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Community interventions and the epidemic prevention potential
Vaccine 20 (2002) 3254–3262
Community interventions and the epidemic prevention potential M. Elizabeth Halloran∗ , Ira M. Longini, David M. Cowart, Azhar Nizam Department of Biostatistics, Rollins School of Public Health, Emory University, 1518 Clifton Road NE, Atlanta, GA 30322, USA Received 8 January 2001; received in revised form 22 April 2002; accepted 18 June 2002
Abstract Evaluation of community-level effects of intervention programs in infectious diseases is receiving increased attention. In this paper, we consider evaluation of the community-level effectiveness measures on the example of vaccination of children against influenza. We introduce the concept of the epidemic prevention potential (EPP) as a measure of the ability of an intervention to either prevent transmission or at least to keep it below a pre-defined limit. As a concept to describe the general ability of an intervention to limit outbreaks to a certain defined size, the term EPP fills a void. We constructed a stochastic simulation model of influenza transmission and vaccination in a structured community to illustrate the effectiveness measures of interest and the epidemic prevention potential. The concepts are general and could be applied to other interventions, such as antivirals and quarantine. © 2002 Elsevier Science Ltd. All rights reserved. Keywords: Community trials; Efficacy; Influenza; Vaccines
1. Introduction Increasing attention is being given to evaluating community-level effects of vaccination strategies [1–5] and other interventions [6–9] in infectious diseases. In light of the current worries about bioterrorism, there is also increased interest in planning interventions in case an infectious agent, such as smallpox, is introduced into a population. Methods to evaluate the effectiveness of such interventions will be required. Much has been written on group randomized studies where whole social units are assigned to treatment groups [9–14]. Considerable theoretical research has focused on establishing threshold conditions for transmission in epidemic models [15–19]. However, little has been written about evaluating interventions when transmission may be a threshold phenomenon. In this paper, we introduce the concept of the epidemic prevention potential (EPP) as one aspect of overall effectiveness when an intervention prevents an outbreak or limits it to less than a certain defined size. We illustrate estimation of effectiveness measures with simulations of vaccination of children against influenza to reduce attack rates in communities [20–24]. We differentiate effectiveness measures that take all communities into account whether or not an epi∗
Corresponding author. Tel.: +1-404-727-7647; fax: +1-404-727-1370. E-mail address: [email protected] (M. Elizabeth Halloran).
demic has occurred from effectiveness measures that condition on an epidemic reaching a pre-defined size threshold. We constructed a stochastic simulation model of influenza transmission and vaccination in a structured community to illustrate evaluation of the indirect, total, and overall effects, and the EPP. Although the focus here is on vaccination, the concepts are general and could be applied to other interventions, such as antivirals and quarantine.
2. Scientific questions of interest Consider a seasonally epidemic infectious agent, such as influenza or cholera. The outcome of interest may be the illness attack rate in the community as a whole or in a stratum within the community, such as in children 18 years old or under. Suppose that several communities receive the intervention of interest and several communities do not receive the intervention. The populations receiving the interventions will be referred to as type A communities, and the control communities as type B communities. We propose three primary questions of interest: 1. What is the effectiveness of the intervention on the attack rates in the communities receiving the intervention? 2. What is the effectiveness of the intervention in reducing the size of epidemics that do occur? 3. What is the effectiveness of the intervention in preventing epidemics?
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Table 1 Measures of vaccination efficacy and effectiveness and comparison groups in an intervention vaccinating children against influenza Comparison group
Effectiveness measure Children Direct VES
Intervention community Baseline community
vac A unvac A
Indirect VEIIA
Total VEIIB
unvac A unvac B
vac A unvac B
Adults
Population
Overall VEIII
Indirect VEIIA
Overall VEIII
All A All B
unvaca
All A All B
A unvaca B
A, intervention community; B, baseline community; vac, vaccinated in intervention; unvac, not vaccinated in intervention. a Adults may be vaccinated before the intervention, but are not a target of this intervention. We could study indirect effects of vaccinating children separately in vaccinated and unvaccinated adults, or any other stratum.
An epidemic will be defined by a threshold attack rate, which could be zero, but generally some small value (see Remark 1). The effectiveness measures in (1) and (2) can be differentiated into indirect, total and overall effectiveness [1–3]. For example, vaccination of children against influenza as a method to reduce cases in adults as well as children has received increasing attention [20–24]. Table 1 summarizes the effectiveness measures of interest for the example of vaccinating children against influenza in communities. The indirect effectiveness (VEIIA ) in children would be estimated from attack rates in the unvaccinated children in the nonintervention communities compared with the attack rates in the unvaccinated children in the intervention communities. The total or combined effectiveness (VEIIB ) in children would be estimated from attack rates in the vaccinated children in the intervention communities compared with the attack rates in the unvaccinated children in the nonintervention communities. The overall effectiveness (VEIII ) in children would be estimated from attack rates in all the children in the intervention communities compared with the attack rates in all the children in the nonintervention communities. Since the intervention program does not target adults, in adults, only the indirect effectiveness (VEIIA ) would be estimated from attack rates in adults in the intervention communities compared with attack rates in adults in the nonintervention communities. For overall effectiveness in the communities as a whole (VEIII ), the attack rates in the whole intervention communities would be compared to the attack rates in the whole nonintervention communities. To provide some notation, assume that the attack rate (ARjk ) in the subgroups k in each community j = A, B in the study is observed. Let RR denote the relative risk based on the attack rates. The subscripts j = 0, 1 denote unvaccinated and vaccinated people within the populations, respectively. The indirect effectiveness, VEIIA is VEIIA = 1 −
ARA0 = 1 − RRIIA . ARB0 ARA1 = 1 − RRIIB . ARB0
VEIII = 1 −
ARA. = 1 − RRIII , ARB.
(3)
where the dot in the subscript denotes the entire population. As in the influenza example, the indirect effects of interest can be estimated in the children and adults separately. Also, the overall effects can be evaluated in just children or in the population as a whole. Subpopulations of interest need to be defined for any particular study. The above effectiveness measures can be estimated either using all communities in the study, or including only those communities with an epidemic above a defined size threshold. The attack rates from several communities can be combined in various ways, which will be touched upon below. For completeness, if only some of the people are vaccinated in communities of type A, then estimation of the direct protective effect of vaccination is possible. For direct effects, the unit of inference is the individual. The direct protective effects, VES , can be estimated by VES = 1 −
ARA1 = 1 − RRS . ARA0
(4)
Interpretation of the VES estimates will need to be cautious, unless the vaccine is allocated randomly to individuals. For scientific question (3), as a measure of the ability of a vaccination strategy to prevent transmission, we define the epidemic prevention potential (EPP) of a vaccination strategy as the relative reduction in the probability of an epidemic in a community compared to what it would be without vaccination. The EPP can also be defined to compare one vaccination strategy with another. Denote the probability of an epidemic in populations type A by Pr(e)A and in populations type B as Pr(e)B . Then the EPP of the vaccination strategy is
(1) EPP = 1 −
The total or combined effectiveness, VEIIB is VEIIB = 1 −
The overall effectiveness, VEIII is
(2)
Pr(e)A . Pr(e)B
We consider the EPP in more detail in Section 3.
(5)
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3. Epidemic prevention potential A few examples illustrate the EPP measure. Suppose that seasonal transmission occurs reliably in each unvaccinated community type B, but it occurs in only about 20% of the intervention communities A. Then Pr(e)B = 1, Pr(e)A = 0.20, and EPP = 1 − 0.20/1 = 0.80. About 80% of epidemics will be prevented by the intervention. Suppose that seasonal outbreaks occur in about 80% of unvaccinated communities, Pr(e)B = 0.80 and 20% of the vaccinated communities, Pr(e)A = 0.20. Then the EPP = 1 − 0.20/0.80 = 0.75. The EPP of the vaccination strategy, being defined relative to baseline, in this second example is lower than in the previous example, even though the probability of an epidemic in the vaccinated communities is the same in both examples. Remark 1. The EPP requires a definition of an epidemic or an outbreak. One possibility is to set a defined size threshold. Transmission of infectious diseases can be quite fickle, as most people who have tried to run vaccine field trials know. There may be no transmission in a given community or there may be very little compared to what would be expected if an epidemic had taken off and become self-sustaining. When a disease is rare, one often tries to define outbreaks as being significantly above baseline. However, in infectious diseases, the definition of an epidemic would most often be related to what would normally have been expected in a particular community. Another option is to define a threshold according to some public health interest. For example, influenza illness attack rates of 15–25% might be expected if an epidemic took off in a community. Then rather than defining zero cases as the public health goal of intervention, holding an epidemic below a certain threshold, say keeping the attack rates below 2.5, 5.0 or even 7.5% might be the epidemic prevention goal. If the number of cases is in fact zero, issues related to analysis are raised (see below). In any event, to determine whether an epidemic has been prevented, a precise definition for the threshold of a successful epidemic is required. The value of the EPP is sensitive to the definition of an epidemic. Remark 2. We define EPP in a relative way, similar to other efficacy and effectiveness measures. However, the absolute difference in the probability that an epidemic will occur, Pr(e)B − Pr(e)A , is also of interest. The relation between the ratio estimator and difference estimator is analogous to other epidemiologic measures. A variety of absolute differences and magnitudes in the probability of an epidemic can yield the same EPP. For instance, if the probability of an epidemic is 0.40 without a vaccination strategy and 0.10 with it, then EPP = 1 − 0.10/0.40 = 0.75, the same as in the second example. However, the absolute difference in the earlier example is 0.20 − 0.80 = −0.60, and in this last example, it is 0.10 − 0.40 = −0.30. Even though the EPP is the same in both examples, the absolute difference
of 0.60 represents a larger public health achievement than 0.30. Remark 3. The EPP of a vaccination strategy is related to the threshold level of vaccination required to prevent transmission. Thresholds for transmission have been of considerable theoretical interest in both deterministic [15] and stochastic [16] modelling of infectious diseases (see also [17–19]). In both types of models, the reproductive number, R, is the expected number of additional infectives produced by one infective in a particular population. In deterministic models, if R ≤ 1, theoretically, transmission will be eliminated, while for R > 1, an epidemic will always take off. But deterministic models do not allow for stochastic effects. In stochastic models, R is related to the probability that an epidemic will take off. If R ≤ 1 in a stochastic system, the epidemic will never take off, in the sense of being self-sustaining. However, a few people may become infected, producing what is called a minor epidemic. If R > 1, there is a positive probability that the epidemic might take off, producing a major epidemic. The probability that the epidemic takes off increases with larger R. As R is reduced close to 1 with high levels of vaccination, the probability of a major epidemic becomes very low. The reproductive number R is generally difficult to estimate from the available data. However, it is easier to establish whether, after an intervention, an epidemic is below or above some defined threshold attack rate. 4. Design and analysis issues In community studies, the unit of inference is the population. Thus, the sample size and power of a study are more closely related to the number of communities in the study than the number of people in each community. The eligible communities need to be transmission-dynamically separate [1,2]. As in community studies of noninfectious diseases, the sample size required to estimate effectiveness has to do with the size of the effect expected and the intercommunity variability. At low vaccination coverage or low efficacies, the indirect and overall effectiveness might be small, requiring a larger number of communities. Estimation of the EPP presents slightly different problems than estimation of the effectiveness measures in Table 1. For each community, the outcome is in a sense 0 or 1, either an epidemic of a certain size occurred or it did not. For any number of control or intervention communities, given an expected probability of an epidemic occurring, one could calculate the expected distribution of the number of epidemics based on the binomial distribution. Using exact statistical methods, one could compute the number of intervention and control communities required to get reliable EPP estimates. The number of communities required will be larger as more baseline communities do not have epidemics and as an intervention is less likely to prevent an epidemic. The number of communities required may be prohibitively large in many
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instances. However, the concept of the EPP of a vaccination strategy has an important public health interpretation as well as an important role in interpreting estimates of indirect, total, and overall effectiveness, as is illustrated below. Given that there are several intervention and several control communities in a study, one can choose among different methods to combine attack rates from different communities to estimate the indirect, total, and overall effects. Transmission-dynamically similar communities could be matched either at the design or at the analysis phase or at both. In a matched analysis, summary measures of effectiveness could be obtained from the weighted average of the risk ratio estimates from all pairs of communities. It is desirable to compute weighted averages, because the populations with more cases contribute more information. Commonly, the inverse variances of the pairwise risk ratios [25] are used. The attack rates have been shown to be approximately binomial proportions when conditioned on the total number of infected people and with mild assumptions about the epidemic process [26,27]. Another paper explores the relation of the number of pairs of communities to the variability of the effectiveness estimates in more detail [27]. Longini et al. [27] also propose a model-based estimator with a random effect on the basic reproductive number. Several problems are raised by the pairwise analysis using the inverse variance weights on the risk ratios. First, transmission may fail to occur in one or both of the communities in a pair. This can be dealt with, though inadequately, by adding a one to the numerator. However, then very large variances may occur. The other problem is that when the indirect or overall effectiveness is low, very negative estimates can occur, destabilizing the summary measures. Also, reciprocal variance weighted averages are efficient, but can be biased. An alternative would be to use the sample sizes as weights. This is less efficient than reciprocal variance weighted averages, but is also less biased [28]. Another alternative is to use the simple average of the attack rates in the intervention communities as the numerator of the summay measure and the simple average of the attack rates in the nonintervention communities as the denominator.
5. Simulated studies of influenza vaccination 5.1. Simulation of intervention We simulated studies of community-level effects of vaccinating children using a stochastic simulator of influenza transmission in a structured population. The effectiveness measures of interest were those described in Table 1 as well as the EPP. The intervention of interest was vaccination of children aged 1–18 years old at different levels of coverage. Each simulated study included four intervention and four baseline communities. For each level of coverage and vaccine efficacy, 100 studies were simulated. For each simulated study, we estimated summary measures of effective-
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ness based on both the matched pair analysis using inverse variance weights of the pairwise risk ratios and the simple average of the attack rates in the intervention compared to the simple average of the attack rates in the nonintervention communities. The matching within pairs in the first analysis would not have been necessary since there was no difference in variation across pairs due to the simulation design. The EPP was computed based on each set of 400 simulations at the different coverages. The direct effects VES were also computed for each set of 400 simulations at the different coverages. Each population of 2000 people was stochastically generated based on the age distribution and approximate household sizes from the US Census 2000 [29]. Each population had four neighborhoods, one high school, one middle school, and two elementary schools. Pre-school children attended either small or large play groups. People mixed within households, neighborhoods, the whole community, and when appropriate, within schools and play groups. Influenza was introduced by randomly assigning 12 initial infectives. The initial infectives were omitted in the analysis. Pre-intervention baseline vaccination coverage was assumed to be 5% in children, 22.9% in adults 19–64 years old, and 68.1% in adults 65 years and over. Vaccination coverage in adults was assumed to remain the same in the intervention communities. The transmission probabilities were calibrated so that, on average, 15% of a baseline population developed influenza illness. Influenza transmission was modeled similarly to that of [20]. Details are in the Appendix A. The pre-intervention vaccine was assumed to have a protective efficacy of VES = 0.70 and efficacy for infectiousness of VEI = 0.80. We simulated two different levels of efficacy of the intervention vaccine. The first intervention vaccine had the same efficacy as the pre-intervention vaccine, that is, VES = 0.70 and VEI = 0.80. The second intervention vaccine had a higher protective efficacy, VES = 0.90, but the same efficacy for infectiousness, VEI = 0.80. The vaccine effects were assumed to work multiplicatively on the transmission probability, both on the susceptibility of vaccinated susceptibles and on the infectiousness of vaccinated people who become infected. The simulated target coverages in children were 30, 50, and 70%, which includes the pre-intervention baseline coverage in children. Vaccination status was randomly assigned based on the probabilities of the target coverage. 5.2. Results Table 2 contains the effectiveness results from the 100 simulated studies of four intervention and four baseline communities each. The indirect, total, and and overall effectiveness estimates increase as coverage and VES increase. In particular, indirect effects in the groups not targeted by the intervention are evident. At 70% coverage of children with VES = 0.70, the mean estimated VEIIA in adults is 0.59. Thus, on average 59% of the cases in adults are prevented
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Table 2 Mean estimates (empirical standard deviation in parentheses) of measures of the effectiveness of vaccinating children based on attack rates at the end of the annual influenza epidemic Coverage (%)
Effectiveness measure Children Directa
VES
Adults
Population
Indirect VEIIA
Total VEIIB
Overall VEIII
Indirect VEIIA
Overall VEIII
30
ave ivw
0.60 (0.19)
0.20 (0.22) 0.16 (0.50)
0.67 (0.11) 0.65 (0.22)
0.25 (0.21) 0.21 (0.45)
0.23 (0.20) 0.21 (0.32)
0.23 (0.20) 0.20 (0.39)
50
ave ivw
0.58 (0.34)
0.41 (0.20) 0.44 (0.33)
0.77 (0.09) 0.77 (0.13)
0.48 (0.18) 0.51 (0.29)
0.42 (0.18) 0.45 (0.27)
0.44 (0.18) 0.46 (0.31)
70
ave ivw
0.63 (0.22)
0.58 (0.14) 0.61 (0.23)
0.85 (0.06) 0.85 (0.09)
0.66 (0.12) 0.69 (0.19)
0.59 (0.13) 0.62 (0.16)
0.62 (0.12) 0.64 (0.17)
70 (VES = 0.9)
ave ivw
0.87 (0.17)
0.71 (0.10) 0.75 (0.15)
0.96 (0.02) 0.95 (0.02)
0.79 (0.08) 0.82 (0.11)
0.72 (0.09) 0.74 (0.11)
0.74 (0.09) 0.76 (0.11)
There were 100 simulated studies with four intervention and four nonintervention communities per study; VES = 0.70, VEI = 0.80, except the last line, where VES = 0.90; ave, simple average; ivw, inverse variance weighted. a VE is included only for completeness. The means and empirical standard deviations are based on 400 separate intervention communities, not S averages of four communities.
by vaccinating the children compared with baseline. The standard deviations of the effectiveness estimates tend to decrease as coverage increases. At low coverages, there are still many negative estimates, even with four intervention and baseline communities each in the study. This leads to high variability of the effectiveness estimates. In fact, it reflects the uncertainty about whether low coverage could provide measurable indirect effects. The means of the simple average estimates approximately equal the means of the inverse variance weighted estimates. However, the standard deviations of the simple average estimates are in general lower than those of the inverse variance weighted estimates, with the difference more pronounced at the lower coverage. The appropriate statistical methods for combining studies of community-level effects of intervention require more research. For completeness and to understand the estimates of total effectiveness VEIIB , the mean of the direct VES estimates from the 400 intervention communities are also included. The VES estimates are lower than the nominal simulated value because the vaccine model is multiplicative (leaky), not all-or-none. The simulated multiplicative VES is for just one exposure to infection. Upon each repeated exposure,
there is an independent probability of becoming infected, so the VES using final attack rates is estimated to be lower than the set multiplicative effect [30,31]. For example, with VES set at 0.70, at 30% coverage, the mean VES estimate is 0.60 in these simulations. This is expected. Thus, the mean estimated total effectiveness in children at 30% coverage can also be less than the set 0.70. However, the mean estimated total effect VEIIB is higher than the mean estimated direct effect. Table 3 shows that both the percent of epidemics taking off and the EPP at different levels of vaccination coverage are sensitive to the definition of an epidemic. Particularly at high coverages and high efficacy, the percent taking off decreases and the EPP increases as the cutoff goes from 2.5 to 7.5%. In the simulated baseline communities, 93.5% have attack rates over 7.5%, while 96.7% have attack rates over 2.5%. One can compute the EPP in Table 3 from the percent taking off. For example, at 70% coverage with VES = 0.90, EPP = 1 − 0.562/0.967 = 0.41 at the 2.5% definition. At the 7.5% definition, EPP = 1 − 0.085/0.935 = 0.91. Thus, the mean EPP of this vaccination strategy at VES = 0.90 varies from 0.41 to 0.91, depending on the definition of an epidemic. If the public health goal were to decrease
Table 3 Percent of epidemics out of 400 taking off and epidemic prevention potential (EPP) based on relative probability of epidemics taking off using different definition of an epidemic at different coverages with VES = 0.70, VEI = 0.80, except the last line, where VES = 0.90 Coverage (%)
Baseline 30 50 70 70 (VES = 0.90)
Percent taking off
Epidemic prevention potential
Definition of an epidemic
Definition of an epidemic
2.5%
5.0%
7.5%
2.5%
5.0%
7.5%
96.7 94.7 84.2 76.0 56.2
94.5 89.5 70.7 51.0 22.0
93.5 80.5 57.2 29.2 8.5
– 0.02 0.13 0.21 0.41
– 0.05 0.25 0.46 0.77
– 0.14 0.39 0.69 0.91
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Table 4 Conditional on an epidemic occurring, mean estimates (empirical standard deviation in parentheses) of measures of the effectiveness of vaccinating children based on attack rates at the end of the annual influenza epidemic Coverage (%)
Def (%)
N
Effectiveness measure Children
Adults
Population
Indirect VEIIA
Total VEIIB
Overall VEIII
Indirect VEIIA
Overall VEIII
30
> 2.5 > 5.0 > 7.5
100 100 100
0.17 (0.20) 0.14 (0.20) 0.06 (0.31)
0.65 (0.12) 0.64 (0.12) 0.62 (0.13)
0.22 (0.19) 0.19 (0.19) 0.12 (0.29)
0.20 (0.19) 0.17 (0.19) 0.11 (0.23)
0.20 (0.19) 0.17 (0.19) 0.11 (0.26)
50
> 2.5 > 5.0 > 7.5
100 99 94
0.30 (0.23) 0.22 (0.23) 0.10 (0.44)
0.73 (0.10) 0.70 (0.10) 0.65 (0.18)
0.38 (0.21) 0.32 (0.20) 0.21 (0.39)
0.33 (0.20) 0.26 (0.20) 0.18 (0.22)
0.52 (0.80) 0.28 (0.20) 0.18 (0.29)
70
> 2.5 > 5.0 > 7.5
95 68 29
0.43 (1.31) – 0.68 (0.26)
0.80 (0.45) – 0.89 (0.09)
0.54 (1.06) – 0.75 (0.20)
0.52 (0.60) 0.48 (0.70) 0.67 (0.22)
0.52 (0.80) 0.43 (1.12) 0.70 (0.21)
There were 100 simulated studies with four intervention and four nonintervention communities per study; VES = 0.70, VEI = 0.80, except the last line, where VES = 0.90; All estimates are based on simple average which may include fewer than four intervention and four nonintervention communities per study. Def, definition of an epidemic; N , number of studies included in analysis.
the attack rates to less than 5.0%, then we would expect between 46 and 77% success with coverage of an influenza vaccine with protective efficacy between VES = 0.70 and VES = 0.90, and a VEI = 0.80. Table 4 shows that the conditional effectiveness measures decrease at 30 and 50% coverage as the cutoff for an epidemic increases. The difference between the unconditional measures in Table 2 and the conditional measures in Table 4 is picked up in the EPP measure. For example, at 50% coverage, using the 5% cutoff, the EPP is 0.25 with a mean conditional overall effectiveness, VEIII = 0.25, in the population. In Table 2, the unconditional average effectiveness is 0.44. In any particular situation, the question is, what is the goal of the intervention, so which is the more appropriate measure? At the higher coverages, increasingly frequently all four intervention communities fail to take off. The conditional effectiveness estimates become very unstable by 70% coverage at VES = 0.70 and are difficult to interpret.
6. Discussion We have introduced the concept of the EPP as an effectiveness measure of interest in community interventions in infectious diseases. As a concept to describe the general ability of an intervention to limit or to contain outbreaks to a certain defined threshold size, the term epidemic prevention potential fills a void. In light of the events since 11 September 2001, and worries over bioterrorism, the concept of the EPP has become particularly relevant. We have delineated three possible goals for evaluation. First, what is the average effectiveness of the strategy in reducing transmission, including both communities with prevented epidemics and communities with reduced but positive transmission? Second, conditional on an epidemic exceeding a certain defined threshold, what is the effective-
ness of the vaccination strategy in limiting transmission? Third, what is the reduction in the probability of an epidemic occurring, that is, the EPP of the strategy? Using stochastic simulations of vaccination of children against influenza, we have illustrated evaluation of the effectiveness measures of interest. The lesson is that at lower coverages, either the unconditional estimates as in Table 2 or the EPP combined with conditional estimates in Table 4 makes sense. At higher coverages when the probability of an epidemic is low, the unconditional estimate or the EPP would make more sense, while the conditional estimates could be omitted. In general, one should not rely solely on the conditional estimates, without including the EPP, as this would be throwing away important information. We have discussed prospective studies that have the opportunity to see whether outbreaks occur or not. Often, post-licensure vaccine efficacy studies are conducted only where outbreaks occur, resulting in underestimates of efficacy because of the ascertainment bias [32]. The extent of underestimation is related to the size of the epidemic triggering the study, the vaccination coverage in the community, and the extent of clustering of vaccination failures in the population. Analogously, if only communities in which epidemics occur are used to study the community-level effects of intervention, then effects will be underestimated. The EPP needs to be used in partnership with measures conditional on an epidemic occurring. A fundamental issue is that when dealing with indirect and overall effects of vaccination strategies in a community, there is no obvious underlying true parameter of interest to estimate[33]. As more community studies of indirect and overall effects of intervention strategies go to the field, additional research is needed to deal with the inherent complex issues. We have used two possible methods of combining the attack rates and relative risks to illustrate calculating summary
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measures of effectiveness for several communities. These and other methods of combining results across communities require further research. Our results are not to be considered recommendations. Since the simulated populations are all the same size, the simple average is similar to using sample size weights. Thus, the comparison to using the inverse variance weights on the risk ratios of the matched pairs would need to be compared using simlautions with different sample sizes or different simulation parameters, as in [27]. The simulation results give insight into the potential effects of vaccinating children in current American communities. Trivalent cold-adapted influenza vaccine has been shown to have a high efficacy in children. The estimated vaccine efficacy for susceptibility, VES , against currently circulating strains of influenza was estimated from a multi-center vaccine trial to be around 0.90 [34,35]. For this vaccine, the vaccine efficacy for infectiousness, VEI , could be about 0.80 [36]. These simulations suggest that vaccination of 70% of children with a highly efficacious influenza vaccine could substantially reduce or prevent transmission in the majority of the communities in which the strategy is implemented. Prospective, designed community level studies will be large and expensive to conduct. Transmission might not occur in either intervention or nonintervention communities. If it does not occur in the vaccinated community, an investigator might be tempted to interpret the lack of transmission as being due to the vaccination strategy. However, transmission might have been absent quite by chance. If transmission is lower in the vaccinated community than in the unvaccinated community, but still occurs, it could still be by chance and not due to vaccination. The inherent variability in the transmission systems across communities needs to be taken into account in determining the number of communities required before going to the field to make the best use of the large amounts of time, energy, and money.
neighborhoods. Small play groups have four children each, and there are between four and six small play groups per neighborhood. Large daycare centers can have as many as 50 children, though the average size is 14 children. School-age children are assigned to either an elementary school, middle school, or high school based on their age. Two neighborhoods share one elementary school, and all four neighborhoods share a middle and high school. Elementary schools have on average 79 children per school, middle schools have an average of 141 students, and high schools have an average of 110 students. For the purposes of our simulator, the ages of children were assumed to be uniform over the intervals 0–4 and 5–18 years of age. Young adults (19–64 years) and older adults (65 years and over) were also uniformly distributed within their respective age groups. On average, in each generated population, 6.92% were under 5 years, 22.08% were 5–18 years, 58.48% were adults 19–64 years old, and 12.52% were adults 65 years or older. The probability that a household has one person is 0.33, two people is 0.34, three is 0.13, four is 0.10, five is 0.07, six is 0.02, and seven is 0.01. The probability that an adult, in a family with children, is 65 years or older is 0.02. The probability that an adult in a household without children is 65 years or older is 0.28. The probability that a two person home has one child and one adult is 0.01. In the simulations in this paper, each population is generated with 12 initial infectives chosen at random. The properties assigned when the population is set up identify each individual. The age-specific vaccination coverage of interest determines the probability that any person will be vaccinated. For each person in the population, a random number between 0 and 1 is generated. If the random number is less than the probability of being vaccinated, the person is vaccinated. A.2. Influenza parameters
Acknowledgements This research was partially supported by NIH grant R01-AI32042.
Appendix A. Simulation model A.1. The population Populations of 2000 people are stochastically generated based on the age distribution and approximate household size published by the United States Census Bureau [29]. A family is a group of up to seven people living together with one or two adults. Each person in the population is assigned to a family within one of four neighborhoods, an age, a gender, an initial disease status indicator, and a vaccination status indicator. Pre-school age children are assigned to either small play groups or large daycare centers within their
The computer code used to run these epidemics is very general and could be used to simulate a great variety of directly transmitted diseases. The simulator has been tailored to simulate influenza. Many of the influenza parameters are adopted from Elveback et al. [20] or the Vespers program [37]. Table 5 shows the distribution of the latent and infectious periods. The probability that an individual will be symptomatic given that person has been infected is 0.67. An asymptomatic infection is assumed to be 50% as infectious as a symptomatic infection. Additionally, this model allows for people to withdraw from all of their mixing groups except Table 5 Baseline influenza parameters (from Elveback et al. [20])
Number of days Probability (all ages) Mean days
Latent period
Infectious period
1 0.3 1.9
3 0.3 4.1
2 0.5
3 0.2
4 0.4
5 0.2
6 0.1
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Table 6 Withdrawal information (Elveback et al. [20]) Age group
Pre-school School age Adults
Probability of withdrawal given illness
Withdrawal day, given withdrawal Day 0
Day 1
Day 2
0.80 0.75 0.50
0.38 0.27 0.20
0.50 0.53 0.60
0.12 0.20 0.20
Table 7 Transmission probabilities in simulated populations Contact group
Children
Adults
Pre-school
School
Younger
Older
Small play group
Large daycare
Elementary
Middle
High
Small play groups Large daycare centers Elementary school Middle school High school
0.0680 – – – –
– 0.0275 – – –
– – 0.00577 – –
– – – 0.00477 –
– – – – 0.00177
– – – – –
– – – – –
Family Child Adult Neighborhood Community
0.08 0.03 0.00001 0.000005
0.08 0.03 0.00001 0.000005
0.08 0.03 0.000027 0.000015
0.08 0.03 0.000027 0.000015
0.08 0.03 0.000027 0.000015
0.03 0.04 0.00012 0.000115
0.03 0.04 0.00012 0.000115
the family unit if they become infected. Table 6 shows the probability of withdrawal as well as the distribution of the number of days before withdrawal given a person does withdraw from the mixing groups. The household transmission probabilities in Table 7 are from estimates in [38,39]. The other transmission probabilities were chosen to calibrate the model to certain illness attack rates in each age group. The overall illness attack rate at baseline was 0.15, 95% CI [0.02, 0.22], with an average attack rate in young children of 0.27, 95% CI [0.02, 0.42], older children 0.29, 95% CI [0.03,0.40], and adults 0.10, 95% CI [0.02,0.14]. The probability of becoming infected was calculated each day for each individual depending on his or her contact groups. The probability of infection each day is based on the binomial model and the escape probabilities for each potentially infectious contact. This approach is more exact than the approximation that Elveback et al. [20] had used in the early 1970s (1976) to conserve computer time. The source code for the population generation and simulation is written in C and is available upon request, although without support or documentation.
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[13] Klar N, Gyorkos T, Donner A. Cluster randomization trials in tropical medicine: a case study. Trans R Soc Trop Med Hyg 1995;89:454–9. [14] Murray DM. Design and Analysis of Group-Randomized Trials. New York: Oxford University Press, 1998. [15] Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc London Ser A 1927;115:700–21. [16] Whittle P. The outcome of a stochastic epidemic—a note on Bailey’s paper. Biometrika 1955;42:116–22. [17] Bailey NTJ. The mathematical theory of epidemics. London: Griffin, 1957. [18] Becker NG. Analysis of infectious disease data. London: Chapman & Hall, 1989. [19] Anderson RM, May RM. Infectious diseases of human: dynamics and control. Oxford: Oxford University Press, 1991. [20] Elveback LR, Fox JP, Ackerman E, Langworthy A, Boyd M, Gatewood L. An influenza simulation model for immunization studies. Am J Epidemiol 1976;103:152–65. [21] Longini IM, Ackerman E, Elveback LR. An optimization model for influenza A epidemics. Math Biosci 1978;38:141–57. [22] Longini IM. Modeling influenza epidemics. In: Kendal AP, Patriarca PA, editors. Proceedings of the UCLA Symposia on Molecular and Cellular Biology, Options for the control of influenza, New Series, vol. 36. New York: Alan Liss, 1986. p. 89–105. [23] Poland GA, Hall CB. Influenza immunization of schoolchildren: can we interrupt community epidemics? Pediatrics 1999;103:1280–1. [24] Piedra PA, Gaglani M, Herschler G, Kozinetz C, Glezen WP. Safety and effectiveness of the trivalent, cold-adapted influenza virus (CAIV-T) vaccine in a community-based, non-randomized, open-label trial in children. Pediatr Res 47, 2000 [abstract]. [25] Longini IM, Halloran ME, Haber MJ, Chen RT. Measuring vaccine efficacy from epidemics of acute infectious agents. Stat Med 1993;12:249–63. [26] Ball F, Clancy D. The final size and severity of a generalised stochastic multitype epidemic model. Adv Appl Probability 1993;25:721–36. [27] Longini IM, Halloran ME, Nizam A. Model-based estimation of vaccine effects from community vaccine trials. Stat Med 2002;21:481–96.
[28] Lin Z. An issue of statistical analysis in controlled multi-centre studies: how shall we weight the centres? Stat Med 1999;18:365–73. [29] United States Census Bureau. Census 2000, retrieved 25 July 2001 [http://www.census.gov/]. [30] Smith PG, Rodrigues LC, Fine PEM. Assessment of the protective efficacy of vaccines against common diseases using case–control and cohort studies. Int J Epidemiol 1984;13(1):87–93. [31] Halloran ME, Haber MJ, Longini IM. Interpretation and estimation of vaccine efficacy under heterogeneity. Am J Epidemiol 1992;136:328– 43. [32] Fine PE, Zell ER. Outbreaks in highly vaccinated populations: implications for studies of vaccine performance. Am J Epidemiol 1994;139:77–90. [33] Halloran ME, Struchiner CJ. Causal inference for infectious diseases. Epidemiology 1995;6:142–51. [34] Longini IM, Halloran ME, Nizam A, Wolff M, Mendelman PM, Fast P, et al. Estimation of the efficacy of live, attenuated influenza vaccine from a two-year, multi-center vaccine trial: implications for influenza epidemic control. Vaccine 2000;18:1902–9. [35] Belshe RB, Gruber WC, Mendelman PM, Cho I, Reisinger K, Block SL, et al. Efficacy of vaccination with live attenuated, cold-adapted, trivalent, intranasal influenza virus vaccine against a variant (A/Sydney) not contained in the vaccine. J Pediatr 2000;136:168–175. [36] Belshe RB, Gruber WC, Mendelman PM, et al. Correlates of immune protection induced by live attenuated, cold-adapted, trivalent, intranasal influenza virus vaccine in children. J Inf Dis 2000;181:1133–7. [37] University of Minnesota National Micropopulation Simulation Resource. Simex simulation resource. World Wide Web/Internet, December 1998 [http://www.labmed.umn.edu/nmsr/simex]. [38] Longini IM, Koopman JS, Haber M, Cotsonis GA. Statistical inference for infectious diseases: risk-specified household and community transmission parameters. Am J Epidemiol 1988;128(4):845–59. [39] Addy CL, Longini IM, Haber MS. A generalized stochastic model for the analysis of infectious disease final size data. Biometrics 1991;47:961–74.
Community interventions and the epidemic prevention potential M. Elizabeth Halloran∗ , Ira M. Longini, David M. Cowart, Azhar Nizam Department of Biostatistics, Rollins School of Public Health, Emory University, 1518 Clifton Road NE, Atlanta, GA 30322, USA Received 8 January 2001; received in revised form 22 April 2002; accepted 18 June 2002
Abstract Evaluation of community-level effects of intervention programs in infectious diseases is receiving increased attention. In this paper, we consider evaluation of the community-level effectiveness measures on the example of vaccination of children against influenza. We introduce the concept of the epidemic prevention potential (EPP) as a measure of the ability of an intervention to either prevent transmission or at least to keep it below a pre-defined limit. As a concept to describe the general ability of an intervention to limit outbreaks to a certain defined size, the term EPP fills a void. We constructed a stochastic simulation model of influenza transmission and vaccination in a structured community to illustrate the effectiveness measures of interest and the epidemic prevention potential. The concepts are general and could be applied to other interventions, such as antivirals and quarantine. © 2002 Elsevier Science Ltd. All rights reserved. Keywords: Community trials; Efficacy; Influenza; Vaccines
1. Introduction Increasing attention is being given to evaluating community-level effects of vaccination strategies [1–5] and other interventions [6–9] in infectious diseases. In light of the current worries about bioterrorism, there is also increased interest in planning interventions in case an infectious agent, such as smallpox, is introduced into a population. Methods to evaluate the effectiveness of such interventions will be required. Much has been written on group randomized studies where whole social units are assigned to treatment groups [9–14]. Considerable theoretical research has focused on establishing threshold conditions for transmission in epidemic models [15–19]. However, little has been written about evaluating interventions when transmission may be a threshold phenomenon. In this paper, we introduce the concept of the epidemic prevention potential (EPP) as one aspect of overall effectiveness when an intervention prevents an outbreak or limits it to less than a certain defined size. We illustrate estimation of effectiveness measures with simulations of vaccination of children against influenza to reduce attack rates in communities [20–24]. We differentiate effectiveness measures that take all communities into account whether or not an epi∗
Corresponding author. Tel.: +1-404-727-7647; fax: +1-404-727-1370. E-mail address: [email protected] (M. Elizabeth Halloran).
demic has occurred from effectiveness measures that condition on an epidemic reaching a pre-defined size threshold. We constructed a stochastic simulation model of influenza transmission and vaccination in a structured community to illustrate evaluation of the indirect, total, and overall effects, and the EPP. Although the focus here is on vaccination, the concepts are general and could be applied to other interventions, such as antivirals and quarantine.
2. Scientific questions of interest Consider a seasonally epidemic infectious agent, such as influenza or cholera. The outcome of interest may be the illness attack rate in the community as a whole or in a stratum within the community, such as in children 18 years old or under. Suppose that several communities receive the intervention of interest and several communities do not receive the intervention. The populations receiving the interventions will be referred to as type A communities, and the control communities as type B communities. We propose three primary questions of interest: 1. What is the effectiveness of the intervention on the attack rates in the communities receiving the intervention? 2. What is the effectiveness of the intervention in reducing the size of epidemics that do occur? 3. What is the effectiveness of the intervention in preventing epidemics?
0264-410X/02/$ – see front matter © 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 4 - 4 1 0 X ( 0 2 ) 0 0 3 1 6 - X
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Table 1 Measures of vaccination efficacy and effectiveness and comparison groups in an intervention vaccinating children against influenza Comparison group
Effectiveness measure Children Direct VES
Intervention community Baseline community
vac A unvac A
Indirect VEIIA
Total VEIIB
unvac A unvac B
vac A unvac B
Adults
Population
Overall VEIII
Indirect VEIIA
Overall VEIII
All A All B
unvaca
All A All B
A unvaca B
A, intervention community; B, baseline community; vac, vaccinated in intervention; unvac, not vaccinated in intervention. a Adults may be vaccinated before the intervention, but are not a target of this intervention. We could study indirect effects of vaccinating children separately in vaccinated and unvaccinated adults, or any other stratum.
An epidemic will be defined by a threshold attack rate, which could be zero, but generally some small value (see Remark 1). The effectiveness measures in (1) and (2) can be differentiated into indirect, total and overall effectiveness [1–3]. For example, vaccination of children against influenza as a method to reduce cases in adults as well as children has received increasing attention [20–24]. Table 1 summarizes the effectiveness measures of interest for the example of vaccinating children against influenza in communities. The indirect effectiveness (VEIIA ) in children would be estimated from attack rates in the unvaccinated children in the nonintervention communities compared with the attack rates in the unvaccinated children in the intervention communities. The total or combined effectiveness (VEIIB ) in children would be estimated from attack rates in the vaccinated children in the intervention communities compared with the attack rates in the unvaccinated children in the nonintervention communities. The overall effectiveness (VEIII ) in children would be estimated from attack rates in all the children in the intervention communities compared with the attack rates in all the children in the nonintervention communities. Since the intervention program does not target adults, in adults, only the indirect effectiveness (VEIIA ) would be estimated from attack rates in adults in the intervention communities compared with attack rates in adults in the nonintervention communities. For overall effectiveness in the communities as a whole (VEIII ), the attack rates in the whole intervention communities would be compared to the attack rates in the whole nonintervention communities. To provide some notation, assume that the attack rate (ARjk ) in the subgroups k in each community j = A, B in the study is observed. Let RR denote the relative risk based on the attack rates. The subscripts j = 0, 1 denote unvaccinated and vaccinated people within the populations, respectively. The indirect effectiveness, VEIIA is VEIIA = 1 −
ARA0 = 1 − RRIIA . ARB0 ARA1 = 1 − RRIIB . ARB0
VEIII = 1 −
ARA. = 1 − RRIII , ARB.
(3)
where the dot in the subscript denotes the entire population. As in the influenza example, the indirect effects of interest can be estimated in the children and adults separately. Also, the overall effects can be evaluated in just children or in the population as a whole. Subpopulations of interest need to be defined for any particular study. The above effectiveness measures can be estimated either using all communities in the study, or including only those communities with an epidemic above a defined size threshold. The attack rates from several communities can be combined in various ways, which will be touched upon below. For completeness, if only some of the people are vaccinated in communities of type A, then estimation of the direct protective effect of vaccination is possible. For direct effects, the unit of inference is the individual. The direct protective effects, VES , can be estimated by VES = 1 −
ARA1 = 1 − RRS . ARA0
(4)
Interpretation of the VES estimates will need to be cautious, unless the vaccine is allocated randomly to individuals. For scientific question (3), as a measure of the ability of a vaccination strategy to prevent transmission, we define the epidemic prevention potential (EPP) of a vaccination strategy as the relative reduction in the probability of an epidemic in a community compared to what it would be without vaccination. The EPP can also be defined to compare one vaccination strategy with another. Denote the probability of an epidemic in populations type A by Pr(e)A and in populations type B as Pr(e)B . Then the EPP of the vaccination strategy is
(1) EPP = 1 −
The total or combined effectiveness, VEIIB is VEIIB = 1 −
The overall effectiveness, VEIII is
(2)
Pr(e)A . Pr(e)B
We consider the EPP in more detail in Section 3.
(5)
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3. Epidemic prevention potential A few examples illustrate the EPP measure. Suppose that seasonal transmission occurs reliably in each unvaccinated community type B, but it occurs in only about 20% of the intervention communities A. Then Pr(e)B = 1, Pr(e)A = 0.20, and EPP = 1 − 0.20/1 = 0.80. About 80% of epidemics will be prevented by the intervention. Suppose that seasonal outbreaks occur in about 80% of unvaccinated communities, Pr(e)B = 0.80 and 20% of the vaccinated communities, Pr(e)A = 0.20. Then the EPP = 1 − 0.20/0.80 = 0.75. The EPP of the vaccination strategy, being defined relative to baseline, in this second example is lower than in the previous example, even though the probability of an epidemic in the vaccinated communities is the same in both examples. Remark 1. The EPP requires a definition of an epidemic or an outbreak. One possibility is to set a defined size threshold. Transmission of infectious diseases can be quite fickle, as most people who have tried to run vaccine field trials know. There may be no transmission in a given community or there may be very little compared to what would be expected if an epidemic had taken off and become self-sustaining. When a disease is rare, one often tries to define outbreaks as being significantly above baseline. However, in infectious diseases, the definition of an epidemic would most often be related to what would normally have been expected in a particular community. Another option is to define a threshold according to some public health interest. For example, influenza illness attack rates of 15–25% might be expected if an epidemic took off in a community. Then rather than defining zero cases as the public health goal of intervention, holding an epidemic below a certain threshold, say keeping the attack rates below 2.5, 5.0 or even 7.5% might be the epidemic prevention goal. If the number of cases is in fact zero, issues related to analysis are raised (see below). In any event, to determine whether an epidemic has been prevented, a precise definition for the threshold of a successful epidemic is required. The value of the EPP is sensitive to the definition of an epidemic. Remark 2. We define EPP in a relative way, similar to other efficacy and effectiveness measures. However, the absolute difference in the probability that an epidemic will occur, Pr(e)B − Pr(e)A , is also of interest. The relation between the ratio estimator and difference estimator is analogous to other epidemiologic measures. A variety of absolute differences and magnitudes in the probability of an epidemic can yield the same EPP. For instance, if the probability of an epidemic is 0.40 without a vaccination strategy and 0.10 with it, then EPP = 1 − 0.10/0.40 = 0.75, the same as in the second example. However, the absolute difference in the earlier example is 0.20 − 0.80 = −0.60, and in this last example, it is 0.10 − 0.40 = −0.30. Even though the EPP is the same in both examples, the absolute difference
of 0.60 represents a larger public health achievement than 0.30. Remark 3. The EPP of a vaccination strategy is related to the threshold level of vaccination required to prevent transmission. Thresholds for transmission have been of considerable theoretical interest in both deterministic [15] and stochastic [16] modelling of infectious diseases (see also [17–19]). In both types of models, the reproductive number, R, is the expected number of additional infectives produced by one infective in a particular population. In deterministic models, if R ≤ 1, theoretically, transmission will be eliminated, while for R > 1, an epidemic will always take off. But deterministic models do not allow for stochastic effects. In stochastic models, R is related to the probability that an epidemic will take off. If R ≤ 1 in a stochastic system, the epidemic will never take off, in the sense of being self-sustaining. However, a few people may become infected, producing what is called a minor epidemic. If R > 1, there is a positive probability that the epidemic might take off, producing a major epidemic. The probability that the epidemic takes off increases with larger R. As R is reduced close to 1 with high levels of vaccination, the probability of a major epidemic becomes very low. The reproductive number R is generally difficult to estimate from the available data. However, it is easier to establish whether, after an intervention, an epidemic is below or above some defined threshold attack rate. 4. Design and analysis issues In community studies, the unit of inference is the population. Thus, the sample size and power of a study are more closely related to the number of communities in the study than the number of people in each community. The eligible communities need to be transmission-dynamically separate [1,2]. As in community studies of noninfectious diseases, the sample size required to estimate effectiveness has to do with the size of the effect expected and the intercommunity variability. At low vaccination coverage or low efficacies, the indirect and overall effectiveness might be small, requiring a larger number of communities. Estimation of the EPP presents slightly different problems than estimation of the effectiveness measures in Table 1. For each community, the outcome is in a sense 0 or 1, either an epidemic of a certain size occurred or it did not. For any number of control or intervention communities, given an expected probability of an epidemic occurring, one could calculate the expected distribution of the number of epidemics based on the binomial distribution. Using exact statistical methods, one could compute the number of intervention and control communities required to get reliable EPP estimates. The number of communities required will be larger as more baseline communities do not have epidemics and as an intervention is less likely to prevent an epidemic. The number of communities required may be prohibitively large in many
M.E. Halloran et al. / Vaccine 20 (2002) 3254–3262
instances. However, the concept of the EPP of a vaccination strategy has an important public health interpretation as well as an important role in interpreting estimates of indirect, total, and overall effectiveness, as is illustrated below. Given that there are several intervention and several control communities in a study, one can choose among different methods to combine attack rates from different communities to estimate the indirect, total, and overall effects. Transmission-dynamically similar communities could be matched either at the design or at the analysis phase or at both. In a matched analysis, summary measures of effectiveness could be obtained from the weighted average of the risk ratio estimates from all pairs of communities. It is desirable to compute weighted averages, because the populations with more cases contribute more information. Commonly, the inverse variances of the pairwise risk ratios [25] are used. The attack rates have been shown to be approximately binomial proportions when conditioned on the total number of infected people and with mild assumptions about the epidemic process [26,27]. Another paper explores the relation of the number of pairs of communities to the variability of the effectiveness estimates in more detail [27]. Longini et al. [27] also propose a model-based estimator with a random effect on the basic reproductive number. Several problems are raised by the pairwise analysis using the inverse variance weights on the risk ratios. First, transmission may fail to occur in one or both of the communities in a pair. This can be dealt with, though inadequately, by adding a one to the numerator. However, then very large variances may occur. The other problem is that when the indirect or overall effectiveness is low, very negative estimates can occur, destabilizing the summary measures. Also, reciprocal variance weighted averages are efficient, but can be biased. An alternative would be to use the sample sizes as weights. This is less efficient than reciprocal variance weighted averages, but is also less biased [28]. Another alternative is to use the simple average of the attack rates in the intervention communities as the numerator of the summay measure and the simple average of the attack rates in the nonintervention communities as the denominator.
5. Simulated studies of influenza vaccination 5.1. Simulation of intervention We simulated studies of community-level effects of vaccinating children using a stochastic simulator of influenza transmission in a structured population. The effectiveness measures of interest were those described in Table 1 as well as the EPP. The intervention of interest was vaccination of children aged 1–18 years old at different levels of coverage. Each simulated study included four intervention and four baseline communities. For each level of coverage and vaccine efficacy, 100 studies were simulated. For each simulated study, we estimated summary measures of effective-
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ness based on both the matched pair analysis using inverse variance weights of the pairwise risk ratios and the simple average of the attack rates in the intervention compared to the simple average of the attack rates in the nonintervention communities. The matching within pairs in the first analysis would not have been necessary since there was no difference in variation across pairs due to the simulation design. The EPP was computed based on each set of 400 simulations at the different coverages. The direct effects VES were also computed for each set of 400 simulations at the different coverages. Each population of 2000 people was stochastically generated based on the age distribution and approximate household sizes from the US Census 2000 [29]. Each population had four neighborhoods, one high school, one middle school, and two elementary schools. Pre-school children attended either small or large play groups. People mixed within households, neighborhoods, the whole community, and when appropriate, within schools and play groups. Influenza was introduced by randomly assigning 12 initial infectives. The initial infectives were omitted in the analysis. Pre-intervention baseline vaccination coverage was assumed to be 5% in children, 22.9% in adults 19–64 years old, and 68.1% in adults 65 years and over. Vaccination coverage in adults was assumed to remain the same in the intervention communities. The transmission probabilities were calibrated so that, on average, 15% of a baseline population developed influenza illness. Influenza transmission was modeled similarly to that of [20]. Details are in the Appendix A. The pre-intervention vaccine was assumed to have a protective efficacy of VES = 0.70 and efficacy for infectiousness of VEI = 0.80. We simulated two different levels of efficacy of the intervention vaccine. The first intervention vaccine had the same efficacy as the pre-intervention vaccine, that is, VES = 0.70 and VEI = 0.80. The second intervention vaccine had a higher protective efficacy, VES = 0.90, but the same efficacy for infectiousness, VEI = 0.80. The vaccine effects were assumed to work multiplicatively on the transmission probability, both on the susceptibility of vaccinated susceptibles and on the infectiousness of vaccinated people who become infected. The simulated target coverages in children were 30, 50, and 70%, which includes the pre-intervention baseline coverage in children. Vaccination status was randomly assigned based on the probabilities of the target coverage. 5.2. Results Table 2 contains the effectiveness results from the 100 simulated studies of four intervention and four baseline communities each. The indirect, total, and and overall effectiveness estimates increase as coverage and VES increase. In particular, indirect effects in the groups not targeted by the intervention are evident. At 70% coverage of children with VES = 0.70, the mean estimated VEIIA in adults is 0.59. Thus, on average 59% of the cases in adults are prevented
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Table 2 Mean estimates (empirical standard deviation in parentheses) of measures of the effectiveness of vaccinating children based on attack rates at the end of the annual influenza epidemic Coverage (%)
Effectiveness measure Children Directa
VES
Adults
Population
Indirect VEIIA
Total VEIIB
Overall VEIII
Indirect VEIIA
Overall VEIII
30
ave ivw
0.60 (0.19)
0.20 (0.22) 0.16 (0.50)
0.67 (0.11) 0.65 (0.22)
0.25 (0.21) 0.21 (0.45)
0.23 (0.20) 0.21 (0.32)
0.23 (0.20) 0.20 (0.39)
50
ave ivw
0.58 (0.34)
0.41 (0.20) 0.44 (0.33)
0.77 (0.09) 0.77 (0.13)
0.48 (0.18) 0.51 (0.29)
0.42 (0.18) 0.45 (0.27)
0.44 (0.18) 0.46 (0.31)
70
ave ivw
0.63 (0.22)
0.58 (0.14) 0.61 (0.23)
0.85 (0.06) 0.85 (0.09)
0.66 (0.12) 0.69 (0.19)
0.59 (0.13) 0.62 (0.16)
0.62 (0.12) 0.64 (0.17)
70 (VES = 0.9)
ave ivw
0.87 (0.17)
0.71 (0.10) 0.75 (0.15)
0.96 (0.02) 0.95 (0.02)
0.79 (0.08) 0.82 (0.11)
0.72 (0.09) 0.74 (0.11)
0.74 (0.09) 0.76 (0.11)
There were 100 simulated studies with four intervention and four nonintervention communities per study; VES = 0.70, VEI = 0.80, except the last line, where VES = 0.90; ave, simple average; ivw, inverse variance weighted. a VE is included only for completeness. The means and empirical standard deviations are based on 400 separate intervention communities, not S averages of four communities.
by vaccinating the children compared with baseline. The standard deviations of the effectiveness estimates tend to decrease as coverage increases. At low coverages, there are still many negative estimates, even with four intervention and baseline communities each in the study. This leads to high variability of the effectiveness estimates. In fact, it reflects the uncertainty about whether low coverage could provide measurable indirect effects. The means of the simple average estimates approximately equal the means of the inverse variance weighted estimates. However, the standard deviations of the simple average estimates are in general lower than those of the inverse variance weighted estimates, with the difference more pronounced at the lower coverage. The appropriate statistical methods for combining studies of community-level effects of intervention require more research. For completeness and to understand the estimates of total effectiveness VEIIB , the mean of the direct VES estimates from the 400 intervention communities are also included. The VES estimates are lower than the nominal simulated value because the vaccine model is multiplicative (leaky), not all-or-none. The simulated multiplicative VES is for just one exposure to infection. Upon each repeated exposure,
there is an independent probability of becoming infected, so the VES using final attack rates is estimated to be lower than the set multiplicative effect [30,31]. For example, with VES set at 0.70, at 30% coverage, the mean VES estimate is 0.60 in these simulations. This is expected. Thus, the mean estimated total effectiveness in children at 30% coverage can also be less than the set 0.70. However, the mean estimated total effect VEIIB is higher than the mean estimated direct effect. Table 3 shows that both the percent of epidemics taking off and the EPP at different levels of vaccination coverage are sensitive to the definition of an epidemic. Particularly at high coverages and high efficacy, the percent taking off decreases and the EPP increases as the cutoff goes from 2.5 to 7.5%. In the simulated baseline communities, 93.5% have attack rates over 7.5%, while 96.7% have attack rates over 2.5%. One can compute the EPP in Table 3 from the percent taking off. For example, at 70% coverage with VES = 0.90, EPP = 1 − 0.562/0.967 = 0.41 at the 2.5% definition. At the 7.5% definition, EPP = 1 − 0.085/0.935 = 0.91. Thus, the mean EPP of this vaccination strategy at VES = 0.90 varies from 0.41 to 0.91, depending on the definition of an epidemic. If the public health goal were to decrease
Table 3 Percent of epidemics out of 400 taking off and epidemic prevention potential (EPP) based on relative probability of epidemics taking off using different definition of an epidemic at different coverages with VES = 0.70, VEI = 0.80, except the last line, where VES = 0.90 Coverage (%)
Baseline 30 50 70 70 (VES = 0.90)
Percent taking off
Epidemic prevention potential
Definition of an epidemic
Definition of an epidemic
2.5%
5.0%
7.5%
2.5%
5.0%
7.5%
96.7 94.7 84.2 76.0 56.2
94.5 89.5 70.7 51.0 22.0
93.5 80.5 57.2 29.2 8.5
– 0.02 0.13 0.21 0.41
– 0.05 0.25 0.46 0.77
– 0.14 0.39 0.69 0.91
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Table 4 Conditional on an epidemic occurring, mean estimates (empirical standard deviation in parentheses) of measures of the effectiveness of vaccinating children based on attack rates at the end of the annual influenza epidemic Coverage (%)
Def (%)
N
Effectiveness measure Children
Adults
Population
Indirect VEIIA
Total VEIIB
Overall VEIII
Indirect VEIIA
Overall VEIII
30
> 2.5 > 5.0 > 7.5
100 100 100
0.17 (0.20) 0.14 (0.20) 0.06 (0.31)
0.65 (0.12) 0.64 (0.12) 0.62 (0.13)
0.22 (0.19) 0.19 (0.19) 0.12 (0.29)
0.20 (0.19) 0.17 (0.19) 0.11 (0.23)
0.20 (0.19) 0.17 (0.19) 0.11 (0.26)
50
> 2.5 > 5.0 > 7.5
100 99 94
0.30 (0.23) 0.22 (0.23) 0.10 (0.44)
0.73 (0.10) 0.70 (0.10) 0.65 (0.18)
0.38 (0.21) 0.32 (0.20) 0.21 (0.39)
0.33 (0.20) 0.26 (0.20) 0.18 (0.22)
0.52 (0.80) 0.28 (0.20) 0.18 (0.29)
70
> 2.5 > 5.0 > 7.5
95 68 29
0.43 (1.31) – 0.68 (0.26)
0.80 (0.45) – 0.89 (0.09)
0.54 (1.06) – 0.75 (0.20)
0.52 (0.60) 0.48 (0.70) 0.67 (0.22)
0.52 (0.80) 0.43 (1.12) 0.70 (0.21)
There were 100 simulated studies with four intervention and four nonintervention communities per study; VES = 0.70, VEI = 0.80, except the last line, where VES = 0.90; All estimates are based on simple average which may include fewer than four intervention and four nonintervention communities per study. Def, definition of an epidemic; N , number of studies included in analysis.
the attack rates to less than 5.0%, then we would expect between 46 and 77% success with coverage of an influenza vaccine with protective efficacy between VES = 0.70 and VES = 0.90, and a VEI = 0.80. Table 4 shows that the conditional effectiveness measures decrease at 30 and 50% coverage as the cutoff for an epidemic increases. The difference between the unconditional measures in Table 2 and the conditional measures in Table 4 is picked up in the EPP measure. For example, at 50% coverage, using the 5% cutoff, the EPP is 0.25 with a mean conditional overall effectiveness, VEIII = 0.25, in the population. In Table 2, the unconditional average effectiveness is 0.44. In any particular situation, the question is, what is the goal of the intervention, so which is the more appropriate measure? At the higher coverages, increasingly frequently all four intervention communities fail to take off. The conditional effectiveness estimates become very unstable by 70% coverage at VES = 0.70 and are difficult to interpret.
6. Discussion We have introduced the concept of the EPP as an effectiveness measure of interest in community interventions in infectious diseases. As a concept to describe the general ability of an intervention to limit or to contain outbreaks to a certain defined threshold size, the term epidemic prevention potential fills a void. In light of the events since 11 September 2001, and worries over bioterrorism, the concept of the EPP has become particularly relevant. We have delineated three possible goals for evaluation. First, what is the average effectiveness of the strategy in reducing transmission, including both communities with prevented epidemics and communities with reduced but positive transmission? Second, conditional on an epidemic exceeding a certain defined threshold, what is the effective-
ness of the vaccination strategy in limiting transmission? Third, what is the reduction in the probability of an epidemic occurring, that is, the EPP of the strategy? Using stochastic simulations of vaccination of children against influenza, we have illustrated evaluation of the effectiveness measures of interest. The lesson is that at lower coverages, either the unconditional estimates as in Table 2 or the EPP combined with conditional estimates in Table 4 makes sense. At higher coverages when the probability of an epidemic is low, the unconditional estimate or the EPP would make more sense, while the conditional estimates could be omitted. In general, one should not rely solely on the conditional estimates, without including the EPP, as this would be throwing away important information. We have discussed prospective studies that have the opportunity to see whether outbreaks occur or not. Often, post-licensure vaccine efficacy studies are conducted only where outbreaks occur, resulting in underestimates of efficacy because of the ascertainment bias [32]. The extent of underestimation is related to the size of the epidemic triggering the study, the vaccination coverage in the community, and the extent of clustering of vaccination failures in the population. Analogously, if only communities in which epidemics occur are used to study the community-level effects of intervention, then effects will be underestimated. The EPP needs to be used in partnership with measures conditional on an epidemic occurring. A fundamental issue is that when dealing with indirect and overall effects of vaccination strategies in a community, there is no obvious underlying true parameter of interest to estimate[33]. As more community studies of indirect and overall effects of intervention strategies go to the field, additional research is needed to deal with the inherent complex issues. We have used two possible methods of combining the attack rates and relative risks to illustrate calculating summary
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measures of effectiveness for several communities. These and other methods of combining results across communities require further research. Our results are not to be considered recommendations. Since the simulated populations are all the same size, the simple average is similar to using sample size weights. Thus, the comparison to using the inverse variance weights on the risk ratios of the matched pairs would need to be compared using simlautions with different sample sizes or different simulation parameters, as in [27]. The simulation results give insight into the potential effects of vaccinating children in current American communities. Trivalent cold-adapted influenza vaccine has been shown to have a high efficacy in children. The estimated vaccine efficacy for susceptibility, VES , against currently circulating strains of influenza was estimated from a multi-center vaccine trial to be around 0.90 [34,35]. For this vaccine, the vaccine efficacy for infectiousness, VEI , could be about 0.80 [36]. These simulations suggest that vaccination of 70% of children with a highly efficacious influenza vaccine could substantially reduce or prevent transmission in the majority of the communities in which the strategy is implemented. Prospective, designed community level studies will be large and expensive to conduct. Transmission might not occur in either intervention or nonintervention communities. If it does not occur in the vaccinated community, an investigator might be tempted to interpret the lack of transmission as being due to the vaccination strategy. However, transmission might have been absent quite by chance. If transmission is lower in the vaccinated community than in the unvaccinated community, but still occurs, it could still be by chance and not due to vaccination. The inherent variability in the transmission systems across communities needs to be taken into account in determining the number of communities required before going to the field to make the best use of the large amounts of time, energy, and money.
neighborhoods. Small play groups have four children each, and there are between four and six small play groups per neighborhood. Large daycare centers can have as many as 50 children, though the average size is 14 children. School-age children are assigned to either an elementary school, middle school, or high school based on their age. Two neighborhoods share one elementary school, and all four neighborhoods share a middle and high school. Elementary schools have on average 79 children per school, middle schools have an average of 141 students, and high schools have an average of 110 students. For the purposes of our simulator, the ages of children were assumed to be uniform over the intervals 0–4 and 5–18 years of age. Young adults (19–64 years) and older adults (65 years and over) were also uniformly distributed within their respective age groups. On average, in each generated population, 6.92% were under 5 years, 22.08% were 5–18 years, 58.48% were adults 19–64 years old, and 12.52% were adults 65 years or older. The probability that a household has one person is 0.33, two people is 0.34, three is 0.13, four is 0.10, five is 0.07, six is 0.02, and seven is 0.01. The probability that an adult, in a family with children, is 65 years or older is 0.02. The probability that an adult in a household without children is 65 years or older is 0.28. The probability that a two person home has one child and one adult is 0.01. In the simulations in this paper, each population is generated with 12 initial infectives chosen at random. The properties assigned when the population is set up identify each individual. The age-specific vaccination coverage of interest determines the probability that any person will be vaccinated. For each person in the population, a random number between 0 and 1 is generated. If the random number is less than the probability of being vaccinated, the person is vaccinated. A.2. Influenza parameters
Acknowledgements This research was partially supported by NIH grant R01-AI32042.
Appendix A. Simulation model A.1. The population Populations of 2000 people are stochastically generated based on the age distribution and approximate household size published by the United States Census Bureau [29]. A family is a group of up to seven people living together with one or two adults. Each person in the population is assigned to a family within one of four neighborhoods, an age, a gender, an initial disease status indicator, and a vaccination status indicator. Pre-school age children are assigned to either small play groups or large daycare centers within their
The computer code used to run these epidemics is very general and could be used to simulate a great variety of directly transmitted diseases. The simulator has been tailored to simulate influenza. Many of the influenza parameters are adopted from Elveback et al. [20] or the Vespers program [37]. Table 5 shows the distribution of the latent and infectious periods. The probability that an individual will be symptomatic given that person has been infected is 0.67. An asymptomatic infection is assumed to be 50% as infectious as a symptomatic infection. Additionally, this model allows for people to withdraw from all of their mixing groups except Table 5 Baseline influenza parameters (from Elveback et al. [20])
Number of days Probability (all ages) Mean days
Latent period
Infectious period
1 0.3 1.9
3 0.3 4.1
2 0.5
3 0.2
4 0.4
5 0.2
6 0.1
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Table 6 Withdrawal information (Elveback et al. [20]) Age group
Pre-school School age Adults
Probability of withdrawal given illness
Withdrawal day, given withdrawal Day 0
Day 1
Day 2
0.80 0.75 0.50
0.38 0.27 0.20
0.50 0.53 0.60
0.12 0.20 0.20
Table 7 Transmission probabilities in simulated populations Contact group
Children
Adults
Pre-school
School
Younger
Older
Small play group
Large daycare
Elementary
Middle
High
Small play groups Large daycare centers Elementary school Middle school High school
0.0680 – – – –
– 0.0275 – – –
– – 0.00577 – –
– – – 0.00477 –
– – – – 0.00177
– – – – –
– – – – –
Family Child Adult Neighborhood Community
0.08 0.03 0.00001 0.000005
0.08 0.03 0.00001 0.000005
0.08 0.03 0.000027 0.000015
0.08 0.03 0.000027 0.000015
0.08 0.03 0.000027 0.000015
0.03 0.04 0.00012 0.000115
0.03 0.04 0.00012 0.000115
the family unit if they become infected. Table 6 shows the probability of withdrawal as well as the distribution of the number of days before withdrawal given a person does withdraw from the mixing groups. The household transmission probabilities in Table 7 are from estimates in [38,39]. The other transmission probabilities were chosen to calibrate the model to certain illness attack rates in each age group. The overall illness attack rate at baseline was 0.15, 95% CI [0.02, 0.22], with an average attack rate in young children of 0.27, 95% CI [0.02, 0.42], older children 0.29, 95% CI [0.03,0.40], and adults 0.10, 95% CI [0.02,0.14]. The probability of becoming infected was calculated each day for each individual depending on his or her contact groups. The probability of infection each day is based on the binomial model and the escape probabilities for each potentially infectious contact. This approach is more exact than the approximation that Elveback et al. [20] had used in the early 1970s (1976) to conserve computer time. The source code for the population generation and simulation is written in C and is available upon request, although without support or documentation.
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