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ABSTRACT |
 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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Within-host models for individual P. falciparum infections have considered how partial immunity affects parasite densities as an explicit consequence of differential survival of the circulating asexual parasites as the infection develops.3,4 However, models for the dynamics of P. falciparum in populations, including those modeling acquired immunity,5–7 have generally not considered the densities of asexual parasites. We are aware of only one population model for P. falciparum malaria that explicitly considered the effects of immunity on parasite densities.8 The lack of models of parasite density is surprising because effects on densities are the clearest evidence for an effect of naturally acquired immunity. The existence of such natural immunity represents some of the strongest evidence that development of an efficacious malaria vaccine is possible.
One of the major effects of immunity in most epidemiologic models of malaria6,7,9 is to reduce the duration of infections and consequently the number of infections of the vector resulting from one human infection. This quantity, following the report of Macdonald,10 is generally assumed to be proportional to infection duration, which leads to a formula that makes the basic reproductive number proportional to the infection duration. However, there is little empirical evidence on the effect of acquired immunity on duration of infection.11 Some models12 and empirical studies13,14 have even suggested that in the semi-immune host, chronic malaria infection may persist longer than in naive hosts.
We now propose a new model for natural immunity to the asexual blood stages of P. falciparum that specifically focuses on the effect on parasite densities, since it is undoubtedly the case that the major impact of acquired immunity is to reduce the overall parasite load in infected individuals. This model makes predictions of the age patterns of patent parasitemia and of the geometric mean parasite density. Infections in naive hosts mimic the levels of parasitemia reached by P. falciparum infections induced to treat neurosyphilis in the United States (Milledgeville Hospital, Milledgeville, GA and National Institutes of Health Laboratories, Columbia, SC from 1940 to 1963, Figure 1
).15 In previously exposed hosts, we model a reduction in the densities that depends on the host’s history of infection. This simulated acquired immunity does not affect the duration of the infections although it can reduce the period for which parasite density is above the detection limits used in field malariology.
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Here, we present how the model has been fitted to prevalence, density, and multiplicity data of P. falciparum obtained from different epidemiologic settings in Ghana, Nigeria, and Tanzania, and show that it gives realistic predictions of age-prevalence and age-density relationships across a range of transmission intensities. This model forms one component of a comprehensive dynamic model for the transmission cycle of the P. falciparum parasite, and of malaria morbidity, and mortality, as well as of cost-effectiveness of control strategies.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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and 2) that were chosen to represent a range of transmission intensities. In addition, datasets from Saradidi, Kenya and from interventions carried out in Matsari village (Garki project), Nigeria were used to fit the model for the incidence of infection, as described in the accompanying paper.16 Data from the pre-intervention phase of the Garki project collected in Rafin Marke, Matsari, and Sugungum villages, and from Navrongo, Idete, and Namawala were used to fit the model for three distinct cross-sectional outcomes in the absence of intervention: 1) age-specific prevalence of patent parasitemia, 2) age-specific geometric mean density of parasitemia in positive individuals, and 3) age-specific multiplicity measured by polymerase chain reaction–restriction fragment length polymorphisms (PCR-RFLP) of the merozoite surface protein 216 gene (data from Navrongo only).
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and Table 1
) and these entomologic data were used as input to the simulation model. In all sites except Idete, the health system at the time of the surveys treated only a small proportion of the clinical malaria episodes; in the Idete study, we calculate from the published results17,18 that the village dispensary treated approximately 64% of clinical malaria attacks and our simulations assume this coverage of effective treatment. Parasite densities in the Garki dataset were recorded by scanning a predetermined number of microscope fields on the thick blood film and recording how many had one or more asexual parasites visible. We converted these values to numbers of parasites/microliter by assuming a Poisson distribution for the number of parasites per field and a blood volume of 0.5 mm3 per 200 fields.9 In the other field studies, parasites were counted against leukocytes and converted to nominal parasites/microliter assuming the usual (though biased) standard of 8,000 leukocytes/microliter.19 The biases in density estimates resulting from these different techniques was accounted for by multiplying the observed parasite densities with constant values estimated for Garki (
0) and non-Garki (1) field studies to rescale them to the values in the malariatherapy patients (determined using the methods of Earle and Perez20).
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Durations of infection. We treat each infection as monoclonal, but note that the infection process allows multiple infections during the same time interval. Each new infection j, initiated in individual i at time t0, is assigned a duration, i.e., tmax(i,j). The duration is defined as the time interval between the first and last days of patent parasitemia that would be observed in the absence of adjustments to allow for host immunity. The present model does not allow the immune status of the host to affect the simulated period for which the infection persists, although the immune status does affect the duration of patent infection by modifying the proportion of time points for which the parasites are at detectable densities.
The simulated infection starts after the pre-erythrocytic latent period lp, set to three time intervals, i.e., 15 days that correspond to the hepatic stage of the life cycle plus the pre-patent blood stage infection. We use the index 
to denote the time since the start of patent infection, i.e., = 1 corresponds to the first five-day interval when the infection becomes patent, three time units after the infection event.
The duration of infection is randomly sampled from a distribution equivalent to that of malariatherapy patients treated in Milledgeville, Georgia.15 Infections in patients from this hospital persisted for longer than those in South Carolina, and this appears to reflect more conservative policies (more treatment) in the latter hospital. We therefore assume that the Georgia infections were more similar to untreated natural infections in a typical malaria-endemic setting of Africa. The period of follow-up after the last positive slide also varied. Thus, confidence that an infection was spontaneously cleared also varied. We considered only patients who did not receive any anti-malarial treatment on their last day of positivity, and for each of these determined t1 (i), the duration of follow-up after the last recorded day of positivity. For a series of cutoffs, t1*, we considered that subset of patients for whom t1 (i)> t1*, and calculated the mean of the logarithm of the observed durations for those patients who were included. We surmise that the subset giving the highest value of this mean best approximates the behavior of untreated natural infections. The maximum value of this mean was 5.13 (corresponding to a geometric mean of 169.0 days), which was computed from a subset comprising those 47 patients for whom t1 (i) > 2 months. In this set of patients, the standard deviation of the natural log of the duration was 0.80, and the distribution of durations in this subset of patients was approximately log normal. In the simulations we therefore sampled the durations of untreated infections using
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Expected densities of single infections.
At each time point, = 0,1, ... max; (i,j), the density, y(i,j,), of the infection j in host i, is set by first determining the expected log density, E(ln(y0(i,j,))), that would apply in the absence of previous exposure. To determine E(ln(y0(i,j,))), we first determine the mean logarithm of the densities of malariatherapy patients in the Georgia hospital specific for the age of the infection () and for the pre-defined duration (tmax). The distributions of densities for the Georgia patients were summarized by grouping the data into five-day categories according to the time since the first day of patent asexual parasitemia and according to durations, and then using moving average smoothing of the logarithmically transformed densities to obtain a function, yG(,tmax) (Figure 1), which is a simple description of average densities experienced in the course of a single malaria infection of a previously naive host.
The actual densities experienced by different malaria therapy patients varied widely even when the inocula were of the same strain, and were not greatly affected by the size of the inoculum.21 In our model this between-host variation is captured by assigning to each individual i in the simulated population a value, (d(i)), drawn from a log-normal distribution (geometric mean 1, variance 
i2), which multiplies the densities so that
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This empirically determined function provides a description of single infections that thus captures the effects of blood stage immunity that the infection stimulates against itself. This includes effects of innate immunity, antigenic variation, and variant independent anti-merozoite immunity.3
Adjustments are then applied to E(ln(y0(i,j,))) to allow for immunity acquired as a result of exposure to previous infections and for co-infection in hosts with multiple infections. These determine the adjusted expected value, E(ln(y(i,j,))). The simulated density y(i,j,) is then determined by sampling its logarithm using normal distributions centered on E(ln(y(i,j,))).
Effects of acquired immunity on the expected parasite density. Acquired immunity to erythrocytic stages of the parasite is related to cumulative exposure to asexual parasites. In our model the extent of acquired immunity depends both on the diversity of the parasites to which the host has been exposed and on the cumulative density of total parasitemia. In the current version we assume no decay of this component of natural immunity in the absence of infections.
The first trigger function is the cumulative density (parasites/microliter of blood x days) of asexual parasitemia since birth up to time t for individual i, i.e.,
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where Y(i,) is the total parasite density of individual i at time , and a is the age at time t. Xy(i,j,t) measures the total antigenic stimulus to which the host has been exposed less the exposure due to infection j, which is measured by the term 
t0,jt y(i, j, ) d. The latter term must be subtracted to avoid allowing twice for acquired immunity stimulated by infection j itself.
We parameterize the effect (Dy) of this exposure on parasite densities with a Hill function of the form
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where we omit the indices from Xy(i,j,t) to simplify the notation, and where Xy* is a parameter to be estimated, which takes a real positive value, and where Dy consequently takes a value 1 when Xy = 0, and is small but positive when Xy is large.
The second trigger function is the cumulative number of prior infections that the host has experienced, i.e.,
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This term monotonically increases with the size of the repertoire of non-variant polymorphic antigens to which the host has been exposed, and therefore provides a measure of the antigenic stimulus that they provide. We assume that the effect of this on parasite densities can be measured by a further, one-parameter sigmoidal function where the further parameter Xh* is constrained to take a positive value, i.e.,
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The third trigger function is the age of the host a, which is inversely related to the extent of maternally derived protection. We treat the level of maternal protection as independent of maternal exposure. In case of a low transmission level, few infants will be exposed during the first few months of life, so maternal immunity is irrelevant. If transmission is frequent, then all mothers will have similar immune status.22
We parameterize the multiplication factor that models the effect of maternal immunity Dm with a decay function that lies between 0 (maximal effect, corresponding to the status at birth), and 1 (no reduction in density), such that
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where am*, the half-life of the maternal immunity, and 
m, the maternal protection at birth, are additional parameters to be estimated. We then compute the expected density for single infections as
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Effects of concurrent infections. When the host is infected with more than one infection at the same time, innate immunologic responses, together with other density-dependent regulatory mechanisms, lead to a reduction in the overall density to below that expected if the infections did not interact. We estimate this effect by assuming the density of each component infection to be multiplied by a term constrained to take a value between 1 (no interaction between co-infections) and 1/M(t), where M (t) is the total multiplicity of concurrent infections at time t. The value 1/M(t) corresponds to reduction of the total density to that expected if there was only a single infection. To achieve this we define a further parameter Dx, constrained to be between 0 and 1 and set
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Distributions of parasite densities and determination of expected prevalence.
The predicted densities for each individual in the simulated population, y(i,j,), are sampled around the expectations of their logarithms, using log-normal distributions. The variance of these log-normal distributions, which is estimated from the field data during the fitting process, comprises both variation between hosts, and variation within individual hosts.
Variation between hosts is quantified by the term i2, while variation within individual hosts is quantified by a term y2(i, j, ), which includes an effect of the cumulative exposure of the host, and is necessary to obtain a good fit. Thus,
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where the parameters 02 and Xv* need to be estimated.
It follows that the simulated densities are distributed as
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The total density at time t in host i is then the sum of the densities of the various co-infections j.
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Entomologic inoculation rate. The input to the model is the annual cycle of the EIR. In the case of the Garki sites in Nigeria, the EIR was assessed using human night bait collections and dissections of the salivary glands of the Anopheles mosquitoes to give the sporozoite rate. A random effects logistic model was used to give season- and village-specific estimates of the sporozoite rate. The EIR values were then rescaled so the annual totals corresponded to the published values of 18, 68, and 132 for the villages of Rafin Marke, Matsari, and Sugungum, respectively9.
In the cases of Idete (annual EIR = 584) and Namawala (annual EIR = 329) in Tanzania and Navrongo in Ghana (annual EIR = 418), the human biting rate was calculated using light trap collections, with adjustment for the relative numbers of mosquitoes caught by human landing and light traps.23,24 The sporozoite rates were estimated by using an enzyme-linked immunosorbent assay (ELISA) to test the heads and thoraces of the captured anophelines for sporozoites.
For each of the sites, which were assumed to be at equilibrium, an estimate of the daily inoculation rate was assigned to each five-day period, based on re-analysis of the original data, and averaging estimates for the same season, where data were available from multiple years. Where the observed dry season biting rate was too low for an estimate of the EIR to be made, a value was assigned equal to 1% of the average rate for the rest of the year. The inoculation rate estimates for Saradidi in Kenya, based on human landing collections and sporozoite ELISA techniques, were taken directly from the report by Beier and others.25
Implementation and fitting of the model.
Since the model parameters are not identifiable from the data from single transmission settings or from single outcome measures, estimates were made by maximizing the joint likelihood for all of the quantities listed in Table 1. Each likelihood evaluation required a stochastic individual-based simulation of a human population comprising 10,000 individuals for each of the 6 datasets (corresponding to the malariologic patterns in 6 different African villages). These simulations were implemented in Fortran 95 using five-day time steps, with as input the EIR for each five-day period. Each simulation began by simulating the exposure of the population over a whole lifetime prior to the period for which the data applied (to ensure the correct values of the cumulative exposure variables at the start of the monitoring period). For this warm-up period the same average annual cycle of the EIR (estimated from the available entomologic data) was assumed to have recurred since the birth of the oldest member of the simulated population. A demographic model was used that ensured that the simulated population remained stable with the same age-distribution throughout (Appendix).
The actual surveys carried out in the six villages were simulated, and predictions thus made for the distribution of parasite densities in each age group at each survey. Simulated prevalence was defined by comparing each predicted parasite density with the limit of detection used in the actual field study. By comparing observed and simulated geometric mean densities we were able to optimize the parameters of our model for immunologic control of parasite densities. The comparison of the observed with simulated prevalence allowed us at the same time to estimate the variances of the parasite density distributions.
The likelihood was computed for each of the outcomes listed in Table 1 separately for each age group and each survey. Binomial likelihoods were used for the prevalence of patent parasitemia, and a normal likelihood for the mean log parasite density among slides positive by microscopy. For the Navrongo dataset Poisson likelihoods were calculated for the total numbers of distinct parasite infections detected by polymerase chain reaction–restriction fragment length polymorphism in the sampled individuals in each age group, and at each survey.
The overall loss function was computed as a weighted sum of the negative log likelihoods for each of these components across all age-groups, surveys, and sites (Tables 1 and 2). Each age-specific prevalence or multiplicity assessment was given unit weight, but we weighted the log likelihoods for parasite densities by a factor of 10 (so that the prevalence and density had roughly equal weight in the final loss function). We also included in the fitting process simulation of the infection process during the intervention phase in Matsari village16 (model B). The log-likelihood for the prevalence data from this simulation was also subtracted from the overall loss function (Table 2). The goodness of fit of the model to the data was assessed graphically.
Simulated annealing26,27 was used to identify the values of the parameters that minimized the loss function (Table 3). Approximate confidence intervals were obtained by estimating the Fisher information for the parameters. This was done by least squares fitting of local quadratic approximations to the (stochastic) log likelihood surface.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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), with the critical value of the curve Xy*, which correspond to a mean age of 7.1 years (SD = 10.1) in the lowest EIR setting (Rafin Marke) and 5.3 years (SD = 8.5) in Navrongo.
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) with the critical value of this curve Xh*, which corresponded to a mean age of 27.5 years (SD = 2.5) in Rafin Marke and 11.4 years (SD = 1.2) in Navrongo. The predicted age pattern of the effects of maternal immunity on the densities (equation 7) (Figure 3c) shows the expected decay of this component of immunity in the first few months of life. The distributions of the densities vary by individual host (equation 2), and there is also stochastic within-host variation in the densities by exposure (equation 10) (Figure 3d). The model did not include any explicit effect of parasite density on the infections emerging from the liver (see Smith and others16), and the best fitting value of Dx, explicitly quantifying interactions between co-infections (equation 9), was 0, which limited the interactions between concurrent infections to those induced by acquired immunity. We explored several different parameterizations to try to capture interactions between co-infections, but we were not able to improve the fit of our models.
The model also assumed that each inoculation behaves as a single parasite clone. This is a simplification since many oo-cysts in the field are heterozygous29,30 and result in mixed populations of sporozoites (Ranford-Cartwright L, unpublished data). However, we were not able to improve the fit of the model by allowing for diversity in the inoculum.
A good fit of the model to the parasite densities could only be achieved by including parameters to allow for biases introduced by different methods for quantifying parasite densities. We aimed to adjust all the modeled densities to the same scale as that in the malariatherapy studies by introducing two parameters , one for the Garki dataset (where the nominal densities were much lower than our model predicted), and one for the other field studies, where the nominal densities were higher than our predictions (Table 3). These adjustments ensured that our predicted densities in naive individuals corresponded on average to those in malariatherapy patients but do not allow us to separate ethnic variations from effects of differences among study sites in the quantification of parasitemia. In all settings, P. falciparum showed the characteristic age-prevalence pattern found in malaria-endemic areas, with the highest prevalence in young children, reaching almost 100% in those sites with EIRs > 200 infectious bites per person per year (Figure 4).
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Despite the highest recorded EIR in Navrongo, the prevalence of P. falciparum was relatively low in the young children; thus, this setting had an anomalously high age of peak prevalence (Figure 4).31 The age at which the multiplicity reaches a peak is also anomalously high in Navrongo, in comparison to other sites that have been studied (Figure 5).13,31
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). There was little peak shift in the geometric mean densities in either the data or the model predictions, which suggested that the highest densities should be found in the 1–2-year-old children. Comparison of the model predictions with the summary of field studies of prevalence in children less than five years found a good fit, with only a small tendency for the predicted prevalence to be below that observed (Figure 7).
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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The realism of the model presented here is limited by a number of gaps in our current knowledge of malaria. For example, no attempt has been made to simulate detailed immunologic processes. This is justified by the lack of agreed immunologic proxy markers for protection in malaria. Thus, models of malaria immunity that aim to predict morbidity and mortality cannot be validated against immunologic data. Modeling T cell dynamics or antibody levels would thus serve only to increase the complexity of the models without adding to the validity. Our model assumed that the main mechanisms controlling parasite densities do not decay in the absence of stimulation because we have no comprehensive database from which to estimate rates of decay. It would be possible to analyze predictions from sets of models fitted to the same data, assuming different rates of decay, but this would require substantially more computing power than the analysis of a single model.
Inhabitants of endemic areas who temporarily move away are more vulnerable to malaria when they return, but it is unclear whether this results from loss of anti-parasite immunity or of parasitologic tolerance. Some specific immune responses against P. falciparum have been shown to be highly labile (e.g., Kinyanjui and others33), but the epidemiologic consequences of this are yet to be elucidated. Even after several decades free of malaria, people in the central highlands of Madagascar retained some protective immunity to P. falciparum.34
To reproduce realistic age-prevalence and age-density curves, we required only to simulate effects of acquired immunity on parasite densities, without any decay in immunity over time or any explicit effect on duration of infection. In common with many other diseases with acquired immunity,35,36 as P. falciparum transmission increases, there is a characteristic decrease in the age at which the maximum prevalence is reached and increases in the actual peak prevalence. Our model was also able to reproduce these shifts.
Terms both in the cumulative parasite load, Xy, and cumulative number of infections, Xh, were needed, which confirmed that models are inadequate in which acquired immunity is a simple function only of the number of strains to which the host has been exposed. The quantitative load of parasites resulting from repeated inoculations is also important in our proposed model. This could reflect effects of the repertoire of antigen variants that have been expressed in the host, as well as undifferentiated effects of the actual number of parasites. Our formulation differs from models that assume the parasite population to comprise a limited number of strains, or that strain-specific immunity can lead to complete protection.
As with all mathematical models for infectious diseases, we made a number of simplifications. First, we assumed that ethnic differences in the response to P. falciparum malaria between sites are small, although such differences are known to occur in sub-Saharan Africa.37,38
Although our model showed satisfactory overall fits to the various datasets across a wide range of transmission intensities, a somewhat poorer fit was observed with the Navrongo data. A likely explanation for this relatively poor fit to this more recently collected dataset is that we assumed that the impact of anti-malaria treatment was negligible. This assumption is clearly appropriate in the case of the historical data from sites where there was little or no effective treatment, and enabled us to avoid the need to model treatment effects. However, in sub-Saharan Africa early diagnosis and treatment is currently the backbone of malaria control.39 We needed to use recent data to fit the model to age-multiplicity patterns because genotyping techniques have only recently become available. The data from Navrongo that we used for this come from an area with generally poor access to primary health services, although there are four health centers and a hospital. A fraction of the population sleeps under mosquito nets after a successful trial carried out a decade ago,40 and a pilot community health program has treated many of the acute illnesses, including fever episodes, in young children.41 The poorer fit of our model to the Navrongo data than to the other datasets suggests that these interventions have reduced levels of malaria infection in Navrongo below those to be expected at the high inoculation rates observed there.
Other limitations arise because our model was fitted to cross-sectional data, rather than to patterns of variation in parasitemia within individuals, and therefore it is more successful in reproducing the former than the latter. Longitudinal patterns within individuals might be better reproduced by fitting within-host models explicitly incorporating effects of antigenic variation to repeated assessments of parasitemia with short time intervals. Such models have been fitted to data from malariatherapy patients3,4,42 but there is no agreement that any particular such model is appropriate. By basing the time course of parasitemia on an empirical description of the malariatherapy data, we attempted to incorporate the dynamics of antigenic variation without explicitly modeling the processes that lead to it. Because our model is individual based and therefore makes predictions of sequential patterns of parasitemia within simulated hosts, it would be possible to amend it to include a more realistic model of within-host dynamics but there are few datasets from endemic areas to which longitudinal patterns of parasitemia with short time-intervals can be fitted.
We conclude that this model appears to reproduce reasonably well the parasitologic patterns seen in malariologic surveys in endemic areas. Development of improved models will require fitting to data on longitudinal patterns of parasitemia in semi-immune individuals, in particular to data from people whose exposure is interrupted by well-documented periods of protection from infection. Such models are needed to understand the consequences of malaria interventions in areas of infrequent or unstable transmission. The present model is appropriate for predicting the impact of interventions such as vaccination in the context of continual exposure to P. falciparum.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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Real populations are created by processes that vary over time, and in sub-Saharan Africa they are growing.43 In a simulated population it is not in general possible to keep the age distribution stable by applying current birth and death rates. Usually a stable age distribution can only be maintained by removing more simulated individuals at each age and time point than the deaths predicted by a realistic mortality model. We separate processes maintaining the age-distribution of the human population from the demographic effects of malaria, which we do not aim to model, by simulating out migration. We use age-specific out-migration rates to those needed to maintain a stable population. In fitting our parasitologic model we do not simulate any mortality. The same approach is used in models that do simulate mortality, in which the out migration rates are reduced to compensate for the deaths.
Age distribution of the simulated human population. We simulate age distributions based on those recorded for African demographic surveillance sites.44 Stable life tables cannot be constructed for these age distributions without including age-specific immigration. We avoid simulating immigration because this would require us to clone simulated individuals, which would alter the variance structure. We fit a continuous parametric curve to smooth the age distribution because the distributions summarized in published data are themselves smooth, and fewer in-migrations are needed. A bathtub function for mortality rates (higher in infants and older people) gives a good fit to historical European data (Safan M, unpublished data) and separates parameters of the age distribution from those of population growth.
If 
is the Malthusian parameter, corresponding to the exponent of an exponential growth model, then a stationary age-distribution is given by f(a) where
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Fitting the parameters of the age distribution.
We set , the population growth rate, to 0 and estimate the parameters µ0, µ1, 0, and 1 and by least squares fitting of the logarithmically transformed percentages of the population in each age group to those of real African populations. Because the observed age distribution is aggregated into a limited number of groups, the parameters µ0 and 0 are difficult to fit. We require a high number of removals in the first year of life, so models with high infant mortality do not require immigration of infants. We specify a high removal rate of infants and constrain the age-specific removal rates in the first year of life so that there are four neonatal removals for every six post-neonatal one. The age-specific removal rates of the simulated population are then µ(a) = µ0e–oa+ µ1e1a.
At each five-day time point, we compare the age-specific cumulative numbers of individuals to that of the target population, and excess individuals are out migrated. After a run-in period, the age distribution based on Tanzanian data (Figure 8a) is approximately stable (Figure 8b).
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Received September 18, 2005. Accepted for publication February 7, 2006.
Acknowledgments: We thank Dan Anderegg for editorial assistance and Paulette Rosé for assistance in retrieving the Garki data. We also thank the members of the Technical Advisory Group (Michael Alpers, Paul Coleman, David Evans, Brian Greenwood, Carol Levin, Kevin Marsh, F. Ellis McKenzie, Mark Miller, and Brian Sharp), the Project Management Team at the Program for Appropriate Technology in Health (PATH) Malaria Vaccine Initiative, and GlaxoSmith-Kline Biologicals S.A for their assistance.
Financial support: The mathematical modeling study was supported by the PATH Malaria Vaccine Initiative and GlaxoSmithKline Biologicals S.A.
Disclaimer: Publication of this report and the contents hereof do not necessarily reflect the endorsement, opinion, or viewpoints of the PATH Malaria Vaccine Initiative or GlaxoSmithKline Biologicals S.A.
* Address correspondence to Thomas Smith, Swiss Tropical Institute, Socinstrasse 57, PO Box, CH-4002, Basel, Switzerland. E-mail: nicolas.maire{at}unibas.ch, Thomas-A.Smith{at}unibas.ch 
Authors’ addresses: Nicolas Maire, Thomas Smith, and Amanda Ross, Swiss Tropical Institute, Socinstrasse 57, PO Box, CH-4002, Basel, Switzerland, Telephone: 41-61-284-8273, Fax: 41-61-284-8105, E-mails: nicolas.maire{at}unibas.ch, Thomas-A.Smith{at}unibas.ch, and amanda.ross{at}unibas.ch. Seth Owusu-Agyei, Navrongo Health Research Center, Ministry of Health, Navrongo Upper East Region, Ghana. Klaus Dietz, Department of Medical Biometry, University of Tübingen, Westbahnhofstrasse 55, 72070 Tübingen, Germany, Telephone: 49-7071-29-78253, Fax: 49-7071-29-5075. E-mail: klaus.dietz{at}uni-tuebingen.de. Louis Molineaux, Peney-Dessus, CH-1242 Satigny, Geneva, Switzerland.
Reprint requests: Thomas Smith, Swiss Tropical Institute, Socinstrasse 57, PO Box, CH-4002, Basel, Switzerland. E-mail: Thomas-A.Smith{at}unibas.ch.
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXREFERENCES |
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= data (error bars show 95% confidence intervals); 
= model predictions.
= field data reported by Beier and others.