Impact of Acquired Immunity and Dose‐Dependent Probability of Illness on Quantitative Microbial Risk Assessment

Dose‐response models in microbial risk assessment consider two steps in the process ultimately leading to illness: from exposure to (asymptomatic) infection, and from infection to (symptomatic) illness. Most data and theoretical approaches are available for the exposure‐infection step; the infection‐illness step has received less attention. Furthermore, current microbial risk assessment models do not account for acquired immunity. These limitations may lead to biased risk estimates. We consider effects of both dose dependency of the conditional probability of illness given infection, and acquired immunity to risk estimates, and demonstrate their effects in a case study on exposure to Campylobacter jejuni. To account for acquired immunity in risk estimates, an inflation factor is proposed. The inflation factor depends on the relative rates of loss of protection over exposure. The conditional probability of illness given infection is based on a previously published model, accounting for the within‐host dynamics of illness. We find that at low (average) doses, the infection‐illness model has the greatest impact on risk estimates, whereas at higher (average) doses and/or increased exposure frequencies, the acquired immunity model has the greatest impact. The proposed models are strongly nonlinear, and reducing exposure is not expected to lead to a proportional decrease in risk and, under certain conditions, may even lead to an increase in risk. The impact of different dose‐response models on risk estimates is particularly pronounced when introducing heterogeneity in the population exposure distribution.     

Methodology for Determining the Appropriateness of a Linear Dose‐Response Function

Microbial food safety risk assessment models can often at times be simplified by eliminating the need to integrate a complex dose‐response relationship across a distribution of exposure doses. This is possible if exposure pathways lead to pathogens at exposure that consistently have a small probability of causing illness. In this situation, the probability of illness will follow an approximately linear function of dose. Consequently, the predicted probability of illness per serving across all exposures is linear with respect to the expected value of dose. The majority of dose‐response functions are approximately linear when the dose is low. Nevertheless, what constitutes “low” is dependent on the parameters of the dose‐response function for a particular pathogen. In this study, a method is proposed to determine an upper bound of the exposure distribution for which the use of a linear dose‐response function is acceptable. If this upper bound is substantially larger than the expected value of exposure doses, then a linear approximation for probability of illness is reasonable. If conditions are appropriate for using the linear dose‐response approximation, for example, the expected value for exposure doses is two to three logs₁₀ smaller than the upper bound of the linear portion of the dose‐response function, then predicting the risk‐reducing effectiveness of a proposed policy is trivial. Simple examples illustrate how this approximation can be used to inform policy decisions and improve an analyst's understanding of risk.     

The Effects of Revealed Information on Catastrophe Loss Projection Models’ Characterization of Risk: Damage Vulnerability Evidence from Florida

We examine whether the risk characterization estimated by catastrophic loss projection models is sensitive to the revelation of new information regarding risk type. We use commercial loss projection models from two widely employed modeling firms to estimate the expected hurricane losses of Florida Atlantic University's building stock, both including and excluding secondary information regarding hurricane mitigation features that influence damage vulnerability. We then compare the results of the models without and with this revealed information and find that the revelation of additional, secondary information influences modeled losses for the windstorm‐exposed university building stock, primarily evidenced by meaningful percent differences in the loss exceedance output indicated after secondary modifiers are incorporated in the analysis. Secondary risk characteristics for the data set studied appear to have substantially greater impact on probable maximum loss estimates than on average annual loss estimates. While it may be intuitively expected for catastrophe models to indicate that secondary risk characteristics hold value for reducing modeled losses, the finding that the primary value of secondary risk characteristics is in reduction of losses in the “tail” (low probability, high severity) events is less intuitive, and therefore especially interesting. Further, we address the benefit‐cost tradeoffs that commercial entities must consider when deciding whether to undergo the data collection necessary to include secondary information in modeling. Although we assert the long‐term benefit‐cost tradeoff is positive for virtually every entity, we acknowledge short‐term disincentives to such an effort.     

Dose-Response and Risk Assessment of Airborne Hexavalent Chromium and Lung Cancer Mortality

This study evaluates the dose-response relationship for inhalation exposure to hexavalent chromium [Cr(VI)] and lung cancer mortality for workers of a chromate production facility, and provides estimates of the carcinogenic potency. The data were analyzed using relative risk and additive risk dose-response models implemented with both Poisson and Cox regression. Potential confounding by birth cohort and smoking prevalence were also assessed. Lifetime cumulative exposure and highest monthly exposure were the dose metrics evaluated. The estimated lifetime additional risk of lung cancer mortality associated with 45 years of occupational exposure to 1 microg/m3 Cr(VI) (occupational exposure unit risk) was 0.00205 (90%CI: 0.00134, 0.00291) for the relative risk model and 0.00216 (90%CI: 0.00143, 0.00302) for the additive risk model assuming a linear dose response for cumulative exposure with a five-year lag. Extrapolating these findings to a continuous (e.g., environmental) exposure scenario yielded an environmental unit risk of 0.00978 (90%CI: 0.00640, 0.0138) for the relative risk model [e.g., a cancer slope factor of 34 (mg/kg-day)-1] and 0.0125 (90%CI: 0.00833, 0.0175) for the additive risk model. The relative risk model is preferred because it is more consistent with the expected trend for lung cancer risk with age. Based on statistical tests for exposure-related trend, there was no statistically significant increased lung cancer risk below lifetime cumulative occupational exposures of 1.0 mg-yr/m3, and no excess risk for workers whose highest average monthly exposure did not exceed the current Permissible Exposure Limit (52 microg/m3). It is acknowledged that this study had limited power to detect increases at these low exposure levels. These cancer potency estimates are comparable to those developed by U.S. regulatory agencies and should be useful for assessing the potential cancer hazard associated with inhaled Cr(VI).     

A Risk Analysis for Airborne Pathogens with Low Infectious Doses: Application to Respirator Selection Against Coccidioides immitis Spores

Probability models incorporating a deterministic versus stochastic infectious dose are described for estimating infection risk due to airborne pathogens that infect at low doses. Such pathogens can be occupational hazards or candidate agents for bioterrorism. Inputs include parameters for the infectious dose model, distribution parameters for ambient pathogen concentrations, the breathing rate, the duration of an exposure period, the anticipated number of exposure periods, and, if a respirator device is used, distribution parameters for respirator penetration values. Application of the models is illustrated with a hypothetical scenario involving exposure to Coccidioides immitis, a fungus present in soil in areas of the southwestern United States Inhaling C. immitis spores causes a respiratory tract infection and is a recognized occupational hazard in jobs involving soil dust exposure in endemic areas An uncertainty analysis is applied to risk estimation in the context of selecting respiratory protection with a desired degree of efficacy.     

A Dynamic Model to Assess Microbial Health Risks Associated with Beneficial Uses of Biosolids

There is increasing interest in the development of a microbial risk assessment methodology for regulatory and operational decision making. This document presents a methodology for assessing risks to human health from pathogen exposure using a population-based model that explicitly accounts for properties unique to an infectious disease process, specifically secondary transmission and immunity. To demonstrate the applicability of this risk-based method, numerical simulations were carried out for a case study example in which the route of exposure was direct consumption of biosolids-amended soil and the pathogen present in the soil was enterovirus. The output from the case study yielded a decision tree that differentiates between conditions in which the relative risk from biosolids exposure is high and those conditions in which the relative risk from biosolids is low. This decision tree illustrates the interaction among the important factors in quantifying risk. For the case study example, these factors include biosolids treatment processes, the pathogen shedding rate of infectious individuals, secondary transmission, and immunity. Further refinement in methods for determining biosolids exposures under field conditions would certainly increase the utility of these approaches.     

Model Averaging Using the Kullback Information Criterion in Estimating Effective Doses for Microbial Infection and Illness

Since the National Food Safety Initiative of 1997, risk assessment has been an important issue in food safety areas. Microbial risk assessment is a systematic process for describing and quantifying a potential to cause adverse health effects associated with exposure to microorganisms. Various dose-response models for estimating microbial risks have been investigated. We have considered four two-parameter models and four three-parameter models in order to evaluate variability among the models for microbial risk assessment using infectivity and illness data from studies with human volunteers exposed to a variety of microbial pathogens. Model variability is measured in terms of estimated ED01s and ED10s, with the view that these effective dose levels correspond to the lower and upper limits of the 1% to 10% risk range generally recommended for establishing benchmark doses in risk assessment. Parameters of the statistical models are estimated using the maximum likelihood method. In this article a weighted average of effective dose estimates from eight two- and three-parameter dose-response models, with weights determined by the Kullback information criterion, is proposed to address model uncertainties in microbial risk assessment. The proposed procedures for incorporating model uncertainties and making inferences are illustrated with human infection/illness dose-response data sets.     

QMRA for Drinking Water: 1. Revisiting the Mathematical Structure of Single‐Hit Dose‐Response Models

Dose‐response models are essential to quantitative microbial risk assessment (QMRA), providing a link between levels of human exposure to pathogens and the probability of negative health outcomes. In drinking water studies, the class of semi‐mechanistic models known as single‐hit models, such as the exponential and the exact beta‐Poisson, has seen widespread use. In this work, an attempt is made to carefully develop the general mathematical single‐hit framework while explicitly accounting for variation in (1) host susceptibility and (2) pathogen infectivity. This allows a precise interpretation of the so‐called single‐hit probability and precise identification of a set of statistical independence assumptions that are sufficient to arrive at single‐hit models. Further analysis of the model framework is facilitated by formulating the single‐hit models compactly using probability generating and moment generating functions. Among the more practically relevant conclusions drawn are: (1) for any dose distribution, variation in host susceptibility always reduces the single‐hit risk compared to a constant host susceptibility (assuming equal mean susceptibilities), (2) the model‐consistent representation of complete host immunity is formally demonstrated to be a simple scaling of the response, (3) the model‐consistent expression for the total risk from repeated exposures deviates (gives lower risk) from the conventional expression used in applications, and (4) a model‐consistent expression for the mean per‐exposure dose that produces the correct total risk from repeated exposures is developed.     

Multi‐Exposure Pathway Model to Compare Escherichia coli O157 Risks and Interventions

The relative contributions of exposure pathways associated with cattle‐manure‐borne Escherichia coli O157:H7 on public health have yet to be fully characterized. A stochastic, quantitative microbial risk assessment (QMRA) model was developed to describe a hypothetical cattle farm in order to compare the relative importance of five routes of exposure, including aquatic recreation downstream of the farm, consumption of contaminated ground beef processed with limited interventions, consumption of leafy greens, direct animal contact, and the recreational use of a cattle pasture. To accommodate diverse environmental and hydrological pathways, existing QMRAs were integrated with novel and simplistic climate and field‐level submodels. The model indicated that direct animal contact presents the greatest risk of illness per exposure event during the high pathogen shedding period. However, when accounting for the frequency of exposure, using a high‐risk exposure‐receptor profile, consumption of ground beef was associated with the greatest risk of illness. Additionally, the model was used to evaluate the efficacy of hypothetical interventions affecting one or more exposure routes; concurrent evaluation of multiple routes allowed for the assessment of the combined effect of preharvest interventions across exposure pathways—which may have been previously underestimated—as well as the assessment of the effect of additional downstream interventions. This analysis represents a step towards a full evaluation of the risks associated with multiple exposure pathways; future incorporation of variability associated with environmental parameters and human behaviors would allow for a comprehensive assessment of the relative contribution of exposure pathways at the population level.     

Model Averaging in Microbial Risk Assessment Using Fractional Polynomials

The alleviation of food-borne diseases caused by microbial pathogen remains a great concern in order to ensure the well-being of the general public. The relation between the ingested dose of organisms and the associated infection risk can be studied using dose-response models. Traditionally, a model selected according to a goodness-of-fit criterion has been used for making inferences. In this article, we propose a modified set of fractional polynomials as competitive dose-response models in risk assessment. The article not only shows instances where it is not obvious to single out one best model but also illustrates that model averaging can best circumvent this dilemma. The set of candidate models is chosen based on biological plausibility and rationale and the risk at a dose common to all these models estimated using the selected models and by averaging over all models using Akaike's weights. In addition to including parameter estimation inaccuracy, like in the case of a single selected model, model averaging accounts for the uncertainty arising from other competitive models. This leads to a better and more honest estimation of standard errors and construction of confidence intervals for risk estimates. The approach is illustrated for risk estimation at low dose levels based on Salmonella typhi and Campylobacter jejuni data sets in humans. Simulation studies indicate that model averaging has reduced bias, better precision, and also attains coverage probabilities that are closer to the 95% nominal level compared to best-fitting models according to Akaike information criterion.     

An Integrated Model of Infection Risk in a Health-Care Environment

Certain respiratory tract infections can be transmitted by hand-to-mucous-membrane contact, inhalation, and/or direct respiratory droplet spray. In a room occupied by a patient with such a transmissible infection, pathogens present on textile and nontextile surfaces, and pathogens present in the air, provide sources of exposure for an attending health-care worker (HCW); in addition, close contact with the patient when the latter coughs allows for droplet spray exposure. We present an integrated model of pertinent source-environment-receptor pathways, and represent physical elements in these pathways as "states" in a discrete-time Markov chain model. We estimate the rates of transfer at various steps in the pathways, and their relationship to the probability that a pathogen in one state has moved to another state by the end of a specified time interval. Given initial pathogen loads on textile and nontextile surfaces and in room air, we use the model to estimate the expected pathogen dose to a HCW's mucous membranes and respiratory tract. In turn, using a nonthreshold infectious dose model, we relate the expected dose to infection risk. The system is illustrated with a hypothetical but plausible scenario involving a viral pathogen emitted via coughing. We also use the model to show that a biocidal finish on textile surfaces has the potential to substantially reduce infection risk via the hand-to-mucous-membrane exposure pathway.     

Point Estimates of Cancer Risk at Low Doses

There has been considerable discussion regarding the conservativeness of low-dose cancer risk estimates based upon linear extrapolation from upper confidence limits. Various groups have expressed a need for best (point) estimates of cancer risk in order to improve risk/benefit decisions. Point estimates of carcinogenic potency obtained from maximum likelihood estimates of low-dose slope may be highly unstable, being sensitive both to the choice of the dose–response model and possibly to minimal perturbations of the data. For carcinogens that augment background carcinogenic processes and/or for mutagenic carcinogens, at low doses the tumor incidence versus target tissue dose is expected to be linear. Pharmacokinetic data may be needed to identify and adjust for exposure-dose nonlinearities. Based on the assumption that the dose response is linear over low doses, a stable point estimate for low-dose cancer risk is proposed. Since various models give similar estimates of risk down to levels of 1%, a stable estimate of the low-dose cancer slope is provided by ŝ = 0.01/ED01, where ED01 is the dose corresponding to an excess cancer risk of 1%. Thus, low-dose estimates of cancer risk are obtained by, risk = ŝ × dose. The proposed procedure is similar to one which has been utilized in the past by the Center for Food Safety and Applied Nutrition, Food and Drug Administration. The upper confidence limit, s , corresponding to this point estimate of low-dose slope is similar to the upper limit, q ₁ obtained from the generalized multistage model. The advantage of the proposed procedure is that ŝ provides stable estimates of low-dose carcinogenic potency, which are not unduly influenced by small perturbations of the tumor incidence rates, unlike ₁.     

基于VaR的多阶段金融资产配置模型

本文提出了基于VaR的多阶段金融资产配置模型。进一步以我国经济环境为依托,考虑了未来各种资产收益、工资变动及物价变动的不确定性,对这一模型进行了仿真计算,并与静态模型在最优性上进行了比较,得出了动态模型优于静态模型的结论。在期望财富相同的情况下,基于VaR的多阶段资产配置模型比静态模型的期望损失成本低,承担的风险更小。     

Landslide Risk Models for Decision Making

This contribution presents a quantitative procedure for landslide risk analysis and zoning considering hazard, exposure (or value of elements at risk), and vulnerability. The method provides the means to obtain landslide risk models (expressing expected damage due to landslides on material elements and economic activities in monetary terms, according to different scenarios and periods) useful to identify areas where mitigation efforts will be most cost effective. It allows identifying priority areas for the implementation of actions to reduce vulnerability (elements) or hazard (processes). The procedure proposed can also be used as a preventive tool, through its application to strategic environmental impact analysis (SEIA) of land-use plans. The underlying hypothesis is that reliable predictions about hazard and risk can be made using models based on a detailed analysis of past landslide occurrences in connection with conditioning factors and data on past damage. The results show that the approach proposed and the hypothesis formulated are essentially correct, providing estimates of the order of magnitude of expected losses for a given time period. Uncertainties, strengths, and shortcomings of the procedure and results obtained are discussed and potential lines of research to improve the models are indicated. Finally, comments and suggestions are provided to generalize this type of analysis.     

Development of a novel machine learning-based weighted modeling approach to incorporate Salmonella enterica heterogeneity on a genetic scale in a dose–response modeling framework

Estimating microbial dose–response is an important aspect of a food safety risk assessment. In recent years, there has been considerable interest to advance these models with potential incorporation of gene expression data. The aim of this study was to develop a novel machine learning model that considers the weights of expression of Salmonella genes that could be associated with illness, given exposure, in hosts. Here, an elastic net-based weighted Poisson regression method was proposed to identify Salmonella enterica genes that could be significantly associated with the illness response, irrespective of serovar. The best-fit elastic net model was obtained by 10-fold cross-validation. The best-fit elastic net model identified 33 gene expression–dose interaction terms that added to the predictability of the model. Of these, nine genes associated with Salmonella metabolism and virulence were found to be significant by the best-fit Poisson regression model (p < 0.05). This method could improve or redefine dose–response relationships for illness from relative proportions of significant genes from a microbial genetic dataset, which would help in refining endpoint and risk estimations.     

Modeling Microbial Growth Within Food Safety Risk Assessments

Risk estimates for food-borne infection will usually depend heavily on numbers of microorganisms present on the food at the time of consumption. As these data are seldom available directly, attention has turned to predictive microbiology as a means of inferring exposure at consumption. Codex guidelines recommend that microbiological risk assessment should explicitly consider the dynamics of microbiological growth, survival, and death in foods. This article describes predictive models and resources for modeling microbial growth in foods, and their utility and limitations in food safety risk assessment. We also aim to identify tools, data, and knowledge sources, and to provide an understanding of the microbial ecology of foods so that users can recognize model limits, avoid modeling unrealistic scenarios, and thus be able to appreciate the levels of confidence they can have in the outputs of predictive microbiology models. The microbial ecology of foods is complex. Developing reliable risk assessments involving microbial growth in foods will require the skills of both microbial ecologists and mathematical modelers. Simplifying assumptions will need to be made, but because of the potential for apparently small errors in growth rate to translate into very large errors in the estimate of risk, the validity of those assumptions should be carefully assessed. Quantitative estimates of absolute microbial risk within narrow confidence intervals do not yet appear to be possible. Nevertheless, the expression of microbial ecology knowledge in "predictive microbiology" models does allow decision support using the tools of risk assessment.     

Dose‐Response Modeling for Inhalational Anthrax in Rabbits Following Single or Multiple Exposures

There is a need to advance our ability to characterize the risk of inhalational anthrax following a low‐dose exposure. The exposure scenario most often considered is a single exposure that occurs during an attack. However, long‐term daily low‐dose exposures also represent a realistic exposure scenario, such as what may be encountered by people occupying areas for longer periods. Given this, the objective of the current work was to model two rabbit inhalational anthrax dose‐response data sets. One data set was from single exposures to aerosolized Bacillus anthracis Ames spores. The second data set exposed rabbits repeatedly to aerosols of B. anthracis Ames spores. For the multiple exposure data the cumulative dose (i.e., the sum of the individual daily doses) was used for the model. Lethality was the response for both. Modeling was performed using Benchmark Dose Software evaluating six models: logprobit, loglogistic, Weibull, exponential, gamma, and dichotomous‐Hill. All models produced acceptable fits to either data set. The exponential model was identified as the best fitting model for both data sets. Statistical tests suggested there was no significant difference between the single exposure exponential model results and the multiple exposure exponential model results, which suggests the risk of disease is similar between the two data sets. The dose expected to cause 10% lethality was 15,600 inhaled spores and 18,200 inhaled spores for the single exposure and multiple exposure exponential dose‐response model, respectively, and the 95% lower confidence intervals were 9,800 inhaled spores and 9,200 inhaled spores, respectively.     

A Bayesian Semiparametric Model for Radiation Dose‐Response Estimation

In evaluating the risk of exposure to health hazards, characterizing the dose‐response relationship and estimating acceptable exposure levels are the primary goals. In analyses of health risks associated with exposure to ionizing radiation, while there is a clear agreement that moderate to high radiation doses cause harmful effects in humans, little has been known about the possible biological effects at low doses, for example, below 0.1 Gy, which is the dose range relevant to most radiation exposures of concern today. A conventional approach to radiation dose‐response estimation based on simple parametric forms, such as the linear nonthreshold model, can be misleading in evaluating the risk and, in particular, its uncertainty at low doses. As an alternative approach, we consider a Bayesian semiparametric model that has a connected piece‐wise‐linear dose‐response function with prior distributions having an autoregressive structure among the random slope coefficients defined over closely spaced dose categories. With a simulation study and application to analysis of cancer incidence data among Japanese atomic bomb survivors, we show that this approach can produce smooth and flexible dose‐response estimation while reasonably handling the risk uncertainty at low doses and elsewhere. With relatively few assumptions and modeling options to be made by the analyst, the method can be particularly useful in assessing risks associated with low‐dose radiation exposures.     

A Parametric Model for Detecting Hormetic Effects in Developmental Toxicity Studies

Hormetic effects have been observed at low exposure levels based on the dose-response pattern of data from developmental toxicity studies. This indicates that there might actually be a reduced risk of exhibiting toxic effects at low exposure levels. Hormesis implies the existence of a threshold dose level and there are dose-response models that include parameters that account for the threshold. We propose a function that introduces a parameter to account for hormesis. This function is a subset of the set of all functions that could represent a hormetic dose-response relationship at low exposure levels to toxic agents. We characterize the overall dose-response relationship with a piecewise function that consists of a hormetic u-shape curve at low dose levels and a logistic curve at high dose levels. We apply our model to a data set from an experiment conducted at the National Toxicology Program (NTP). We also use the beta-binomial distribution to model the litter response data. It can be seen by observing the structure of these data that current experimental designs for developmental studies employ a limited number of dose groups. These designs may not be satisfactory when the goal is to illustrate the existence of hormesis. In particular, increasing the number of low-level doses improves the power for detecting hormetic effects. Therefore, we also provide the results of simulations that were done to characterize the power of current designs in detecting hormesis and to demonstrate how this power can be improved upon by altering these designs with the addition of only a few low exposure levels.     

Modeling the Dose Response Relationship of Waterborne Acanthamoeba

This study developed dose response models for determining the probability of eye or central nervous system infections from previously conducted studies using different strains of Acanthamoeba spp. The data were a result of animal experiments using mice and rats exposed corneally and intranasally to the pathogens. The corneal inoculations of Acanthamoeba isolate Ac 118 included varied amounts of Corynebacterium xerosis and were best fit by the exponential model. Virulence increased with higher levels of C. xerosis. The Acanthamoeba culbertsoni intranasal study with death as an endpoint of response was best fit by the beta‐Poisson model. The HN‐3 strain of A. castellanii was studied with an intranasal exposure and three different endpoints of response. For all three studies, the exponential model was the best fit. A model based on pooling data sets of the intranasal exposure and death endpoint resulted in an LD₅₀ of 19,357 amebae. The dose response models developed in this study are an important step towards characterizing the risk associated with free‐living amoeba like Acanthamoeba in drinking water distribution systems. Understanding the human health risk posed by free‐living amoeba will allow for quantitative microbial risk assessments that support building design decisions to minimize opportunities for pathogen growth and survival.