## Abstract

**The status of measles elimination is best summarized by evaluation of the effective reproduction number R; maintaining R < 1 is necessary and sufficient to achieve elimination. Previously described methods for estimating R from the sizes and durations of chains of measles transmission and the proportion of cases imported were applied to the measles data reported for the United States in 1997–1999. These comprised 338 cases, forming 165 chains of transmission, of which 43 had >1 case. One hundred seven cases were classified as importations. All 3 methods suggested that R was in the range 0.6–0.7. Results were not sensitive to the minimum size and duration of outbreak considered (so long as single-case chains were excluded) or to exclusion of chains without a known imported source. These results demonstrate that susceptibility to measles was beneath the epidemic threshold and that endemic transmission was eliminated.**

During 1997–1999 in the United States, 338 measles cases were reported; for 120 of these cases, a link to importation could not be identified [1]. Thus, the question arises: Has the goal of measles elimination been achieved? For elimination to be achieved, there must be no sustained chains of endemic transmission; that is, all cases must be linked to an importation. However, even the most robust surveillance system cannot detect every such link. Thus, a method of assessing elimination from surveillance data that does not require that the system detect all links to importation is an essential tool.

Because zero measles incidence cannot be sustained in the presence of imported disease (and cases will continue to be imported until the disease is eradicated globally), 3 of us (N.J.G., G.D.S., and C.P.F.) have previously proposed that elimination be defined as "a situation in which endemic transmission has stopped, sustained transmission cannot occur, and secondary spread from importations will end naturally, without intervention" [2, p. 1041]. This definition is equivalent to a requirement to maintain the effective reproduction number, *R*, of measles below the threshold at *R* = 1 [3]. We demonstrated 3 methods for estimating *R* from surveillance data once the disease has been eliminated in a country or region. Using these methods, we analyzed US measles data for 1995–1997 and concluded that endemic transmission had indeed been eliminated [2].

Here we present estimates of *R* for 1997–1999, which show that endemic transmission of measles in the United States continued to be eliminated during these years.

## METHODS

** Confirmed measles cases, 1997–1999.** Confirmed measles cases are reported to the National Immunization Program at the Centers for Disease Control and Prevention (CDC) with accompanying epidemiological information. National Immunization Program classifies the importation status of cases and groups them into chains of transmission based on either known links found during the investigations or close temporal and geographical clustering. The quality and completeness of these data have been assessed elsewhere [4].

** Assessing R from measles case data.** When endemic transmission of measles has stopped, imported cases drive the observed epidemiology. Some importations will produce no secondary cases, whereas others will cause some limited spread to susceptible persons. The extent of this spread depends on the effective reproduction number,

*R*, the average number of secondary cases produced by each case. When

*R*< 1, this spread will always be limited and endemic transmission cannot become reestablished.

When endemic transmission has been interrupted, data on measles cases may be used to estimate the value of .R by 3 methods: from the proportion of cases imported **( R = I** proportion of cases imported); from the distribution of sizes of chains of transmission; and from the distribution of durations of chains of transmission [2].

To estimate *R* by use of these methods, one must assume that all chains of transmission are finite. If this assumption is made, the estimates of *R* obtained with these methods will always be <1 (although the upper confidence limit on *R* may exceed 1). If the methods are inappropriately applied to data from a period when endemic transmission occurred (but was reported as a series of separate chains because of undetected links in the chain of transmission), the value of *R* obtained would be marginally <1, but the confidence interval on *R* would be expected to include 1 (thereby not excluding the possibility of endemic transmission).

The first method requires a conservative definition of "importation"; otherwise, *R* may be underestimated. Therefore, we classified as an importation any case in a person who traveled outside the United States in the 18 days before rash onset, unless onset occurred ⩾7 days after onset in a traveling companion.

For example, if a family returned from a holiday outside the United States, and one member developed measles 5 days after returning and another 15 days after returning, we would classify the first case as an importation and the second as spread from this importation. Our definition is more conservative than the National Immunization Program definition, which would classify both of these sample cases as importations, solely on the basis of the time between foreign travel and disease onset for each individual.

For the other 2 methods of estimating *R*, a "chain of transmission" or "chain" is defined as the entire series of cases that can be linked to the same source. This includes single-case chains, which are not linked to any other cases. We calculated the duration of a chain of transmission as the difference between the dates of disease onset of the first and last cases. If this was 0–6 days, cases were considered as being in the same generation; 7–14 days was considered as 1 generation of spread; 15–24 days was considered as 2 generations; and another generation was added for every extra 10 days.

These methods of estimating *R* are based on a model of the spread of infection that predicts the distribution of sizes and durations of chains of transmission that arise from an importation. The model assumes that the number of secondary cases caused by a single infectious individual has a Poisson distribution with mean *R*. Figures 1 and 2 show the expected distributions of chain size and duration for a range of values of *R*. For example, for *R* = 0.7, 50% of chains are expected to be limited to a single case, 32% to have 2–4 cases, 11% to have 5–9 cases, and 7% to have ⩾10 cases (figure 1). For a given population, the 3 estimates of *R* (and associated confidence intervals) are best obtained by analysis of the data by means of a maximum likelihood approach (see Appendix).

An important methodological consideration is the minimum size and duration of chains that should be considered in these analyses. Smaller chains, especially single-case chains, are less likely to be detected by surveillance but more likely to be composed of false-positive cases. Discarding chains of transmission below a minimum size or duration may reduce bias from these factors but has the disadvantage of reducing the data available for analysis. To reduce bias but retain sufficient data, we arbitrarily based our analysis of chain size on those involving ⩾3 cases and our analysis of chain duration on those with at least

2 generations of secondary spread. We investigated the sensitivity of our results to the minimum size and duration considered by estimating *R* for minimum chain sizes of 1–5 cases and minimum chain duration from 0 to 4 generations of spread.

Chains of transmission with no identified source present another challenge for analysis. One possibility is to adjust the data by adding missing cases. However, for each chain with no identified source, there are 3 possible explanations: it may be linked to an unidentified importation, it may have an unidentified link to another identified chain, or, especially for single-case chains, it may be the result of a false-positive laboratory test. The appropriate adjustment would require adding an imported case, adding an indigenous case, and deleting the chain, respectively. Because it is not known which of these possibilities applies to each chain, it is not possible to adjust the data in this way. In the sensitivity analyses for minimum chain size and duration, we conducted all analyses twice, either including or excluding chains without an identified imported source.

## RESULTS

** Importation status.** During 1997–1999, 338 confirmed cases were reported to CDC: 138 in 1997 and 100 in each of 1998 and 1999. Of these cases, we classified 52, 24, and 31 as importations (9 fewer than CDC's classification) (table 1). The proportions of imported cases were 38%, 24%, and 31% in 1997, 1998, and 1999, respectively, giving an overall proportion of 32% over the 3 years.

** Data on chains of transmission of measles.** During the 3 years, measles cases were reported from a total of 165 chains of transmission (table 1): 122 were single-case chains. Of the single-case chains, 79 were imported and 43 could not be linked to importation. There were 43 chains of transmission with >1 case; 30 of these had at least 3 cases and only 4 had ⩾10 cases (table 1). Of the 43 chains with >1 case, 21 had ⩾2 generations of spread; only 4 of these had ⩾5 generations of spread (table 2). For 2 chains, the interval between the first and last cases was <7 days, so cases were categorized in the same generation (i.e., no spread); neither of these chains was linked to an imported case.

** Estimates of R.** The value of

*R*for measles in the United States during 1997–1999 was estimated as 0.68 from the proportion of cases imported and as 0.63 from both the distributions of chain sizes and chain durations (table 3). The observed distribution of chain sizes and that predicted by the model with the maximum likelihood estimate of

*R*= 0.63 are shown in figure 3. Similarly, figure 4 shows the fit to the distribution of chain duration. For all 3 estimates, the value of

*R*was lowest in 1997 and highest in 1998, but this year-to-year variation was not significant. Estimates of

*R*from the proportion of cases imported showed the least year-to-year variation.

Sensitivity analyses showed that estimates of *R* were considerably lower if single-case chains were included in the analysis based on size and if the chains with no spread were included in the analysis based on duration but were otherwise fairly consistent (tables 4 and 5). Estimates of *R* did not change significantly when the analysis was limited to chains with an identified imported source; however, the confidence intervals on *R* widened.

## DISCUSSION

Our analyses suggest that during 1997–1999, the true value of the reproduction number *R* for measles in the United States lay in the range 0.6–0.7. These results demonstrate that in the United States, susceptibility to measles was beneath the epidemic threshold and that endemic transmission was eliminated. Furthermore, this value of *R* is lower than the 0.85 previously estimated for the period 1995–1997 [2>], suggesting an improvement in control of measles. Elimination of endemic measles can be sustained in the United States if low levels of susceptibility are maintained through high coverage with 2 doses of vaccine.

In general, and as expected, the estimate of *R* derived from the proportion of imported cases (which tends to overestimate the true value [2]) is higher than those derived from the distribution of chain size and duration (which tend to underestimate the true value [2]). However, the 3 estimates are remarkably similar. The estimates were also robust to reanalyses of subsets of the data; varying the minimum size of chain considered (as long as single-case chains were excluded) and varying the minimum duration (as long a chains with 0 generations of spread were excluded) had little impact. Considering only chains with an identified imported source also produced similar estimates of *R* but widened the confidence intervals around these estimates.

For our base analysis, the distributions of chain size and duration did not differ greatly from that predicted by the model, which is based on the assumption that the average number of secondary cases caused by an infectious individual is constant throughout the population. Heterogeneity in the distribution of susceptible persons would tend to produce bigger chains in the more susceptible pockets of the population and smaller ones in better-protected groups. The only suggestion of heterogeneity was the larger-than-expected number of single-case chains, which could alternatively be explained by false-positive laboratory diagnoses. There were no large outbreaks reported from susceptible pockets of unvaccinated persons. However, such outbreaks have occurred in other countries with excellent measles control and are likely to occur in the United States in the future. Communities with high proportions of unvaccinated persons can support large outbreaks, even though the rest of the population is well protected [5]. However, such communities do not present the potential for maintaining endemic transmission unless they reach the critical community size, which for measles is of the order of several hundred thousand people [3, 6–8].

In the elimination phase of a disease control program, surveillance systems should be capable of detecting any impending failure of the elimination strategy, a failure to implement this strategy correctly, or foci of transmission in which additional measures may be needed. Active search for every isolated case gives little if any benefit, because only large clusters of cases would provide evidence of the ineffectiveness of the program. As a minimum, surveillance should determine whether each reported case has been imported, to enable a single estimate of *R* to be made. To avoid underestimating *R* by use of this method, it is important that the proportion of cases imported is not overestimated. In this respect, the CDC definition of an imported case was not appropriate for our analysis, and it was therefore necessary to reclassify some cases.

Estimating *R* from the distribution of chain size and duration is beyond the "minimum" requirement suggested above. To permit such analyses, surveillance must emphasize linking cases into chains of transmission. In practice, some links between cases will not be identified, even for diseases that always induce medical consultation. Therefore, it is reasonable to assume that cases of measles occurring in temporal and geographic clusters are part of the same chain.

The methods used here to estimate *R* from cases are necessarily retrospective. They evaluate the past but do not predict the future. Prediction of future values of *R* requires methods for monitoring the susceptibility of the population, such as measurement of vaccine coverage and serological surveillance. Accurate and timely monitoring of vaccine coverage to ensure high coverage should be a priority. In cohorts with little exposure to natural infection, the proportion susceptible can be estimated from data on vaccination status (proportions who have received no dose, 1 dose only, and 2 doses) and vaccination effectiveness. This enables projection of future levels of susceptibility by age from which the reproduction number *R* can be calculated through the use of age-structured transmission models [9]. Such analyses can identify the potential for resurgences before they occur, allowing time to implement supplementary strategies to prevent them [10].

Clear communication of goals and achievements to the public, the media, and politicians is an important aspect of disease control programs. Whatever definition of elimination is adopted by epidemiologists and public health professionals, in the public mind the word "elimination" will imply the absence of cases. Use of the expression "elimination of endemic measles transmission" rather than "elimination of measles" may help to inform the public that although measles cases still occur, the epidemiology in the United States is now driven by cases imported from other countries. Most imported cases will produce little secondary spread, and some will lead to large outbreaks, but none will reestablish endemic transmission. It is important to distinguish "elimination of endemic transmission" from "absence of endemic transmission"; the latter implies that no endemic transmission has occurred but does not comment on the potential for endemic transmission to be reestablished. When delivering the message that elimination of endemic measles transmission has been achieved, public health authorities should clearly state that they expect outbreaks to continue to occur as a result of imported cases and that some of these outbreaks will be large. Public health authorities should also point out that endemic transmission may be reestablished if high vaccination coverage levels are not maintained.

Through the methods we have described, surveillance of the degree of secondary spread from imported cases provides an opportunity for monitoring the susceptibility of the population to ensure that it remains below the population susceptibility that could sustain endemic transmission. Successful application of these methods requires that sufficient imported cases occur, which may not happen consistently in small or isolated populations. In the period we studied, the number of imported cases reported each year in the United States ranged from 24 to 52, with >100 imported cases total during 1997–1999. Applying these methods to assess elimination in settings with many fewer imported cases may require aggregation of data for several countries and/or years as appropriate. Countries without imported measles cases should ensure that their measles surveillance system is sufficiently sensitive to detect such cases. Imported cases and limited secondary spread do not indicate a failure of a country's elimination efforts. Elimination of endemic measles is the highest level of elimination a country can achieve before global eradication of measles.

## APPENDIX

We assume a Poisson distribution for the number of secondary cases produced by an infected individual, with mean *R*, and we assume that this does not change during the course of a chain of transmission. With this assumption, we derive the log likelihood function, L, for each method of estimating *R*. The best estimate of *R* is the value that maximizes the log likelihood. Approximate 95% confidence intervals can be obtained from the profile log likelihood as the range of values of *R* giving a log likelihood within 1.92 of the maximum.

** Proportion of cases imported.** The log likelihood of

*I*imported cases generating a total of C cases is given by

[2]. This is maximized when *R* = 1 - *I/C.*

** Distribution of chain size.** Following a single importation, the probability,

*S*of a chain with

_{p}*j*cases (including the initial importation) is given by

[2]. If *m ^{j}* is the observed number of chains with

*j*cases, and only chains with at least / cases are considered, the log likelihood is given by

** Distribution of chain duration.** The probability of a chain with at most

*k*generations of secondary spread,

*G*is given by

_{b}where *E _{k}(x)* denotes the iterated exponential function (the number

*x*the power of

*x*to the power of

*x... k*times, so that

*E*1,

_{0}(x) =*E*

_{1}(x) = x,*E*etc.) [11].

_{2}{x) = x^{x},If *n _{k}* is the observed number of chains with

*k*generations of spread, and only chains with at least

*K*generations of spread are considered, the log likelihood

*L*is given by