Modeling malaria vaccines. I: New uses for old ideas

Starting from a modification of the model of malaria transmission developed for the Garki project, this paper develops a model containing variables relevant to the stimulation of malaria vaccination programs. Modifications include (1) integration of maintenance of immunity dependent on boosting and the possibility of loss of immunity; (2) introduction of a boosting factor distinct from susceptibility to infection; (3) reinterpretation of the epidemiological compartments of positive immunes and nonimmunes in terms of severity of disease rather than just infection; (4) interpretation of the different stage-specific levels of immunity; (5) discrimination between different susceptibilities for the immune and nonimmune classes; (6) reformulation of the expression for acquisition of immunity to be biologically more acceptable. Simulations using the Garki model, Nedelman's modification of it, and our Basic model compare the similarities and differences in the predictive behavior of the models. Simulations using the Basic model reproduce observed periodic fluctuations of malaria attributed to the interplay of transmission-blocking immunity and loss of immunity in the absence of boosting in areas of unstable malaria transmission.     

Census error and the detection of density dependence

1. Studies aiming to identify the prevalence and nature of density dependence in ecological populations have often used statistical analysis of ecological time-series of population counts. Such time-series are also being used increasingly to parameterize models that may be used in population management. 2. If time-series contain measurement errors, tests that rely on detecting a negative relationship between log population change and population size are biased and prone to spuriously detecting density dependence (Type I error). This is because the measurement error in density for a given year appears in the corresponding change in population density, with equal magnitude but opposite sign. 3. This effect introduces bias that may invalidate comparisons of ecological data with density-independent time-series. Unless census error can be accounted for, time-series may appear to show strongly density-dependent dynamics, even though the density-dependent signal may in reality be weak or absent. 4. We distinguish two forms of census error, both of which have serious consequences for detecting density dependence. 5. First, estimates of population density are based rarely on exact counts, but on samples. Hence there exists sampling error, with the level of error depending on the method employed and the number of replicates on which the population estimate is based. 6. Secondly, the group of organisms measured is often not a truly self-contained population, but part of a wider ecological population, defined in terms of location or behaviour. Consequently, the subpopulation studied may effectively be a sample of the population and spurious density dependence may be detected in the dynamics of a single subpopulation. In this case, density dependence is detected erroneously, even if numbers within the subpopulation are censused without sampling error. 7. In order to illustrate how process variation and measurement error may be distinguished we review data sets (counts of numbers of birds by single observers) for which both census error and long-term variance in population density can be estimated. 8. Tests for density dependence need to obviate the problem that measured population sizes are typically estimates rather than exact counts. It is possible that in some cases it may be possible to test for density dependence in the presence of unknown levels of census error, for example by uncovering nonlinearities in the density response. However, it seems likely that these may lack power compared with analyses that are able to explicitly include census error and we review some recently developed methods.     

Parasite prevalence and the size of host populations: an experimental test

Although important in epidemiological theory, the relationship between the size of host populations and the prevalence of parasites has not been investigated empirically. Commonly used models suggest no relationship, but this prediction is sensitive to assumptions about parasite transmission. In laboratory populations, I manipulated the size of Tribolium castaneum flour beetle populations and measured the prevalence and distribution of a parasitic mite, Acarophenax tribolii. I found that parasite prevalence did not vary for a wide range of host population sizes. However, prevalence was lower in populations with less than 40 hosts. This effect cannot be attributed to changes in host population density because host density was held constant among treatments. The reduction in prevalence of small populations below a threshold that I observed is predicted by the extinction debt model, but it is not expected from models of host-parasite interactions that assume density-dependent transmission. The distribution of parasites, measured using Lloyd's patchiness index, was not affected by host population size. The mean crowding of parasites, however, was negatively related with host density. Finally, the prevalence of parasites in large populations did not differ from that found in sets of smaller patches as long as the smaller populations in aggregate were equivalent in size to the large population.     

The need for double-sampling designs in survival studies: an application to monitor PEPFAR

Summary .  In 2007, there were 33.2 million people around the world living with HIV/AIDS ( UNAIDS/WHO, 2007 ). In May 2003, the U.S. President announced a global program, known as the President's Emergency Plan for AIDS Relief (PEPFAR), to address this epidemic. We seek to estimate patient mortality in PEPFAR in an effort to monitor and evaluate this program. This effort, however, is hampered by loss to follow-up that occurs at very high rates. As a consequence, standard survival data and analysis on observed nondropout data are generally biased, and provide no objective evidence to correct the potential bias. In this article, we apply double-sampling designs and methodology to PEPFAR data, and we obtain substantially different and more plausible estimates compared with standard methods (1-year mortality estimate of 9.6% compared to 1.7%). The results indicate that a double-sampling design is critical in providing objective evidence of possible nonignorable dropout and, thus, in obtaining accurate data in PEPFAR. Moreover, we show the need for appropriate analysis methods coupled with double-sampling designs.     

On Bivariate Cumulative Damage Models With Application

Bivariate cumulative damage models are proposed where the responses given the damages are independent random variables. The bivariate damage process can be either bivariate Poisson or bivariate gamma. A bivariate continuous cumulative damage model is investigated in which the responses given the damages have gamma distributions. In this case evaluation of the joint density function and bivariate tail probability function is facilitated by expanding the gamma distributions of the conditional responses by Laguerre polynomials. This approach also leads to evaluation of associated survival models. Moments and estimating equations are discussed. In addition, a bivariate discrete cumulative damage model is investigated in which the responses given the damages have a distribution chosen from a class that includes the negative binomial, the Neyman Type‐A, the Polya‐Aeppli, and the Lagrangian Poisson. Probabilities are obtained from recursive formulas which do not involve cancellation error as all quantities are non‐negative. Moments and estimating equations are presented for these models also. The continuous and the discrete models are applied to describe the rise of systolic and diastolic blood pressure with age.     

Expected estimating equations for missing data, measurement error, and misclassification, with application to longitudinal nonignorable missing data

Summary .   Missing data, measurement error, and misclassification are three important problems in many research fields, such as epidemiological studies. It is well known that missing data and measurement error in covariates may lead to biased estimation. Misclassification may be considered as a special type of measurement error, for categorical data. Nevertheless, we treat misclassification as a different problem from measurement error because statistical models for them are different. Indeed, in the literature, methods for these three problems were generally proposed separately given that statistical modeling for them are very different. The problem is more challenging in a longitudinal study with nonignorable missing data. In this article, we consider estimation in generalized linear models under these three incomplete data models. We propose a general approach based on expected estimating equations (EEEs) to solve these three incomplete data problems in a unified fashion. This EEE approach can be easily implemented and its asymptotic covariance can be obtained by sandwich estimation. Intensive simulation studies are performed under various incomplete data settings. The proposed method is applied to a longitudinal study of oral bone density in relation to body bone density.     

Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models

Line transect sampling is one of the most widely used methods for animal abundance assessment. Standard estimation methods assume certain detection on the transect, no animal movement, and no measurement errors. Failure of the assumptions can cause substantial bias. In this work, the effect of error measurement on line transect estimators is investigated. Based on considerations of the process generating the errors, a multiplicative error model is presented and a simple way of correcting estimates based on knowledge of the error distribution is proposed. Using beta models for the error distribution, the effect of errors and of the proposed correction is assessed by simulation. Adequate confidence intervals for the corrected estimates are obtained using a bootstrap variance estimate for the correction and the delta method. As noted by Chen (1998, Biometrics 54, 899-908), even unbiased estimators of the distances might lead to biased density estimators, depending on the actual error distribution. In contrast with the findings of Chen, who used an additive model, unbiased estimation of distances, given a multiplicative model, lead to overestimation of density. Some error distributions result in observed distance distributions that make efficient estimation impossible, by removing the shoulder present in the original detection function. This indicates the need to improve field methods to reduce measurement error. An application of the new methods to a real data set is presented.     

Mammal density and patterns of ectoparasite species richness and abundance

Patterns of species richness, prevalence and abundance of ectoparasites have rarely been investigated at both the levels of populations and species of hosts. Here, we investigated the effects in changes in small mammal density on species richness, abundance and prevalence of ectoparasitic fleas. The comparative analyses were conducted for different small mammal species and among several populations during a long-term survey. We tested the hypothesis that an increase in host density should be linked with an increase in parasite species richness both among host species and among populations within host species, as predicted by epidemiological models. We also used host species density data from literature. We found that host density has a major influence on the species richness of ectoparasite communities of small mammals among host populations. We found no relationship between data of host density from the literature and parasite species richness. In contrast with epidemiological hypotheses, we found no relationships between abundance, or prevalence, and host density, either among host species or among host populations. Moreover, a decrease in abundance of fleas in relation with an increase in host density was observed for two mammal species (Apodemus agrarius and A. flavicollis). The decrease or the lack of increase in flea abundance in relation with an increase in host density suggests anti-parasitic behavioural activities such as grooming.     

Structural risk minimization: a robust method for density-dependence detection and model selection

Statistically distinguishing density‐dependent from density‐independent populations and selecting the best demographic model for a given population are problems of primary importance. Traditional approaches are PBLR (parametric bootstrapping of likelihood ratios) and Information criteria (IC), such as the Schwarz information criterion (SIC), the Akaike information criterion (AIC) or the Final prediction error (FPE). While PBLR is suitable for choosing from a couple of models, ICs select the best model from among a set of candidates. In this paper, we use the Structural risk minimization (SRM) approach. SRM is the model selection criterion developed within the Statistical learning theory (SLT), a theory of great generality for modelling and learning with finite samples. SRM is almost unknown in the ecological literature and has never been used to analyze time series. First, we compare SRM with PBLR in terms of their ability to discriminate between the Malthusian and the density‐dependent Ricker model. We rigorously repeat the experiments described in a previous study and find out that SRM is equally powerful in detecting density‐independence and much more powerful in detecting density‐dependence. Then, we compare SRM against ICs in terms of their ability to select one of several candidate models; we generate, via stochastic simulation, a huge amount of artificial time series both density‐independent and dependent, with and without exogenous covariates, using different dataset sizes, noise levels and parameter values. Our findings show that SRM outperforms traditional ICs, because generally a) it recognizes the model underlying the data with higher frequency, and b) it leads to lower errors in out‐of‐samples predictions. SRM superiority is specially apparent with short time series. We finally apply SRM to the population records of Alpine ibex Capra ibex living in the Gran Paradiso National Park (Italy), already investigated by other authors via traditional statistical methods; we both analyze their models and introduce some novel ones. We show that models that are best according to SRM show also the lowest leave‐one‐out cross‐validation error.     

Analysis of Yield-Density Models for Intercropping Experiments

Yield density models are used to describe the relationship between the yield of one or more crops and densities of planting. In this paper, we propose a correlated error structure for a linear yield-density model for intercopping and competition experiments. Four possible estimators of the parameters of the error structure are evaluated using a Monte Carlo study. The estimators are compared on the basis of gain in efficiency as measured by the generalized variance. An example is provided.     

Detection error influences both temporal seroprevalence predictions and risk factors associations in wildlife disease models

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People, energy and avian species richness

Aim To investigate the inter‐relationships between energy availability, species richness and human population density, particularly whether human population density influences the manner in which species richness responds to energy availability. Location British 10‐km grid cells. Methods Using regressions, we investigate how human population density varies with energy availability and the nature of relationships between the numbers of species, classified by abundance and threat categories, and human population density. We then assess whether the relationships between these species richness measures and energy availability are altered when accounting for human population density. We conduct analyses using both independent error models and ones that control for spatial autocorrelation. Results Human population density was strongly and positively correlated with energy availability. Total species richness, and that of unthreatened, threatened, common and moderately common species, increases in a positive decelerating manner with human density. When human population density was taken into account, these species groups exhibited similar species–energy relationships, but the slopes of these relationships were significantly reduced in independent error models and, in the case of total richness, in spatial models. Main conclusions Positive correlations between human density and species richness probably arise as both increase with energy availability. Our data are compatible with the suggestion that high human population densities reduce the rate at which species richness increases with energy availability, but additional research is required before causality can be confirmed.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

Effect of host populations on the intensity of ticks and the prevalence of tick-borne pathogens: how to interpret the results of deer exclosure experiments

Deer are important blood hosts for feeding Ixodes ricinus ticks but they do not support transmission of many tick-borne pathogens, so acting as dead-end transmission hosts. Mathematical models show their role as tick amplifiers, but also suggest that they dilute pathogen transmission, thus reducing infection prevalence. Empirical evidence for this is conflicting: experimental plots with deer removal (i.e. deer exclosures) show that the effect depends on the size of the exclosure. Here we present simulations of dynamic models that take into account different tick stages, and several host species (e.g. rodents) that may move to and from deer exclosures; models were calibrated with respect to Ixodes ricinus ticks and tick-borne encephalitis (TBE) in Trentino (northern Italy). Results show that in small exclosures, the density of rodent-feeding ticks may be higher inside than outside, whereas in large exclosures, a reduction of such tick density may be reached. Similarly, TBE prevalence in rodents decreases in large exclosures and may be slightly higher in small exclosures than outside them. The density of infected questing nymphs inside small exclosures can be much higher, in our numerical example almost twice as large as that outside, leading to potential TBE infection risk hotspots.     

Estimation of Density-Dependent Mortality of Juvenile Bivalves in the Wadden Sea

We investigated density-dependent mortality within the early months of life of the bivalves Macoma balthica (Baltic tellin) and Cerastoderma edule (common cockle) in the Wadden Sea. Mortality is thought to be density-dependent in juvenile bivalves, because there is no proportional relationship between the size of the reproductive adult stocks and the numbers of recruits for both species. It is not known however, when exactly density dependence in the pre-recruitment phase occurs and how prevalent it is. The magnitude of recruitment determines year class strength in bivalves. Thus, understanding pre-recruit mortality will improve the understanding of population dynamics. We analyzed count data from three years of temporal sampling during the first months after bivalve settlement at ten transects in the Sylt-Rømø-Bay in the northern German Wadden Sea. Analyses of density dependence are sensitive to bias through measurement error. Measurement error was estimated by bootstrapping, and residual deviances were adjusted by adding process error. With simulations the effect of these two types of error on the estimate of the density-dependent mortality coefficient was investigated. In three out of eight time intervals density dependence was detected for M. balthica, and in zero out of six time intervals for C. edule. Biological or environmental stochastic processes dominated over density dependence at the investigated scale.     

Modeling observation error and its effects in a random walk/extinction model

This paper examines the consequences of observation errors for the "random walk with drift", a model that incorporates density independence and is frequently used in population viability analysis. Exact expressions are given for biases in estimates of the mean, variance and growth parameters under very general models for the observation errors. For other quantities, such as the finite rate of increase, and probabilities about population size in the future we provide and evaluate approximate expressions. These expressions explain the biases induced by observation error without relying exclusively on simulations, and also suggest ways to correct for observation error. A secondary contribution is a careful discussion of observation error models, presented in terms of either log-abundance or abundance. This discussion recognizes that the bias and variance in observation errors may change over time, the result of changing sampling effort or dependence on the underlying population being sampled.     

A probabilistic neural network approach for modeling and classification of bacterial growth/no-growth data

In this paper, we propose to use probabilistic neural networks (PNNs) for classification of bacterial growth/no-growth data and modeling the probability of growth. The PNN approach combines both Bayes theorem of conditional probability and Parzen's method for estimating the probability density functions of the random variables. Unlike other neural network training paradigms, PNNs are characterized by high training speed and their ability to produce confidence levels for their classification decision. As a practical application of the proposed approach, PNNs were investigated for their ability in classification of growth/no-growth state of a pathogenic Escherichia coli R31 in response to temperature and water activity. A comparison with the most frequently used traditional statistical method based on logistic regression and multilayer feedforward artificial neural network (MFANN) trained by error backpropagation was also carried out. The PNN-based models were found to outperform linear and nonlinear logistic regression and MFANN in both the classification accuracy and ease by which PNN-based models are developed.     

基于不同空间模型的马尾松林生态系统碳密度研究

以江西省马尾松林生态系统为研究对象,基于样地调查及样品碳含量测定结果计算其碳密度,并选取立地、植被及气象等方面的15个因子,采用多元线性逐步回归方法筛选出对生态系统碳密度影响显著的因子,然后分别利用最小二乘模型(OLS)、空间误差模型(SEM)、空间滞后模型(SLM)和地理加权回归模型(GWR)构建生态系统碳密度与其影响因子之间的关系模型,筛选出最优的拟合模型。结果表明:对马尾松林生态系统碳密度影响显著的因子分别为海拔、坡度、土层厚度、胸径、年均温度和年均降水量。4种模型拟合结果均显示碳密度与坡度呈负相关,与海拔、土层厚度、胸径呈正相关。模型的决定系数(R²)由大到小分别为GWR(0.8043)>SEM(0.6371)>SLM(0.6364)>OLS(0.6321),模型均方误差(MSE)与赤池信息准则(AIC)最大的均为OLS模型,最小的均为GWR模型;残差检验表明GWR模型能有效降低模型残差的空间自相关性。综合分析得出GWR模型的拟合效果最优,更适用于江西省马尾松林生态系统碳密度的估测。     

Density-structured models for plant population dynamics

Density-structured models are structured population models in which the state variable is the proportion of populations or sites in a small number of discrete density states. Although such models have rarely been used, they have the advantage that they are straightforward to parameterize, make few assumptions about population dynamics, and permit rapid data collection using coarse density assessment. In this article, we highlight their use in relating population dynamics to environmental variation and their robustness to measurement error. We show that density-structured models are able to accurately represent population dynamics under a wide range of conditions. We look at the effects of including a persistent seedbank and describe numerical approximations for the mean and variance of population size. For simulated data, we determine the extent to which the underlying continuous process may be inferred from density-structured data. Finally, we discuss issues of parameter estimation and applications for which these types of models may be useful.