Large-scale vaccination campaigns (SIAs) and improved routine immunization (RI) have greatly reduced measles incidence in low-income countries. However, the interval between SIAs required to maintain these gains over the long term is not clear. We developed a dynamic model of measles transmission to assess measles vaccination strategies in Cambodia, Ghana, India, Morocco, Nigeria, and Uganda. We projected measles cases from 2008 to 2050 under (a) holding SIAs every 2, 4, 6, or 8 years, (b) improvements in first dose routine measles vaccine (MCV1) coverage of 0%, 1%, 3% annually, and (c) introducing MCV2 once MCV1 coverage reaches 70%, 80%, 90%. If MCV1 continues improving, then India and Nigeria could hold SIAs every 4 years without significant probability of large outbreaks, and the other countries every 6-8 years. If RI remains stagnant, India and Nigeria should hold SIAs every 2 years, and the other countries every 4-6 years.

This paper analyzes the effectiveness of various monotonic discretizations of an ODE in a parameter range reduction algorithm. Several properties of discretizations are given, and five classes of discretizations are defined for various step numbers s. The range reduction algorithm that employs these discretizations is described. Using both analytical results based on the prototypical model x ′ = λx, and empirical results based on two more complicated models, it is shown that one particular class of discretizations (the A1OUT class) results in the tightest bounds on the parameters. This result is shown to be attributed to a certain characteristic value, A 0 , of the discretization. Accumulation of these discretizations is also defined, and its usefulness in the range reduction algorithm is described.

Abstract. An algorithm that constructs a continuous piecewise linear representation, subject to a certain slope/segment length constraint, of a given set of data is described. The algorithm yields an optimal or near optimal representation, subject to this constraint, in the L∞ norm and does so in at worst O(nK) time, where n is the number of data points and K the number of segments. The constraint is determined by a user-specified parameter, t min , which dictates a lower bound for the distance between opposite-sign slope discontinuities. For reasonable t min values, the resulting representation of the data captures the signal and both smooths the noise and provides a measure of it. This representation is useful for applications that require a specified interval for the data values and also allows easy and continuous interpolation of ranges between recorded time points. The algorithm is described, some of its properties are proven, and its capabilities demonstrated with several examples. Comparisons are made with alternative techniques. , a primary concern is a reduction in the number of points. In other applications such as image processing [19,13], experimental sciences [21], computer science problems [9], and parameter estimation methods for ordinary differential equation (ODE) models [28], accurate representations of the data allow for more efficient and effective analysis.Over the past 50 years, there has been considerable work on the problem of representing discrete data by a (piecewise) continuous function and the related problem of approximating one (piecewise) continuous function with another. In virtually all of this work, algorithms are presented that compute approximations that are best or at least good with respect to some metric and satisfy some list of constraints. The approximations are usually piecewise polynomials and often linear; in the latter case we call them piecewise linear representations (PLRs). The metric is generally either the L 2 norm (the least squares criterion) or the L ∞ norm (the minimax criterion). Typically the constraint is the number of segments permitted. A complementary (and easier) problem that is frequently addressed is the determination of a piecewise poly-

BackgroundDynamic models of infection transmission can project future disease burden within a population. Few dynamic measles models have been developed for low-income countries, where measles disease burden is highest. Our objective was to review the literature on measles epidemiology in low-income countries, with a particular focus on data that are needed to parameterize dynamic models.MethodsWe included age-stratified case reporting and seroprevalence studies with fair to good sample sizes for mostly urban African and Indian populations. We emphasized studies conducted before widespread immunization. We summarized age-stratified attack rates and seroprevalence profiles across these populations. Using the study data, we fitted a "representative" seroprevalence profile for African and Indian settings. We also used a catalytic model to estimate the age-dependent force of infection for individual African and Indian studies where seroprevalence was surveyed. We used these data to quantify the effects of population density on the basic reproductive number R0.ResultsThe peak attack rate usually occurred at age 1 year in Africa, and 1 to 2 years in India, which is earlier than in developed countries before mass vaccination. Approximately 60% of children were seropositive for measles antibody by age 2 in Africa and India, according to the representative seroprevalence profiles. A statistically significant decline in the force of infection with age was found in 4 of 6 Indian seroprevalence studies, but not in 2 African studies. This implies that the classic threshold result describing the critical proportion immune (pc) required to eradicate an infectious disease, pc = 1-1/R0, may overestimate the required proportion immune to eradicate measles in some developing country populations. A possible, though not statistically significant, positive relation between population density and R0 for various Indian and African populations was also found. These populations also showed a similar pattern of waning of maternal antibodies. Attack rates in rural Indian populations show little dependence on vaccine coverage or population density compared to urban Indian populations. Estimated R0 values varied widely across populations which has further implications for measles elimination.ConclusionsIt is possible to develop a broadly informative dynamic model of measles transmission in low-income country settings based on existing literature, though it may be difficult to develop a model that is closely tailored to any given country. Greater efforts to collect data specific to low-income countries would aid in control efforts by allowing highly population-specific models to be developed.

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