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.     

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.     

Cryptosporidium Infection Risk: Results of New Dose‐Response Modeling

Cryptosporidium human dose‐response data from seven species/isolates are used to investigate six models of varying complexity that estimate infection probability as a function of dose. Previous models attempt to explicitly account for virulence differences among C. parvum isolates, using three or six species/isolates. Four (two new) models assume species/isolate differences are insignificant and three of these (all but exponential) allow for variable human susceptibility. These three human‐focused models (fractional Poisson, exponential with immunity and beta‐Poisson) are relatively simple yet fit the data significantly better than the more complex isolate‐focused models. Among these three, the one‐parameter fractional Poisson model is the simplest but assumes that all Cryptosporidium oocysts used in the studies were capable of initiating infection. The exponential with immunity model does not require such an assumption and includes the fractional Poisson as a special case. The fractional Poisson model is an upper bound of the exponential with immunity model and applies when all oocysts are capable of initiating infection. The beta Poisson model does not allow an immune human subpopulation; thus infection probability approaches 100% as dose becomes huge. All three of these models predict significantly (>10x) greater risk at the low doses that consumers might receive if exposed through drinking water or other environmental exposure (e.g., 72% vs. 4% infection probability for a one oocyst dose) than previously predicted. This new insight into Cryptosporidium risk suggests additional inactivation and removal via treatment may be needed to meet any specified risk target, such as a suggested 10^?4 annual risk of Cryptosporidium infection.     

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.     

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.     

QMRA for Drinking Water: 2. The Effect of Pathogen Clustering in Single‐Hit Dose‐Response Models

Spatial and/or temporal clustering of pathogens will invalidate the commonly used assumption of Poisson‐distributed pathogen counts (doses) in quantitative microbial risk assessment. In this work, the theoretically predicted effect of spatial clustering in conventional “single‐hit” dose‐response models is investigated by employing the stuttering Poisson distribution, a very general family of count distributions that naturally models pathogen clustering and contains the Poisson and negative binomial distributions as special cases. The analysis is facilitated by formulating the dose‐response models in terms of probability generating functions. It is shown formally that the theoretical single‐hit risk obtained with a stuttering Poisson distribution is lower than that obtained with a Poisson distribution, assuming identical mean doses. A similar result holds for mixed Poisson distributions. Numerical examples indicate that the theoretical single‐hit risk is fairly insensitive to moderate clustering, though the effect tends to be more pronounced for low mean doses. Furthermore, using Jensen's inequality, an upper bound on risk is derived that tends to better approximate the exact theoretical single‐hit risk for highly overdispersed dose distributions. The bound holds with any dose distribution (characterized by its mean and zero inflation index) and any conditional dose‐response model that is concave in the dose variable. Its application is exemplified with published data from Norovirus feeding trials, for which some of the administered doses were prepared from an inoculum of aggregated viruses. The potential implications of clustering for dose‐response assessment as well as practical risk characterization are discussed.     

Cryptosporidiosis Susceptibility and Risk: A Case Study

Regional estimates of cryptosporidiosis risks from drinking water exposure were developed and validated, accounting for AIDS status and age. We constructed a model with probability distributions and point estimates representing Cryptosporidium in tap water, tap water consumed per day (exposure characterization); dose response, illness given infection, prolonged illness given illness; and three conditional probabilities describing the likelihood of case detection by active surveillance (health effects characterization). The model predictions were combined with population data to derive expected case numbers and incidence rates per 100,000 population, by age and AIDS status, borough specific and for New York City overall in 2000 (risk characterization). They were compared with same-year surveillance data to evaluate predictive ability, assumed to represent true incidence of waterborne cryptosporidiosis. The predicted mean risks, similar to previously published estimates for this region, overpredicted observed incidence-most extensively when accounting for AIDS status. The results suggest that overprediction may be due to conservative parameters applied to both non-AIDS and AIDS populations, and that biological differences for children need to be incorporated. Interpretations are limited by the unknown accuracy of available surveillance data, in addition to variability and uncertainty of model predictions. The model appears sensitive to geographical differences in AIDS prevalence. The use of surveillance data for validation and model parameters pertinent to susceptibility are discussed.     

Human Salmonellosis: Estimation of Dose-Illness from Outbreak Data

The quantification of the relationship between the amount of microbial organisms ingested and a specific outcome such as infection, illness, or mortality is a key aspect of quantitative risk assessment. A main problem in determining such dose-response models is the availability of appropriate data. Human feeding trials have been criticized because only young healthy volunteers are selected to participate and low doses, as often occurring in real life, are typically not considered. Epidemiological outbreak data are considered to be more valuable, but are more subject to data uncertainty. In this article, we model the dose-illness relationship based on data of 20 Salmonella outbreaks, as discussed by the World Health Organization. In particular, we model the dose-illness relationship using generalized linear mixed models and fractional polynomials of dose. The fractional polynomial models are modified to satisfy the properties of different types of dose-illness models as proposed by Teunis et al . Within these models, differences in host susceptibility (susceptible versus normal population) are modeled as fixed effects whereas differences in serovar type and food matrix are modeled as random effects. In addition, two bootstrap procedures are presented. A first procedure accounts for stochastic variability whereas a second procedure accounts for both stochastic variability and data uncertainty. The analyses indicate that the susceptible population has a higher probability of illness at low dose levels when the combination pathogen-food matrix is extremely virulent and at high dose levels when the combination is less virulent. Furthermore, the analyses suggest that immunity exists in the normal population but not in the susceptible population.     

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 Uncertainty and Model Averaging in the Estimation of Infectious Doses for Microbial Pathogens

Food‐borne infection is caused by intake of foods or beverages contaminated with microbial pathogens. Dose‐response modeling is used to estimate exposure levels of pathogens associated with specific risks of infection or illness. When a single dose‐response model is used and confidence limits on infectious doses are calculated, only data uncertainty is captured. We propose a method to estimate the lower confidence limit on an infectious dose by including model uncertainty and separating it from data uncertainty. The infectious dose is estimated by a weighted average of effective dose estimates from a set of dose‐response models via a Kullback information criterion. The confidence interval for the infectious dose is constructed by the delta method, where data uncertainty is addressed by a bootstrap method. To evaluate the actual coverage probabilities of the lower confidence limit, a Monte Carlo simulation study is conducted under sublinear, linear, and superlinear dose‐response shapes that can be commonly found in real data sets. Our model‐averaging method achieves coverage close to nominal in almost all cases, thus providing a useful and efficient tool for accurate calculation of lower confidence limits on infectious doses.     

A Risk Assessment Scheme of Infection Transmission Indoors Incorporating the Impact of Resuspension

A new risk assessment scheme was developed to quantify the impact of resuspension to infection transmission indoors. Airborne and surface pathogenic particle concentration models including the effect of two major resuspension scenarios (airflow‐induced particle resuspension [AIPR] and walking‐induced particle resuspension [WIPR]) were derived based on two‐compartment mass balance models and validated against experimental data found in the literature. The inhalation exposure to pathogenic particles was estimated using the derived airborne concentration model, and subsequently incorporated into a dose‐response model to assess the infection risk. Using the proposed risk assessment scheme, the influences of resuspension towards indoor infection transmission were examined by two hypothetical case studies. In the case of AIPR, the infection risk increased from 0 to 0.54 during 0–0.5 hours and from 0.54 to 0.57 during 0.5–4 hours. In the case of WIPR, the infection risk increased from 0 to 0.87 during 0–0.5 hours and from 0.87 to 1 during 0.5–4 hours. Sensitivity analysis was conducted based on the design‐of‐experiments method and showed that the factors that are related to the inspiratory rate of viable pathogens and pathogen virulence have the most significant effect on the infection probability under the occurrence of AIPR and WIPR. The risk assessment scheme could serve as an effective tool for the risk assessment of infection transmission indoors.     

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.     

Norovirus Dose–Response: Are Currently Available Data Informative Enough to Determine How Susceptible Humans Are to Infection from a Single Virus?

Two forms of single‐hit infection dose‐response models have previously been developed to assess available data from human feeding trials and estimate the norovirus dose‐response relationship. The mechanistic interpretations of these models include strong assumptions that warrant reconsideration: the first study includes an implicit assumption that there is no immunity to Norwalk virus among the specific study population, while the recent second study includes assumptions that such immunity could exist and that the nonimmune have no defensive barriers to prevent infection from exposure to just one virus. Both models addressed unmeasured virus aggregation in administered doses. In this work, the available data are reanalyzed using a generalization of the first model to explore these previous assumptions. It was hypothesized that concurrent estimation of an unmeasured degree of virus aggregation and important dose‐response parameters could lead to structural nonidentifiability of the model (i.e., that a diverse range of alternative mechanistic interpretations yield the same optimal fit), and this is demonstrated using the profile likelihood approach and by algebraic proof. It is also demonstrated that omission of an immunity parameter can artificially inflate the estimated degree of aggregation and falsely suggest high susceptibility among the nonimmune. The currently available data support the assumption of immunity within the specific study population, but provide only weak information about the degree of aggregation and susceptibility among the nonimmune. The probability of infection at low and moderate doses may be much lower than previously asserted, but more data from strategically designed dose‐response experiments are needed to provide adequate information.     

Ebola Virus Dose Response Model for Aerosolized Exposures: Insights from Primate Data

This study develops dose–response models for Ebolavirus using previously published data sets from the open literature. Two such articles were identified in which three different species of nonhuman primates were challenged by aerosolized Ebolavirus in order to study pathology and clinical disease progression. Dose groups were combined and pooled across each study in order to facilitate modeling. The endpoint of each experiment was death. The exponential and exact beta-Poisson models were fit to the data using maximum likelihood estimation. The exact beta-Poisson was deemed the recommended model because it more closely approximated the probability of response at low doses though both models provided a good fit. Although transmission is generally considered to be dominated by person-to-person contact, aerosolization is a possible route of exposure. If possible, this route of exposure could be particularly concerning for persons in occupational roles managing contaminated liquid wastes from patients being treated for Ebola infection and the wastewater community responsible for disinfection. Therefore, this study produces a necessary mathematical relationship between exposure dose and risk of death for the inhalation route of exposure that can support quantitative microbial risk assessment aimed at informing risk mitigation strategies including personal protection policies against occupational exposures.     

Predictive Bayesian Microbial Dose-Response Assessment Based on Suggested Self-Organization in Primary Illness Response: Cryptosporidium parvum

The probability of illness caused by very low doses of pathogens cannot generally be tested due to the numbers of subjects that would be needed, though such assessments of illness dose response are needed to evaluate drinking water standards. A predictive Bayesian dose-response assessment method was proposed previously to assess the unconditional probability of illness from available information and avoid the inconsistencies of confidence-based approaches. However, the method uses knowledge of the conditional dose-response form, and this form is not well established for the illness endpoint. A conditional parametric dose-response function for gastroenteric illness is proposed here based on simple numerical models of self-organized host-pathogen systems and probabilistic arguments. In the models, illnesses terminate when the host evolves by processes of natural selection to a self-organized critical value of wellness. A generalized beta-Poisson illness dose-response form emerges for the population as a whole. Use of this form is demonstrated in a predictive Bayesian dose-response assessment for cryptosporidiosis. Results suggest that a maximum allowable dose of 5.0 x 10(-7) oocysts/exposure (e.g., 2.5 x 10(-7) oocysts/L water) would correspond with the original goals of the U.S. Environmental Protection Agency Surface Water Treatment Rule, considering only primary illnesses resulting from Poisson-distributed pathogen counts. This estimate should be revised to account for non-Poisson distributions of Cryptosporidium parvum in drinking water and total response, considering secondary illness propagation in the population.     

Topics in Microbial Risk Assessment: Dynamic Flow Tree Process

Microbial risk assessment is emerging as a new discipline in risk assessment. A systematic approach to microbial risk assessment is presented that employs data analysis for developing parsimonious models and accounts formally for the variability and uncertainty of model inputs using analysis of variance and Monte Carlo simulation. The purpose of the paper is to raise and examine issues in conducting microbial risk assessments. The enteric pathogen Escherichia coli O157:H7 was selected as an example for this study due to its significance to public health. The framework for our work is consistent with the risk assessment components described by the National Research Council in 1983 (hazard identification; exposure assessment; dose-response assessment; and risk characterization). Exposure assessment focuses on hamburgers, cooked a range of temperatures from rare to well done, the latter typical for fast food restaurants. Features of the model include predictive microbiology components that account for random stochastic growth and death of organisms in hamburger. For dose-response modeling, Shigella data from human feeding studies were used as a surrogate for E. coli O157:H7. Risks were calculated using a threshold model and an alternative nonthreshold model. The 95% probability intervals for risk of illness for product cooked to a given internal temperature spanned five orders of magnitude for these models. The existence of even a small threshold has a dramatic impact on the estimated risk.     

A Distributional Approach to Characterizing Low-Dose Cancer Risk

Since cancer risk at very low doses cannot be directly measured in humans or animals, mathematical extrapolation models and scientific judgment are required. This article demonstrates a probabilistic approach to carcinogen risk assessment that employs probability trees, subjective probabilities, and standard bootstrapping procedures. The probabilistic approach is applied to the carcinogenic risk of formaldehyde in environmental and occupational settings. Sensitivity analyses illustrate conditional estimates of risk for each path in the probability tree. Fundamental mechanistic uncertainties are characterized. A strength of the analysis is the explicit treatment of alternative beliefs about pharmacokinetics and pharmacodynamics. The resulting probability distributions on cancer risk are compared with the point estimates reported by federal agencies. Limitations of the approach are discussed as well as future research directions.     

Diminishing Willingness to Pay per Quality‐Adjusted Life Year: Valuing Acute Foodborne Illness

We design and conduct a stated‐preference survey to estimate willingness to pay (WTP) to reduce foodborne risk of acute illness and to test whether WTP is proportional to the corresponding gain in expected quality‐adjusted life years (QALYs). If QALYs measure utility for health, then economic theory requires WTP to be nearly proportional to changes in both health quality and duration of illness and WTP could be estimated by multiplying the expected change in QALYs by an appropriate monetary value. WTP is elicited using double‐bounded, dichotomous‐choice questions in which respondents (randomly selected from the U.S. general adult population, n = 2,858) decide whether to purchase a more expensive food to reduce the risk of foodborne illness. Health risks vary by baseline probability of illness, reduction in probability, duration and severity of illness, and conditional probability of mortality. The expected gain in QALYs is calculated using respondent‐assessed decrements in health‐related quality of life if ill combined with the duration of illness and reduction in probability specified in the survey. We find sharply diminishing marginal WTP for severity and duration of illness prevented. Our results suggest that individuals do not have a constant rate of WTP per QALY, which implies that WTP cannot be accurately estimated by multiplying the change in QALYs by an appropriate monetary value.     

Issues Related to Time Averaging of Exposure in Modeling Risks Associated with Intermittent Exposures to Lead

Typical exposures to lead often involve a mix of long-term exposures to relatively constant exposure levels (e.g., residential yard soil and indoor dust) and highly intermittent exposures at other locations (e.g., seasonal recreational visits to a park). These types of exposures can be expected to result in blood lead concentrations that vary on a temporal scale with the intermittent exposure pattern. Prediction of short-term (or seasonal) blood lead concentrations arising from highly variable intermittent exposures requires a model that can reliably simulate lead exposures and biokinetics on a temporal scale that matches that of the exposure events of interest. If exposure model averaging times (EMATs) of the model exceed the shortest exposure duration that characterizes the intermittent exposure, uncertainties will be introduced into risk estimates because the exposure concentration used as input to the model must be time averaged to account for the intermittent nature of the exposure. We have used simulation as a means of determining the potential magnitude of these uncertainties. Simulations using models having various EMATs have allowed exploration of the strengths and weaknesses of various approaches to time averaging of exposures and impact on risk estimates associated with intermittent exposures to lead in soil. The International Commission of Radiological Protection (ICRP) model of lead pharmacokinetics in humans simulates lead intakes that can vary in intensity over time spans as small as one day, allowing for the simulation of intermittent exposures to lead as a series of discrete daily exposure events. The ICRP model was used to compare the outcomes (blood lead concentration) of various time-averaging adjustments for approximating the time-averaged intake of lead associated with various intermittent exposure patterns. Results of these analyses suggest that standard approaches to time averaging (e.g., U.S. EPA) that estimate the long-term daily exposure concentration can, in some cases, result in substantial underprediction of short-term variations in blood lead concentrations when used in models that operate with EMATs exceeding the shortest exposure duration that characterizes the intermittent exposure. Alternative time-averaging approaches recommended for use in lead risk assessment more reliably predict short-term periodic (e.g., seasonal) elevations in blood lead concentration that might result from intermittent exposures. In general, risk estimates will be improved by simulation on shorter time scales that more closely approximate the actual temporal dynamics of the exposure.