is against the above background of heterogeneous host demography that our mathematical
model operates.

Geographic distribution of early growth rates

Our model of household-based transmission is relatively simple and parsimonious, and aims
to identify the effects of different household patterns across Great Britain. Two
transmission rates are used: transmission between members of the same household and
transmission to general members of the local population (ward). Transmission between wards
is not included, for model simplicity and transparency of results. While movement between
wards and continuous importation of infection from abroad are likely to have a significant
impact on the behaviour of pandemic influenza, these operate at a different scale from
household-level transmission and so as a first approximation can be ignored. The general
spread of infection within the ward community is modelled as frequency-dependent
transmission, in accordance with general modelling of large human populations [

19,

20],
while transmission within the household is assumed to be density dependent, such that
individuals interact in a pairwise manner irrespective of household size. In practice,
household transmission probably lies between the extremes of frequency and density
dependence [

21], but the precise scaling is
likely depend on the type of household and ages of occupants. By assuming such
density-dependent transmission we are maximizing the degree of heterogeneity –
other assumptions produce weaker results but do not effect the qualitative conclusions.

Although our modelling framework can deal with a range of dynamic aspects of infection,
here we simply consider the early (asymptotic) growth rate of infection within each ward.
In particular, a range of standard theory shows that starting with a low level of
infection within the population, and following some initial short-lived transients, the
disease incidence and prevalence is predicted to increase exponentially [

20,

22,

23]; it is this early exponential growth rate
that is of primary interest here. In particular, we seek the ‘basic reproductive
ratio’,

*r*_{0}, defined such that the early growth of
infected cases (

*I*) is given by

*I*(

*t*)~exp([

*r*_{0}−1]

*gt*)
where 1/

*g* is the average infectious period. We note that this value of

*r*_{0} defined in terms of growth rates differs from the

*R*_{0} defined in terms of number of secondary cases; although
both agree at the invasion threshold

*r*_{0}=

*R*_{0}=1 (see online
Supplementary information for a more detailed discussion). We decided to use

*r*_{0} as its definition most closely matches observations taken
during an epidemic.

While a relatively simple formulation, our model is parameterized to match observables
concerning pandemic influenza. Using the national distribution of household sizes, we fix
the household and community transmission rates to yield a 40% chance of transmission
between any two household members (often called the

*secondary attack
rate*) and a basic reproductive ratio of approximately 2 for Great Britain as a
whole, which is consistent with statistical work in this field [

1,

2]. Our qualitative
conclusions are robust to the precise choice of parameters.

shows the geographical distribution of
basic reproductive ratios (*r*_{0} values) in each ward in Great
Britain. We observe in
*b* an approximate 25% variation in *r*_{0} between
the mean and most extreme wards, which corresponds to a 50% variation in early epidemic
growth rates. In general, high growth rates reflect a greater than average abundance of
large household sizes and high proportion of dependent children, although the precise
relationship is complex and nonlinear. It is clear from both the locations of wards with
highest *r*_{0} (
*a*), and the discussion in the Supplementary information, that high values
of *r*_{0} tend to be associated with the large conurbations of
Great Britain, with the areas of highest *r*_{0} having around 20
times the mean population density of Great Britain. Epidemiologically, those wards with
the highest *r*_{0} will require the greatest levels of control and
therefore may be targeted with high priority during an epidemic; in addition, the fact
that these high *r*_{0} wards are generally in urban areas may mean
that pandemic influenza (or any other novel pathogen) is likely to enter such wards early
in a national epidemic.

Control by vaccination

Prophylactic vaccination may be a key epidemiological tool in combating any future UK
epidemic, either to eliminate completely the risk of a large-scale epidemic or to be used
in conjunction with other methods such a social distancing, antivirals or contact tracing
[

24]. For simplicity, we assume that an
effective vaccine is available. Reducing this efficacy does not change our qualitative
results but will make any vaccination strategy less effective.

*a* considers three methods of targeting the delivery of vaccination within
wards, with the results for each ward displayed as a point. The results of our household
model agree with previous findings in terms of the critical level of vaccination coverage
required to prevent an outbreak [

25].
Vaccinating entire households at random (green) is an inefficient means of targeting. This
is because effective herd immunity at the household level can be achieved without the need
to vaccinate all household members; in essence vaccine is being wasted on individuals who
already have some protection through being in partially vaccinated households. Vaccinating
individuals at random (red) is a simpler and better strategy, and is found to outperform
the expected vaccination threshold (black line) predicted by simpler unstructured models
[

19,

20,

26]. The improvement over the
prediction from unstructured models is because random vaccination of individuals
effectively biases vaccination towards larger households, thereby targeting control at
these most epidemiologically important units. However, an ideally targeted strategy [

25] – prioritizing individuals in
households with the most susceptibles – has even greater benefits with the
required level of critical vaccination never exceeding the random-mixing prediction of
50%. We see overall that ideal targeting can reduce by about 40% the amount of vaccine
required nationally.

Unfortunately, the optimal method of targeted vaccination is both impractical and
unworkable. Therefore we seek an alternative proxy that incorporates insights from the
ideal strategy and readily allows vaccination to be targeted towards a proportion of
individuals in the larger households. From
*b* we predict that vaccinating children biases protection towards the
larger households, yet does not waste vaccine immunizing all members; in addition it is
likely to be both ethically and socially acceptable. With this in mind, we consider three
forms of vaccination at the ward level: (1) vaccination of dependent children (who account
for about 23% of the GB population); (2) heterogeneous random vaccination, where
individuals are vaccinated at random with the proportion vaccinated equal to the
proportion of children within the ward; and (3) homogeneous random vaccination, where
individuals are vaccinated at random in every ward, such that the proportion vaccinated
nationally matches the proportion of dependent children. Alternatively, we can consider
heterogeneous and homogeneous vaccination as randomizations of the vaccination of
dependent children; heterogeneous vaccination randomizes the distribution of vaccine
within each ward, whereas homogeneous vaccination randomizes the distribution of vaccine
over the whole of Great Britain. As such, comparing these three strategies allow us to
access the impact of targeting children, both in terms of efficient deployment within a
ward and also as a means of proportioning vaccine between wards. Even though all three
strategies ultimately vaccinate the same number of individuals (around 23% of the
population), it is clear that targeting has advantages (
*b*). We measure the efficacy of vaccination through
*r*_{V} (the equivalent of *r*_{0}, but
after vaccination). Vaccinating dependent children causes a 35% drop in this reproductive
ratio (from ≈2 to ≈1·3), whereas both homogeneous and
heterogeneous vaccination only cause a reduction of around 25%.

Comparing homogeneous and heterogeneous vaccination in more detail allows us to assess
the impact of targeting wards with the most children, without targeting large families
within those wards. The histograms and box-whisker plots of *r*_{V}
show that the targeting inherent in heterogeneous vaccination offers negligible mean
benefit over homogeneous vaccination (
*b*). However, this ward-level targeting does significantly reduce the
variability in epidemic growth rates bringing those wards with extremely high growth rates
under greater control.

(*c*,
*d*) considers the behaviour at the ward level, with particular focus on
*r*_{0} before vaccination and the equivalent measure
*r*_{V} after a proportion of the population has been vaccinated.
In general targeting vaccination towards dependent children not only reduces the average
reproductive ratio (*r*_{V}) but also significantly reduces much of
the variability (
*c*). Wards that originally had high *r*_{0} values
due to large average household sizes with many children are now brought much closer to the
average. In only nine wards (red circles) out of over 10 000 is heterogeneous random
vaccination predicted to outperform vaccination targeted towards children –
meaning that at a local as well as a national scale vaccinating children is overwhelmingly
effective. The precise socio-demographic characteristics of these nine outliers is
explored more fully in the Supplementary material, but all these wards have either large
student or older adult households, breaking the general rule that large households are
associated with many dependent children.