SUPPLEMENT ARTICLE The Theory of Measles Elimination: Implications for the Design of Elimination Strategies Nigel J. Gay Health Protection Agency, Modelling and Economics Unit, Communicable Disease Surveillance Centre, London, United Kingdom The theory of disease transmission provides a consistent framework within which to design, evaluate, and monitor measles elimination programs. Elimination of measles requires maintaining the effective reproduction number R at !1, by achieving and maintaining low levels of susceptibility. The essential features of different vaccination strategies (e.g., routine versus campaigns, number of doses) can be compared within this framework. Designing an elimination program for a particular population involves setting target levels of susceptibility, establishing the current susceptibility profile, selecting an approach to reduce susceptibility below the target, and selecting an approach to maintain susceptibility below the target. A key indicator of the sustainability of an elimination program is the residual level of susceptibility of a cohort after it has completed its scheduled vaccination opportunities. This can be estimated from vaccination coverage data. The high transmissibility of measles poses a significant challenge to any attempt to eliminate it. Measles elimination goals have been adopted in a range of countries, subregions, and regions adopting a variety of vaccination strategies. Here I present the theoretical concepts relevant to measles elimination, such as the reproduction number and susceptibility threshold; investigate and compare the essential features of routine and campaign vaccination strategies within this framework; present the stages of designing a measles elimination strategy; and discuss implications for surveillance of measles elimination programs. BASIC CONCEPTS Measles is transmitted from person to person. The crucial factor determining the spread of infection is therefore the number of secondary cases caused by each infectious person. Basic reproduction number, R0. The basic reproduction number, R0, is a summary measure of the transmissibility of an infection within a population, defined as the average number of secondary infections pro- Reprints or correspondence: Nigel J. Gay, Health Protection Agency, Modelling and Economics Unit, Communicable Disease Surveillance Centre, 61 Colindale Ave., London NW9 5EQ, United Kingdom ([email protected]). The Journal of Infectious Diseases 2004; 189(Suppl 1):S27–35 2004 by the Infectious Diseases Society of America. All rights reserved. 0022-1899/2004/18909S1-0005$15.00 duced by a typical infective person in a totally susceptible population. It depends on the characteristics of the infectious agent (e.g., infectivity and duration of infectiousness) and of the population (e.g., population density and social mixing patterns). R0 therefore differs between infections in the same population but also for the same infection in different populations. For example, within any given population, the R0 for measles is greater than the R0 for rubella, and, all else being equal, the R0 for measles is greater in a dense, urban population than a sparse, rural population. Because R0 is defined on the basis of the potential for transmission in a totally susceptible population, it does not depend on the level of susceptibility in the population and is unaffected by vaccination. It represents the maximum transmission potential of the infection—the average number of persons with whom an infected person makes effective contact during the infectious period. Effective reproduction number, R. The effective reproduction number, R, is a summary measure of the potential for transmission of an infection within a population, defined as the average number of secondary infections produced by a typical infective person. The value of R depends on the levels of susceptibility in the population and on the basic reproduction number R0. In a completely susceptible population R p R 0. Theory of Measles Elimination • JID 2004:189 (Suppl 1) • S27 When R is 11, each case produces on average 11 secondary case, so the number of cases increases from one generation of cases to the next; when R is !1, the number of cases decreases. Thus, the value R p 1 is an important threshold [1]. Elimination criterion. If R is maintained constantly at !1, the number of cases will decrease (on average) with each generation, and all endemic chains of transmission will eventually die out. Imported cases introduced into the population will not be able reestablish endemic transmission. Elimination of indigenous transmission will therefore be achieved. Susceptibility threshold. To be relevant to vaccination policy decisions, the theoretical concepts of R0 and R must be applicable to practical questions regarding available data. Perhaps the most important question regards the susceptibility threshold: What level of susceptibility corresponds to the R p 1 threshold? The simplest case to consider is that of a homogeneously mixing population. Although such simple models ignore the complexities of real populations, they are worthy of discussion to establish the basic principles, many of which carry over into more complicated models. In a homogeneously mixing population, the basic and effective reproduction numbers are related simply by R p R 0 x, where x is the proportion of the population susceptible to infection. The threshold at R p 1 defines the critical proportion susceptible, x ∗ p 1/R 0. Critical vaccination coverage. Eliminating infection requires maintaining R ! 1 , by keeping the proportion susceptible below the critical value x ∗ p 1/R 0 . Equivalently, the proportion immune must be greater than the critical proportion immune pc p 1 ⫺ x ∗ p 1 ⫺ 1/R 0. Achieving this level of immunity through a vaccination program administered at birth will be sufficient to attain elimination. Two important points are often overlooked. First, the formula assumes that vaccination is given at birth, or as soon as infants become susceptible—if vaccination is delayed until the second or third year of life, a higher proportion immune will be required [2]. Second, this formula refers to the level of immunity that must be achieved, not the vaccination coverage—the efficacy of the vaccine must be taken into account when calculating the coverage required. The duration of protection afforded by successful vaccination is also important—waning of immunity that enabled persons experiencing secondary vaccine failure to become significant transmitters of infection would thwart attempts at elimination. Epidemic cycle. If measles is endemic in a population, it occurs in epidemic cycles (figure 1). During the course of an epidemic cycle, R oscillates around threshold at R p 1, changing constantly as the level of susceptibility fluctuates. An epidemic can begin if R 1 1. After the onset of an epidemic, those infected acquire immunity, so the number of susceptible persons falls and R begins to decline. When infection has depleted the pool of susceptible persons sufficiently, R is reduced to !1 and the number of cases in the outbreak declines. At the end of the outbreak, R begins to increase again because of the addition of new susceptible persons through birth. When R exceeds 1, a new epidemic can begin. If a population is sufficiently large, chains of transmission can be sustained throughout the period when R ! 1; otherwise, an epidemic will not occur until the infection is reintroduced to the population. Critical community size. The concept of a critical community size for sustaining endemic measles transmission arose through work in the prevaccination era, studying the persis- Figure 1. Simple model of measles transmission in a population with 80% routine vaccination coverage (90% efficacy) from year 5. (Two-week time step: number of cases indicated by bars; number of susceptibles by dots.) S28 • JID 2004:189 (Suppl 1) • Gay tence of measles in island and city populations [3, 4]. The issue is whether chains of transmission could persist through the post-epidemic period, when R ! 1 , until the susceptible persons build up to the critical level at which R 1 1, when the next epidemic can begin. At such low levels of transmission, stochastic (chance) effects become paramount. Consider cases as occurring in discrete generations. At each generation there is a finite probability that no secondary infections will be produced and therefore that transmission will die out. This probability depends on the expected number of cases in the next generation (i.e., on the number of cases in the current generation and the current value of R). In small communities, the number of cases in each generation becomes so low in the trough of transmission that the chain almost always breaks at some point, and transmission fades out before the susceptible persons have built up again. In contrast, the probability of fadeout is very low in sufficiently large communities, because the number of cases never becomes critically small. The critical community size is the size of population needed to sustain endemic transmission (i.e., to prevent fade-out). For measles in an unvaccinated population, this is observed to be ∼250,000– 500,000 [3, 4], possibly lower for sparse populations and higher for dense populations [4]. The critical community size may be better expressed in terms of the average number of cases per generation, or equivalently the average input of susceptible persons into the population per generation interval. Routine vaccination (at a level insufficient to achieve elimination) would increase the community size needed to sustain measles transmission, because it reduces the input of new susceptible persons into the population. The concepts of elimination and critical community size should not be confused. In sufficiently small populations, fadeout can occur after only a short period in which R ! 1. However, if susceptible persons are allowed to reaccumulate and R is not maintained at !1, widespread transmission may recommence when the infection is reintroduced. This is not elimination. Elimination requires the (indefinite) maintenance of R ! 1 throughout a population. If this is achieved, by achieving a sufficiently low proportion of susceptible persons in each population subgroup, chains of transmission will eventually die out regardless of the total number of susceptible persons within the population, because each case will, on average, produce !1 secondary case. Moreover, elimination achieves a stable situation—reintroduction of infection will not lead to widespread transmission. Estimating R0. During the course of an epidemic cycle, R oscillates around 1 as the proportion susceptible oscillates around the threshold. If disease remains endemic, the average proportion susceptible to infection remains at the threshold level, even after vaccination is introduced (figure 1). Thus, in any population that is assumed to mix homogeneously in which measles is endemic, R0 can be estimated directly as the reciprocal of the average proportion susceptible. This provides a straightforward method for estimating R0 from seroprevalence data, which can be used after the introduction of vaccination, even for growing populations. Various formulas have been derived to relate the average proportion susceptible before introduction of vaccination to easily observed parameters such as the average age at infection in the absence of vaccination, A; the life expectancy, L; and the average duration of maternal antibody protection, m [5, 6]. The simplest case is for a population with no growth in which no individuals die before acquiring measles. In this scenario, the average time for which an individual is susceptible to measles is A–m (being susceptible between ages m and A), out of a life expectancy L. With no population growth, the average proportion of the population susceptible is given simply by x ∗ p (A–m)/L. Note that the relationship R 0 p 1/x ∗ can be used to express R0 in terms of these parameters; R0 p L/ (A–m) in the above example. This simple formula has been used to estimate R0 for measles at 14–18 in England and Wales and at 12.5 in North America [2]. Thus, assuming homogeneous mixing, the critical level of immunity in England and Wales was calculated as 94% (using the upper estimate of R 0 p 18 ) or 96% if vaccination was delayed until the second birthday [2]. Such high levels of immunity cannot be achieved with a single dose of a vaccine that has 90%–95% efficacy. Heterogeneity. Although homogeneous mixing models illustrate qualitatively the impact of vaccination, many practical applications demand that some of the heterogeneity of the population is accounted for. In the design of vaccination programs, most attention has focused on modeling heterogeneity arising from age-related contact patterns [7–12], but spatial and temporal heterogeneity can also have implications [6, 11]. Such models divide the population into subgroups and specify the degree of mixing within and between these subgroups [6]. Difficulties in estimating the contact patterns arise from the absence of information on “who acquired infection from whom” [6, 9], but further progress has been made by combining information from several infections with similar transmission routes [13]. Given the different contact rates between the various groups in a heterogeneously mixing population, the difficulty in calculating R0 and R lies in defining a “typical” infective person as some suitable average across all subgroups within the population. A mathematically rigorous method for calculating R0 and R from the average number of secondary cases in each group caused by an infective person in each of the groups (the “next-generation matrix”) does not yield any simple formulas [14]. In particular, R0 cannot be estimated as the reciprocal of the average proportion susceptible. However, the rigorous Theory of Measles Elimination • JID 2004:189 (Suppl 1) • S29 definition does preserve the most important results from homogeneous mixing populations for heterogeneously mixing populations. Most notably, R ! 1 remains the criterion for elimination. Also, if a proportion x of every population subgroup is susceptible, then R p R 0 x . Thus, elimination can be achieved by reducing the proportion susceptible in each population subgroup below 1/R0 (pc p 1 ⫺ 1/R 0 ). However, this is not the most efficient way of eliminating the disease. There are many different combinations of susceptibility levels in the various subgroups that produce the threshold R p 1. Achieving levels of susceptibility below 1/R0 in the core groups that contribute most to transmission allows higher levels of susceptibility elsewhere and consequently more susceptible persons overall [15]. The importance of schools in the transmission of measles in developed countries was confirmed by a model that incorporated variable contact rates among school-aged children (higher during school terms than during school holidays) [8], which reproduced the seasonal pattern of measles within the biannual epidemic cycle. Incorporating age-dependent contact rates reduced estimates of pc for measles to 84%–92% [7] (from 94%, using a homogeneous model). However, these early models focused on primary school children and underestimated the potential for measles transmission among older children. Refining the contact rates used for these age groups on the basis of more recent data precludes values of pc !90% [9]. Epidemiology after elimination. After the elimination of endemic measles transmission from a population, all cases of measles must be linked to infections imported from outside the population [16]. As long as R is maintained at !1, importations will not reestablish endemic transmission but may cause limited secondary spread. The expected distribution of the size Figure 2. and duration of such outbreaks depends on R; the larger the value of R, the larger and longer the outbreaks [16]. IMPLICATIONS FOR DESIGN OF MEASLES ELIMINATION PROGRAMS The implications of this theory for elimination strategies are clear. The first step is to reduce the levels of susceptibility in the population so that R ! 1; the task thereafter is to ensure that these low levels of susceptibility are maintained. To design an elimination program for a particular population following this general approach requires setting target levels of susceptibility, establishing the current susceptibility profile, selecting an approach to reduce susceptibility below the target, and selecting an approach to maintain susceptibility below the target. Setting Susceptibility Targets Setting target levels of susceptibility first requires Rmax, the maximum permissible value of R, to be selected. Clearly, for elimination, R max ! 1, but the choice of a particular value is a policy decision that takes into account the safety margin desired, the degree of secondary spread from imported cases that can be tolerated, and the resources available—lower values of Rmax provide more security but require a greater vaccination effort. At the simplest level, all that is then required is an estimate of R0. For example, if R 0 p 16 and Rmax is chosen to be 0.8, the target level below which the proportion susceptible in all age groups and other subgroups must be reduced is 0.8/16 p 5%. More flexibility can be introduced into susceptibility targets by balancing higher levels in some age groups against lower World Health Organization target levels of susceptibility for measles elimination in Europe S30 • JID 2004:189 (Suppl 1) • Gay levels in others, to give the same overall R. The effects of agespecific transmission rates can be accounted for in these calculations. For example, the target levels of susceptibility in the World Health Organization (WHO) strategy for the elimination of measles from the European region were designed for R max p 0.7 by use of a heterogeneous mixing model for which R 0 p 11 (figure 2). A crude calculation suggests a 6.4% susceptibility target for all age groups, but achieving a lower level (5%) in age groups with the highest transmission rates, namely secondary school children and young adults, allows higher levels of susceptibility in preschool (15%) and primary school (10%) children. Establishing Susceptibility Profile The susceptibility profile describes the distribution of susceptibility to measles within a population. Most important is the variation in the proportion susceptible to measles by age, but other relevant variables include vaccination status and, in some cases, population subgroup. Three sources of data are useful in assessing the susceptibility profile: measles case notifications, vaccination coverage reports, and seroprevalence surveys. The most direct way to estimate the susceptibility profile is via a suitably stratified serological survey, interpreting samples negative for measles antibody as indicating susceptibility to measles. It is essential to ensure that the assay used is adequately sensitive and specific, especially in highly vaccinated populations in which many persons protected by vaccination may have low antibody levels. Use of quantitative assays allows standardization of results between different surveys through panels of reference serum samples [17] and against international standards. The proportion of each birth cohort not protected by vaccination can be calculated from vaccination status—the proportions that have received no dose, 1 dose only, or 2 doses— and the efficacy of 1 and 2 doses: proportion not protected by vaccination p proportion unvaccinated + [proportion receiving 1 dose only ⫻ (1 ⫺ efficacy of 1 dose)] + [proportion receiving 2 doses ⫻ (1 ⫺ efficacy of 2 doses)]. Vaccination status may be measured directly (e.g., in surveys) or inferred from coverage data. Because this calculation does not attempt to account for naturally acquired immunity, it is most useful in cohorts with high vaccine coverage. It will significantly overestimate the proportion susceptible in cohorts with low vaccine coverage and a high exposure to natural infection. Case notifications are best used by calculating the age-specific incidence during the most recent epidemic by means of the finest possible age stratification. This provides only a qualitative indication of the relative susceptibility at different ages, because the attack rate among susceptible persons may be age-dependent—it is often higher in school-aged children than in preschool children. It should also be noted that cases may provide a better reflection of the susceptibility before the epidemic than after it. More complex methods that combine information from several types of data are also available, including susceptible reconstruction methods [18] and dynamic transmission models [19]. Reducing Susceptibility below Targets Campaigns. Mass vaccination campaigns aim to immunize a high proportion of the susceptible persons in the population by achieving a high level of coverage across a wide age range, often over a short period of time. The age range for a campaign Figure 3. Simple model of measles transmission in a population with an vaccination campaign in year 5 that immunizes 80% of all susceptible persons and 80% routine vaccination coverage from year 5. Theory of Measles Elimination • JID 2004:189 (Suppl 1) • S31 Figure 4. A, Vaccination status in the year 2000 of children in England born during 1990–1998. These cohorts were too young to be vaccinated in the 1994 national measles vaccination campaign (which targeted 5- to 16-year-olds) and have been vaccinated with measles-mumps-rubella vaccine according to the routine schedule (at 12–15 months and 4 years). B, Estimated susceptibility in 2000 of children in England born during 1990–1998. Incidence of measles virus infection in England was very low during 1990–2000, so all immunity is assumed to be vaccine-derived (10% of children assumed to remain susceptible after 1 dose of vaccine and 1% after 2 doses of vaccine). The second dose (at age 4 years) reduces the susceptibility of the cohort from 20% to 10% within the WHO European region target for the 5- to 9-year age group. However, the 10% residual susceptibility at age 5 years is above the 5% target for older age groups, suggesting that this limit will be exceeded as these cohorts age. Targeting of 0-dose children is needed to reduce susceptibility in these cohorts. Susceptibility in older cohorts targeted by the 1994 campaign, and in adults, is low (!5%). Figure 5. Estimated proportion of children susceptible to measles in England in 2000 by district health authority (DHA). Each point represents 1 DHA and shows the proportion susceptible among children born during 1990–1994 and 1995–1997. Contour lines indicate the values of R associated with the susceptibility levels. There is considerable variation in vaccination coverage between districts: The 20 districts in which R 1 0.85 were provided with some extra resources to conduct additional vaccinations. can be selected by comparing the susceptibility profile of the population against the susceptibility targets—including in the campaign all cohorts in which susceptibility exceeds the targets. By immunizing a high proportion of the susceptible persons in the population, successful campaigns reduce R well below 1. This has a dramatic impact on the incidence of measles, causing chains of transmission to die out rapidly (figure 3). For example, if R is reduced to 0.5, the number of cases will, on average, halve with each new generation of cases (every ∼2 weeks). The potential for endemic transmission will not reemerge until the input of new susceptible persons into the population has restored susceptibility to the R p 1 threshold. The time taken to do this will depend principally on the number of susceptible persons immunized by the campaign and the rate at which new susceptible persons are added [20]. The duration of impact of campaigns in heterogeneously mixing populations can be estimated in a similar fashion as the time taken for R to exceed 1 [19]. Routine programs. Introduction of a routine vaccination program does not have the same rapid impact on the proportion of the population susceptible to infection as does a vaccination campaign. Rather, programs that vaccinate early in life reduce the rate at which susceptible persons are added into the population (figure 1). Protecting some individuals from infection reduces the number of infectious cases, lessening the risk that a susceptible person will contact an infectious person and thereby become infected. Both the direct protection of new cohorts and the reduction in the risk of infection cause the age distribution of susceptible persons to shift toward older age groups [18]. However, unless sufficiently high levels of immunity are achieved in the vaccinated cohorts, the infection will remain endemic and establish a new epidemic cycle oscillating around R p 1 (figure 1). These direct and indirect effects can be investigated by use of dynamic models of the transmission of infection. Use of a single dose of measles vaccine cannot achieve a high enough level of immunity to achieve elimination. A routine 2dose schedule can achieve high levels of immunity (198% efficacy), but it takes many years to feed through all age groups. Outbreaks often occur in the cohorts just too old to have received 2 doses. Most countries that have achieved elimination through high coverage with a routine 2-dose schedule have also conducted specially targeted supplementary vaccination of older age groups who were born before or missed by the 2dose schedule. Maintaining Susceptibility below Targets To maintain R at !1 requires a strategy for preventing the reaccumulation of susceptible persons in the population. This entails achieving high levels of immunity in children too young to be vaccinated during the campaign and those born after it. Two alternative approaches are available: a routine 2-dose schedule or 1 routine dose plus regular follow-up campaigns [21]. Whatever vaccination strategy is adopted, a crucial factor determining success or failure in the long run is the residual level of susceptibility of a cohort after it has completed its scheduled vaccination opportunities. The choice of strategy should largely be determined by the need to minimize this residual proportion susceptible. If the residual susceptibility of each cohort is not reduced below the critical level, the accumulation of susceptible persons will eventually increase R to 11. Vaccination status of each cohort should be assessed once it has completed its scheduled vaccination opportunities to enable the proportion susceptible to be calculated directly (cohorts born after a catch-up campaign will have had little exposure to natural infection). As above, the proportion susceptible is most sensitive to the proportion remaining completely unvaccinated, and it is crucial that the second opportunity minimizes the number of “0-dose” children. The other consideration is the age at which this low level of susceptibility is achieved: the later the age, the more susceptible persons in the population, the greater the value of R, and the greater the risk that R will exceed 1. In this respect, it would be ideal to give the second dose of a 2-dose schedule as soon as possible after the first, for example, at 15–18 months of age. However, the age at vaccination may also affect the coverage Theory of Measles Elimination • JID 2004:189 (Suppl 1) • S33 Figure 6. R for measles in England, 1995–2002, estimated from distribution of outbreak size. R has increased with the reaccumulation of susceptible children after the 1994 campaign, because of the failure to maintain sufficiently high vaccination coverage. Elimination of measles, achieved between 1995 and 2001 [23], appears unlikely to be sustained. achieved; many developed countries find that school entry (at age 4–6 years) provides a good opportunity to achieve high coverage, particularly among previously unvaccinated children. In such settings, the advantage of achieving lower residual susceptibility in a cohort must be balanced against the disadvantage of allowing those with failure of first-dose vaccination to remain susceptible until school entry. Heterogeneity plays a key role in this decision: Contact rates among school-aged children are considerably higher than among preschool children. Provided that first-dose coverage is high, a lower value of R may be achieved by ensuring minimal levels of susceptibility in the age groups with highest contact rates than by providing an early opportunity to protect those with failure of first-dose vaccination. Delaying the second dose further, for example until secondary school entry at 11–12 years, has no such justification, as it is unlikely to result in further improvement in coverage but allows those experiencing vaccination failure to remain susceptible throughout primary school. Surveillance Having selected a strategy for maintaining R at !1, monitoring its implementation is largely a question of ensuring accurate and timely vaccination data. To calculate susceptibility of a birth cohort once it has completed its vaccination opportunities requires that the vaccination status of the cohort is known (particularly the proportion of “0-dose” children) (figure 4), and not just the coverage at each vaccination opportunity independently. This calculation may be best performed at the local (e.g., district) level as a key performance indicator for the vacS34 • JID 2004:189 (Suppl 1) • Gay cination program. When necessary, districts can then implement supplementary measures (e.g., identifying and vaccinating “0-dose” children) to bring susceptibility below the target level (figure 5). Surveillance of measles cases can also be used to monitor the value of R. After the elimination of endemic measles transmission from a population, all cases of measles must be linked to infections imported from outside the population [16]. The expected distribution of the size of outbreaks depends on R; the larger the value of R, the larger and longer the outbreaks [16]. Monitoring the proportion of imported cases and the size and duration of outbreaks enables R to be estimated [16, 22] (figure 6). A successful elimination program should maintain R below the target (Rmax). Obstacles to Elimination Clearly, the success of measles elimination strategies depends on the ability to implement them fully in practice. Potential problems range from the initial difficulty of identifying sufficient resources to the challenge of sustaining high vaccination coverage after the disappearance of endemic disease. Measles transmission in groups who refuse vaccination is emerging as a problem in many developed countries with elimination programs and merits discussion here. Such groups do not present the potential for sustaining endemic transmission unless they reach the critical community size. However, little can be done to prevent large outbreaks when measles is introduced into them. Strictly then, measles is not eliminated from such groups because R is not maintained at !1. However, they cannot sustain endemic transmission if they do not reach the critical community size. What does this mean for the population as a whole? Again, strictly, measles is not eliminated from the population, but it cannot sustain endemic transmission. “Elimination of endemic transmission” may be the most appropriate phrase to describe this situation. What are the implications for a potential global eradication program [24]? Clearly, if every country achieved true elimination and sustained R at !1 everywhere, then global eradication would be achieved. But what if every population (e.g., country, WHO region) reached the state at which it had “eliminated endemic transmission,” but pockets of unvaccinated individuals remained? Could global measles transmission be sustained by transmission between populations? Clearly yes, in theory, if the unvaccinated groups within different populations were sufficiently connected and their combined size was sufficiently large. Investigation of the potential for sustained transmission between groups who refuse vaccination is needed. SUMMARY The theory of disease transmission provides a consistent framework within which to design, evaluate, and monitor measles elimination programs. The key is to identify the susceptibility profile of the population and to plan a vaccination strategy to reduce and maintain susceptibility below the threshold. Nevertheless, the high transmissibility of measles poses a significant challenge to any attempt to eliminate it. References 1. Fine PEM. Herd immunity: history, theory, practice. Epidemiol Rev 1993; 15:265–302. 2. Anderson RM, May RM. Directly transmitted infectious diseases: control by vaccination. Science 1982; 215:1053–60. 3. Bartlett MS. Measles periodicity and community size. J R Stat Soc [Ser A] 1957; 120:48–60. 4. Black FL. Measles endemicity in insular populations: critical community size and its evolutionary implication. J Theor Biol 1966; 11: 207–11. 5. Dietz K. The estimation of the basic reproduction number for infectious diseases. Stat Methods Med Res 1993; 2:23–41. 6. Anderson RM, May RM. Infectious diseases of humans: dynamics and control. Oxford, UK: Oxford University Press, 1991. 7. Anderson RM, May RM. Age-related changes in the rate of disease transmission: implications for the design of vaccination programmes. J Hyg (Lond) 1985; 94:365–435. 8. Schenzle D. An age-structured model of pre- and post-vaccination measles transmission. IMA J Math Appl Med Biol 1984; 1:169–91. 9. Gay NJ, Hesketh LM, Morgan-Capner P, Miller E. Interpretation of serological surveillance data for measles using mathematical models: implications for vaccine strategy. Epidemiol Infect 1995; 115:139–56. 10. Anderson RM, Grenfell BT. Quantitative investigations of different rubella vaccination policies for the control of congenital rubella syndrome (CRS) in the United Kingdom. J Hyg (Lond) 1986; 96:305–33. 11. Hethcote HW. An age-structured model for pertussis transmission. Math Biosci 1997; 145:89–136. 12. Halloran ME, Cochi SL, Lieu TA, Wharton M, Fehrs L. Theoretical epidemiologic and morbidity effects of routine varicella immunization of preschool children in the United States. Am J Epidemiol 1994; 140: 81–104. 13. Farrington CP, Kanaan M, Gay NJ. Estimation of the basic reproduction number for infectious diseases for age-stratified serological survey data. Appl Stat 2001; 50:251–92. 14. Diekmann O, Heesterbeek JAP, Metz JAJ. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol 1990; 28:365–82. 15. Anderson RM, May RM. Spatial, temporal and genetic heterogeneity in host populations and the design of vaccination programmes. IMA J Math Appl Med Biol 1984; 1:233–66. 16. De Serres G, Gay NJ, Farrington CP. Epidemiology of transmissible diseases after elimination. Am J Epidemiol 2000; 151:1039–48. 17. Andrews NJ, Pebody RG, Berbers G, et al. The European Sero-Epidemiology Network: standardising the enzyme immunoassay results for measles, mumps and rubella. Epidemiol Infect 2000; 125:127–41. 18. Fine PEM, Clarkson JA. Measles in England and Wales. II. The impact of the measles vaccination programme on the distribution of immunity in the population. Int J Epidemiol 1982; 11:15–25. 19. Babad HR, Nokes DJ, Gay NJ, Miller E, Morgan-Capner P, Anderson RM. Predicting the impact of measles vaccination in England and Wales: model validation and analysis of policy options. Epidemiol Infect 1995; 114:319–41. 20. Nokes DJ, Swinton J. Vaccination in pulses: a strategy for global eradication of measles and polio? Trends Microbiol 1997; 5:14–9. 21. de Quadros CA, Olivé JM, Hersh BS, et al. Measles elimination in the Americas: evolving strategies. JAMA 1996; 275:224–9. 22. Gay NJ, De Serres G, Farrington CP, Redd SB, Papania MJ. Assessment of the status of measles elimination from reported outbreaks: United States, 1997–1999. J Infect Dis 2004; 189(Suppl 1):S27–35. 23. Ramsay ME, Jin L, White J, Litton P, Cohen B, Brown D. The elimination of indigenous measles transmission in England and Wales. J Infect Dis 2003; 187(Suppl 1):S198–207. 24. Hanratty B, Holt T, Duffell E, et al. UK measles outbreak in nonimmune anthroposophic communities: the implications for the elimination of measles from Europe. Epidemiol Infect 2000; 125:377–83. Theory of Measles Elimination • JID 2004:189 (Suppl 1) • S35
© Copyright 2026 Paperzz