Prioritization for influenza vaccinations and improvement in coverage levels continue to be the subjects of investigation (Longini & Halloran 2005
; Patel et al. 2005
; Halloran & Longini 2006
; Dushoff et al. 2007
; Hadeler & Mueller 2007
). Until 2008, the priority groups for influenza vaccination allocation according to the US Center for Disease Control (CDC) guidelines were children 0.5–4 years old, adults over 50 as well as people with certain medical conditions. Data up to 2007 suggest that adults aged 18–49 had coverage levels on par with children aged 5–17 (CDC 2008
). In 2008, all children aged 0.5–17 were included in the priority group, with no data available yet on the effect of the recent prioritization policy on coverage levels. For the 2009 H1N1 vaccination plan, individuals aged 0.5–24 make up one of the priority cohorts for vaccine allocation (CDC 2009
Motivated in part by the need for evidence-based guidance on those issues, we have devised a method to prioritize the allocation of a sizeable quantity of a vaccine or antivirals in a stratified population. As our strategy is identical for vaccine and antivirals (see the electronic supplementary material), we concentrate on vaccine for the rest of this paper. The setting we consider in this paper is a large population, where interactions between people and influenza dynamics are represented by a network; the explicit example we use is the EpiSims network describing the population of Utah (Barrett et al. 2008
). The population is stratified (usually by age), and vaccine is distributed between the strata and then allocated at random within each stratum—thus, we cannot relate individuals who receive the vaccine to particular nodes on a network describing the population. We use simulations on the network to estimate the next generation matrix, whose entries represent the number of persons in one stratum infected by an average infectious person in another stratum during the exponential growth period of the epidemic. Using this matrix, we present an explicit algorithm for an allocation of a sufficiently large quantity of a vaccine between the population strata, namely we specify the (generally unequal) coverage levels in each stratum as a function of the total vaccine quantity available. In the process we also determine prioritization (ranking) for vaccine allocation among the strata—the available vaccine is distributed simultaneously for all strata, but the strata with the higher ranking get higher coverage levels. The latter prioritization may be the most important practical outcome of our approach given a number of uncertainties related to the exact allocation proportions, such as the total number of people who will get vaccinated, and the precision with which a network describes a real population. We then apply our algorithm to the EpiSims network for the population of Utah, and test its optimality by simulations.
The theoretical framework for our strategy is the optimal vaccination policy for reducing the epidemic's reproductive number in a stratified mass-action model. Equivalently, the initial growth rate of the infection is minimized when the infection is introduced into a population vaccinated (or treated) according to this allocation—all this is explained in detail in the electronic supplementary material. We note that the emphasis is not on the smallest quantity of a vaccine needed to get the reproductive number below 1 (such a quantity may be unavailable, and the precise reproductive number of the epidemic may be unknown before the epidemic starts); rather, the emphasis is on the policy for distributing some quantity of a vaccine between the strata to minimize the initial growth rate of the epidemic. Formulating such a policy with this approach requires knowledge (up to a scaling factor) of the next generation matrix, whose entries are defined as the number of new infections in one stratum caused by an average infected individual in another stratum during the exponential growth period of the epidemic; it also requires knowledge of the initial distribution of susceptibles among the strata. Our method is appropriate for a ‘sizeable’ (sufficiently large) quantity of vaccine; the optimal allocation of a ‘small’ quantity may be different.
A related allocation policy in a stratified mass-action model appeared before in Cairns (1989)
; we have derived our policy following a more general model introduced in Wallinga et al. (submitted)
, which is more flexible and allows for the treatment of a wider class of next generation matrices and varying vaccine efficacies among the strata. A key concept that emerges from the strategy is the least spread line. This is a line in an N
-dimensional space, where N
is the number of strata in the population; points (vectors) on that line represent the distributions of susceptibles to be left in the strata after an optimal allocation of a sufficiently large quantity of a vaccine; movement along the line corresponds to the varying total amounts of vaccine—see §2.2 for more details. The least spread line can be found explicitly using the next generation matrix and the initial distribution of susceptibles between the strata, by solving several systems of linear equations—see the electronic supplementary material. One can then compare the least spread line with the initial distribution of susceptibles, and devise a prioritization scheme for vaccine allocation between the strata.
This paper shows that these theoretical results can be proven for a stratified population that interacts via stratified mass-action-type kinetics, in which individuals within each stratum are exchangeable. Disease transmission in real populations departs from such kinetics in several ways. Specifically, the local depletion of susceptibles among the contacts of infectious persons (owing to the clustering of transmission within infected households or school classrooms) violates exchangeability; moreover, an average infected person differs from an average person, in terms of their contacts. To test the robustness of the method to departures from its assumptions, we tested it on an explicit, individual-based simulation on the EpiSims platform, in which influenza dynamics were simulated in the population of Utah, and in which we have focused on the stratification of the population by age (0–6, 7–13, 14–18, adults). In the absence of information on prior immunity, we simulate a situation in which everyone is susceptible—this may be relevant, possibly with the exception of persons over 50, to the 2009 H1N1 influenza outbreak. To find the least spread line, one needs an estimate of the next generation matrix K = (kij), where kij equals the number of persons in stratum i infected by an average infectious person in stratum j during the exponential growth period of the epidemic. To assess K, we have run 100 epidemic simulations with five randomly chosen persons initially infected in Utah. There was a large degree of initial stochasticity, but in the early exponential growth period, the next generation matrices were similar in different simulations.
We note that the ‘dynamical’ next generation matrices estimated in this fashion using an exponential growth period on a ‘large’ network are not rigorously defined in this paper, in part because there is no explicit analytical structure describing the network (such as a community of households)—we consider this issue further in the discussion. The dynamical next generation matrices estimated by simulations can be compared to the more classical ‘static’ stratified mass-action next generation matrices (Diekmann & Heesterbeek 2000
; Wallinga et al. 2006
; Mossong et al. 2008
), which are obtained by measuring the number of contacts in the model between a typical member of each stratum and the members of all strata, without any simulation of infection dynamics. The dynamical matrix differs from the static matrix because as the epidemic runs, local structure in the network becomes involved (infected individuals will cluster, e.g. in households, so that even at the early phase of the epidemic many contacts will be with immune persons) and because within a stratum, an ‘average’ person has different characteristics from an ‘average infected person’ (e.g. the average infected person in a stratum is likely to live in a bigger household than the average person in the same stratum). To illustrate that point, we have computed both the dynamical and the static next generation matrices and formulated ‘optimal’ allocation strategies for both. The matrices were quite different, giving different prioritization for vaccine allocation for the strata we have chosen. We have shown by simulations that the dynamical one is very nearly optimal, and in particular it gives lower initial growth rate for the epidemic than for the static one.
The dynamic estimate of the least spread line in our simulations consists of all multiples of the vector (0.029, 0.115, 0.041, 0.814). This means that given a number T of susceptibles left after a vaccine allocation, it is optimal for minimizing the initial growth rate of an epidemic to have 0.029T susceptibles in age group 0–6, 0.115T susceptibles in the age group 7–13, etc. This constitutes a simple, explicit guideline for optimal vaccine allocation. The initial distribution of individuals according to those four age groups in Utah is (0.13, 0.125, 0.094, 0.651). This renders an explicit formula for optimal coverage levels in the four strata as a function of the total vaccine quantity to be distributed. In particular, top priority (highest coverage levels) in an allocation of a sizeable quantity of seasonal influenza vaccinations (enough to reach the least spread line, which in our case means coverage of 20.1 per cent or more of Utah's population) goes to young children (0–6), followed by teens (14–18), then children 7–13, with the adult share being quite low. We also note that for a ‘small’ quantity of influenza vaccinations, with no prior immunity, top priority goes to teens, who have a disproportionate number of infections in the early stages of simulated epidemics, with no vaccine used—see the electronic supplementary material for more details.
We want to point out that minimizing the initial growth rate of an epidemic is not equivalent to minimizing the epidemic's final size (see Ball & Clancy 1993
; Britton 2001
for relations between the latter and the reproductive number), though the two are related. We have tested our ‘optimal initial growth rate’ strategy by simulations against other vaccination strategies with the same total quantity of a vaccine used. In one set of simulations, we have allowed for seasonality in transmission parameters. One hundred epidemics were simulated for each vaccination policy, and the functions C
) representing the average cumulative number of persons infected by day t
were plotted. We have found that the ‘optimal’ strategy indeed had the smallest initial growth rate. Moreover compared to policies which were ‘far enough’ from it, it did better in terms of C
) at all times, and correspondlingly in terms of the final size. For some of the policies which were ‘close’ enough to the optimal, the function C
) eventually descended below the one for the optimal policy, with a modest improvement in the final size. Previous work (e.g. Dushoff et al. 2007
) has shown that optimizing the final size is extremely difficult and sensitive to small differences in parameter values. Moreover, the final size calculations (whether or not weighted by severity) rely on the strong assumptions of fixed conditions (no effects of seasonality, behaviour change, control measures, etc. and no further availability of vaccines), which will be violated in practice. Because these are uncertain, a strategy focused on the present (minimizing the initial growth rate) may be justified as more reliable than the one designed to optimize long-term outcomes.
Finally, our findings were compared to the current seasonal influenza vaccination coverage levels in the USA (CDC 2008
). Data up to 2007 suggest that adults aged 18–49 had coverage levels on par with children aged 5–17. Our simulations/analysis, which rely on the structure of the EpiSims network, in particular suggest that if we vaccinated the same number of people as currently receiving the vaccine, but reduced the adult share considerably in favour of children, transmission could be significantly reduced; moreover, incorporating the existence of prior immunity should only strengthen this conclusion, assuming that prior immunity on the average increases with the age for children, and is lower than the one for adults. For the same reason, our prioritization of teens as the second most important compared to 7–13 year olds is questionable, though both groups should receive attention, and the top priority should remain with preschool children. In that context, we are supportive of the CDC decision (CDC 2009
) to include young individuals in the priority cohort for H1N1 influenza vaccination.