Validation of Quantitative Microbial Risk Assessment Using Epidemiological Data from Outbreaks of Waterborne Gastrointestinal Disease

The assumptions underlying quantitative microbial risk assessment (QMRA) are simple and biologically plausible, but QMRA predictions have never been validated for many pathogens. The objective of this study was to validate QMRA predictions against epidemiological measurements from outbreaks of waterborne gastrointestinal disease. I screened 2,000 papers and identified 12 outbreaks with the necessary data: disease rates measured using epidemiological methods and pathogen concentrations measured in the source water. Eight of the 12 outbreaks were caused by Cryptosporidium, three by Giardia, and one by norovirus. Disease rates varied from 5.5 × 10^?6 to 1.1 × 10^?2 cases/person‐day, and reported pathogen concentrations varied from 1.2 × 10^?4 to 8.6 × 10² per liter. I used these concentrations with single‐hit dose–response models for all three pathogens to conduct QMRA, producing both point and interval predictions of disease rates for each outbreak. Comparison of QMRA predictions to epidemiological measurements showed good agreement; interval predictions contained measured disease rates for 9 of 12 outbreaks, with point predictions off by factors of 1.0–120 (median = 4.8). Furthermore, 11 outbreaks occurred at mean doses of less than 1 pathogen per exposure. Measured disease rates for these outbreaks were clearly consistent with a single‐hit model, and not with a “two‐hit” threshold model. These results demonstrate the validity of QMRA for predicting disease rates due to Cryptosporidium and Giardia.     

Outbreak-Based Giardia Dose–Response Model Using Bayesian Hierarchical Markov Chain Monte Carlo Analysis

Giardia is a zoonotic gastrointestinal parasite responsible for a substantial global public health burden, and quantitative microbial risk assessment (QMRA) is often used to forecast and manage this burden. QMRA requires dose–response models to extrapolate available dose–response data, but the existing model for Giardia ignores valuable dose–response information, particularly data from several well-documented waterborne outbreaks of giardiasis. The current study updates Giardia dose–response modeling by synthesizing all available data from outbreaks and experimental studies using a Bayesian random effects dose–response model. For outbreaks, mean doses (D) and the degree of spatial and temporal aggregation among cysts were estimated using exposure assessment implemented via two-dimensional Monte Carlo simulation, while potential overreporting of outbreak cases was handled using published overreporting factors and censored binomial regression. Parameter estimation was by Markov chain Monte Carlo simulation and indicated that a typical exponential dose–response parameter for Giardia is r = 1.6 × 10⁻² [3.7 × 10⁻³, 6.2 × 10⁻²] (posterior median [95% credible interval]), while a typical morbidity ratio is m = 3.8 × 10⁻¹ [2.3 × 10⁻¹, 5.5 × 10⁻¹]. Corresponding (logistic-scale) variance components were σ_r = 5.2 × 10⁻¹ [1.1 × 10⁻¹, 9.6 × 10⁻¹] and σ_m = 9.3 × 10⁻¹ [7.0 × 10⁻², 2.8 × 10⁰], indicating substantial variation in the Giardia dose–response relationship. Compared to the existing Giardia dose–response model, the current study provides more representative estimation of uncertainty in r and novel quantification of its natural variability. Several options for incorporating variability in r (and m) into QMRA predictions are discussed, including incorporation via Monte Carlo simulation as well as evaluation of the current study's model using the approximate beta-Poisson.     

Considering the Risk of Infection by Cryptosporidium via Consumption of Municipally Treated Drinking Water from a Surface Water Source in a Southwestern Ontario Community

Through the use of case‐control analyses and quantitative microbial risk assessment (QMRA), relative risks of transmission of cryptosporidiosis have been evaluated (recreational water exposure vs. drinking water consumption) for a Canadian community with higher than national rates of cryptosporidiosis. A QMRA was developed to assess the risk of Cryptosporidium infection through the consumption of municipally treated drinking water. Simulations were based on site‐specific surface water contamination levels and drinking water treatment log₁₀ reduction capacity for Cryptosporidium. Results suggested that the risk of Cryptosporidium infection via drinking water in the study community, assuming routine operation of the water treatment plant, was negligible (6 infections per 10¹³ persons per day—5th percentile: 2 infections per 10¹⁵ persons per day; 95th percentile: 3 infections per 10¹² persons per day). The risk is essentially nonexistent during optimized, routine treatment operations. The study community achieves between 7 and 9 log₁₀Cryptosporidium oocyst reduction through routine water treatment processes. Although these results do not preclude the need for constant vigilance by both water treatment and public health professionals in this community, they suggest that the cause of higher rates of cryptosporidiosis are more likely due to recreational water contact, or perhaps direct animal contact. QMRA can be successfully applied at the community level to identify data gaps, rank relative public health risks, and forecast future risk scenarios. It is most useful when performed in a collaborative way with local stakeholders, from beginning to end of the risk analysis paradigm.     

A Poultry-Processing Model for Quantitative Microbiological Risk Assessment

A poultry-processing model for a quantitative microbiological risk assessment (QMRA) of campylobacter is presented, which can also be applied to other QMRAs involving poultry processing. The same basic model is applied in each consecutive stage of industrial processing. It describes the effects of inactivation and removal of the bacteria, and the dynamics of cross-contamination in terms of the transfer of campylobacter from the intestines to the carcass surface and the environment, from the carcasses to the environment, and from the environment to the carcasses. From the model it can be derived that, in general, the effect of inactivation and removal is dominant for those carcasses with high initial bacterial loads, and cross-contamination is dominant for those with low initial levels. In other QMRA poultry-processing models, the input-output relationship between the numbers of bacteria on the carcasses is usually assumed to be linear on a logarithmic scale. By including some basic mechanistics, it is shown that this may not be realistic. As nonlinear behavior may affect the predicted effects of risk mitigations; this finding is relevant for risk management. Good knowledge of the variability of bacterial loads on poultry entering the process is important. The common practice in microbiology to only present geometric mean of bacterial counts is insufficient: arithmetic mean are more suitable, in particular, to describe the effect of cross-contamination. The effects of logistic slaughter (scheduled processing) as a risk mitigation strategy are predicted to be small. Some additional complications in applying microbiological data obtained in processing plants are discussed.     

Exposure Factors for Wastewater‐Irrigated Asian Vegetables and a Probabilistic Rotavirus Disease Burden Model for Their Consumption

Many farmers in water‐scarce regions of developing countries use wastewater to irrigate vegetables and other agricultural crops, a practice that may expand with climate change. There are a number of health risks associated with wastewater irrigation for human food crops, particularly with surface irrigation techniques common in the developing world. The World Health Organization (WHO) recommends using quantitative microbial risk assessment (QMRA) to determine if the irrigation scheme meets health standards. However, only a few vegetables have been studied for wastewater risk and little information is known about the disease burden of wastewater‐irrigated vegetable consumption in China. To bridge this knowledge gap, an experiment was conducted to determine volume of water left on Asian vegetables and lettuce after irrigation. One hundred samples each of Chinese chard (Brassica rapa var. chinensis), Chinese broccoli (Brassica oleracea var. alboglabra), Chinese flowering cabbage (Brassica rapa var. parachinensis), and lettuce (Lactuca sativa) were harvested after overhead sprinkler irrigation. Chinese broccoli and flowering cabbage were found to capture the most water and lettuce the least. QMRAs were then constructed to estimate rotavirus disease burden from consumption of wastewater‐irrigated Asian vegetables in Beijing. Results indicate that estimated risks from these reuse scenarios exceed WHO guideline thresholds for acceptable disease burden for wastewater use, signifying that reduction of pathogen concentration or stricter risk management is necessary for safe reuse. Considering the widespread practice of wastewater irrigation for food production, particularly in developing countries, incorporation of water retention factors in QMRAs can reduce uncertainty regarding health risks for consumers worldwide.     

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.     

Development and application of a dose–response model for Elizabethkingia spp

Elizabethkingia spp. are common environmental pathogens responsible for infections in more vulnerable populations. Although the exposure routes of concern are not well understood, some hospital-associated outbreaks have indicated possible waterborne transmission. In order to facilitate quantitative microbial risk assessment (QMRA) for Elizabethkingia spp., this study fit dose–response models to frog and mice datasets that evaluated intramuscular and intraperitoneal exposure to Elizabethkingia spp. The frog datasets could be pooled, and the exact beta-Poisson model was the best fitting model with optimized parameters α  = 0.52 and β = 86,351. Using the exact beta-Poisson model, the dose of Elizabethkingia miricola resulting in a 50% morbidity response (LD₅₀) was estimated to be approximately 237,000 CFU. The model developed herein was used to estimate the probability of infection for a hospital patient under a modeled exposure scenario involving a contaminated medical device and reported Elizabethkingia spp. concentrations isolated from hospital sinks after an outbreak. The median exposure dose was approximately 3 CFU/insertion event, and the corresponding median risk of infection was 3.4E-05. The median risk estimated in this case study was lower than the 3% attack rate observed in a previous outbreak, however, there are noted gaps pertaining to the possible concentrations of Elizabethkingia spp. in tap water and the most likely exposure routes. This is the first dose–response model developed for Elizabethkingia spp. thus enabling future risk assessments to help determine levels of risk and potential effective risk management strategies.     

Development of an Interspecies Nested Dose‐Response Model for Mycobacterium avium subspecies paratuberculosis

Mycobacterium avium subspecies paratuberculosis (MAP) causes chronic inflammation of the intestines in humans, ruminants, and other species. It is the causative agent of Johne's disease in cattle, and has been implicated as the causative agent of Crohn's disease in humans. To date, no quantitative microbial risk assessment (QMRA) for MAP utilizing a dose‐response function exists. The objective of this study is to develop a nested dose‐response model for infection from oral exposure to MAP utilizing data from the peer‐reviewed literature. Four studies amenable to dose‐response modeling were identified in the literature search and optimized to the one‐parameter exponential or two‐parameter beta‐Poisson dose‐response models. A nesting analysis was performed on all permutations of the candidate data sets to determine the acceptability of pooling data sets across host species. Three of four data sets exhibited goodness of fit to at least one model. All three data sets exhibited good fit to the beta‐Poisson model, and one data set exhibited goodness of fit, and best fit, to the exponential model. Two data sets were successfully nested using the beta‐Poisson model with parameters α = 0.0978 and N₅₀ = 2.70 × 10² CFU. These data sets were derived from sheep and red deer host species, indicating successful interspecies nesting, and demonstrate the highly infective nature of MAP. The nested dose‐response model described should be used for future QMRA research regarding oral exposure to MAP.     

Risk of Norovirus Gastroenteritis from Consumption of Vegetables Irrigated with Highly Treated Municipal Wastewater—Evaluation of Methods to Estimate Sewage Quality

Quantitative microbial risk assessment was used to assess the risk of norovirus gastroenteritis associated with consumption of raw vegetables irrigated with highly treated municipal wastewater, using Melbourne, Australia as an example. In the absence of local norovirus concentrations, three methods were developed: (1) published concentrations of norovirus in raw sewage, (2) an epidemiological method using Melbourne prevalence of norovirus, and (3) an adjustment of method 1 to account for prevalence of norovirus. The methods produced highly variable results with estimates of norovirus concentrations in raw sewage ranging from 10⁴ per milliliter to 10⁷ per milliliter and treated effluent from 1 × 10^?3 per milliliter to 3 per milliliter (95th percentiles). Annual disease burden was very low using method 1, from 4 to 5 log₁₀ disability adjusted life years (DALYs) below the 10^?6 threshold (0.005–0.1 illnesses per year). Results of method 2 were higher, with some scenarios exceeding the threshold by up to 2 log₁₀ DALYs (up to 95,000 illnesses per year). Method 3, thought to be most representative of Melbourne conditions, predicted annual disease burdens >2 log₁₀ DALYs lower than the threshold (～4 additional cases per year). Sensitivity analyses demonstrated that input parameters used to estimate norovirus concentration accounted for much of the model output variability. This model, while constrained by a lack of knowledge of sewage concentrations, used the best available information and sound logic. Results suggest that current wastewater reuse behaviors in Melbourne are unlikely to cause norovirus risks in excess of the annual DALY health target.     

The Impact of Consumer Phase Models in Microbial Risk Analysis

In quantitative microbiological risk assessment (QMRA), the consumer phase model (CPM) describes the part of the food chain between purchase of the food product at retail and exposure. Construction of a CPM is complicated by the large variation in consumer food handling practices and a limited availability of data. Therefore, several subjective (simplifying) assumptions have to be made when a CPM is constructed, but with a single CPM their impact on the QMRA results is unclear. We therefore compared the performance of eight published CPMs for Campylobacter in broiler meat in an example of a QMRA, where all the CPMs were analyzed using one single input distribution of concentrations at retail, and the same dose‐response relationship. It was found that, between CPMs, there may be a considerable difference in the estimated probability of illness per serving. However, the estimated relative risk reductions are less different for scenarios modeling the implementation of control measures. For control measures affecting the Campylobacter prevalence, the relative risk is proportional irrespective of the CPM used. However, for control measures affecting the concentration the CPMs show some difference in the estimated relative risk. This difference is largest for scenarios where the aim is to remove the highly contaminated portion from human exposure. Given these results, we conclude that for many purposes it is not necessary to develop a new detailed CPM for each new QMRA. However, more observational data on consumer food handling practices and their impact on microbial transfer and survival are needed to generalize this conclusion.     

Evaluating the Potential for a Helicobacter pylori Drinking Water Guideline

Helicobacter pylori is a microaerophilic, gram‐negative bacterium that is linked to adverse health effects including ulcers and gastrointestinal cancers. The goal of this analysis is to develop the necessary inputs for a quantitative microbial risk assessment (QMRA) needed to develop a potential guideline for drinking water at the point of ingestion (e.g., a maximum contaminant level, or MCL) that would be protective of human health to an acceptable level of risk while considering sources of uncertainty. Using infection and gastric cancer as two discrete endpoints, and calculating dose‐response relationships from experimental data on humans and monkeys, we perform both a forward and reverse risk assessment to determine the risk from current reported surface water concentrations of H. pylori and an acceptable concentration of H. pylori at the point of ingestion. This approach represents a synthesis of available information on human exposure to H. pylori via drinking water. A lifetime risk of cancer model suggests that a MCL be set at <1 organism/L given a 5‐log removal treatment because we cannot exclude the possibility that current levels of H. pylori in environmental source waters pose a potential public health risk. Research gaps include pathogen occurrence in source and finished water, treatment removal rates, and determination of H. pylori risks from other water sources such as groundwater and recreational water.     

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.     

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.     

Demonstrating the Benefits of Predictive Bayesian Dose–Response Relationships Using Six Exposure Studies of Cryptosporidium parvum

A conventional dose–response function can be refitted as additional data become available. A predictive dose–response function in contrast does not require a curve-fitting step, only additional data and presents the unconditional probabilities of illness, reflecting the level of information it contains. In contrast, the predictive Bayesian dose–response function becomes progressively less conservative as more information is included. This investigation evaluated the potential for using predictive Bayesian methods to develop a dose–response for human infection that improves on existing models, to show how predictive Bayesian statistical methods can utilize additional data, and expand the Bayesian methods for a broad audience including those concerned about an oversimplification of dose–response curve use in quantitative microbial risk assessment (QMRA). This study used a dose–response relationship incorporating six separate data sets for Cryptosporidium parvum. A Pareto II distribution with known priors was applied to one of the six data sets to calibrate the model, while the others were used for subsequent updating. While epidemiological principles indicate that local variations, host susceptibility, and organism strain virulence may vary, the six data sets all appear to be well characterized using the Bayesian approach. The adaptable model was applied to an existing data set for Campylobacter jejuni for model validation purposes, which yielded results that demonstrate the ability to analyze a dose–response function with limited data using and update those relationships with new data. An analysis of the goodness of fit compared to the beta-Poisson methods also demonstrated correlation between the predictive Bayesian model and the data.     

Inferring an Augmented Bayesian Network to Confront a Complex Quantitative Microbial Risk Assessment Model with Durability Studies: Application to Bacillus Cereus on a Courgette Purée Production Chain

The Monte Carlo (MC) simulation approach is traditionally used in food safety risk assessment to study quantitative microbial risk assessment (QMRA) models. When experimental data are available, performing Bayesian inference is a good alternative approach that allows backward calculation in a stochastic QMRA model to update the experts’ knowledge about the microbial dynamics of a given food‐borne pathogen. In this article, we propose a complex example where Bayesian inference is applied to a high‐dimensional second‐order QMRA model. The case study is a farm‐to‐fork QMRA model considering genetic diversity of Bacillus cereus in a cooked, pasteurized, and chilled courgette purée. Experimental data are Bacillus cereus concentrations measured in packages of courgette purées stored at different time‐temperature profiles after pasteurization. To perform a Bayesian inference, we first built an augmented Bayesian network by linking a second‐order QMRA model to the available contamination data. We then ran a Markov chain Monte Carlo (MCMC) algorithm to update all the unknown concentrations and unknown quantities of the augmented model. About 25% of the prior beliefs are strongly updated, leading to a reduction in uncertainty. Some updates interestingly question the QMRA model.     

Uncertainty in Cancer Risk Estimates

Several existing databases compiled by Gold et al.^(1–3) for carcinogenesis bioassays are examined to obtain estimates of the reproducibility of cancer rates across experiments, strains, and rodent species. A measure of carcinogenic potency is given by the TD₅₀ (daily dose that causes a tumor type in 50% of the exposed animals that otherwise would not develop the tumor in a standard lifetime). The lognormal distribution can be used to model the uncertainty of the estimates of potency (TD₅₀) and the ratio of TD₅₀'s between two species. For near-replicate bioassays, approximately 95% of the TD₅₀'s are estimated to be within a factor of 4 of the mean. Between strains, about 95% of the TD₅₀'s are estimated to be within a factor of 11 of their mean, and the pure genetic component of variability is accounted for by a factor of 6.8. Between rats and mice, about 95% of the TD₅₀'s are estimated to be within a factor of 32 of the mean, while between humans and experimental animals the factor is 110 for 20 chemicals reported by Allen et al.⁽⁴⁾ The common practice of basing cancer risk estimates on the most sensitive rodent species-strain-sex and using interspecies dose scaling based on body surface area appears to overestimate cancer rates for these 20 human carcinogens by about one order of magnitude on the average. Hence, for chemicals where the dose-response is nearly linear below experimental doses, cancer risk estimates based on animal data are not necessarily conservative and may range from a factor of 10 too low for human carcinogens up to a factor of 1000 too high for approximately 95% of the chemicals tested to date. These limits may need to be modified for specific chemicals where additional mechanistic or pharmacokinetic information may suggest alterations or where particularly sensitive subpopu-lations may be exposed. Supralinearity could lead to anticonservative estimates of cancer risk. Underestimating cancer risk by a specific factor has a much larger impact on the actual number of cancer cases than overestimates of smaller risks by the same factor. This paper does not address the uncertainties in high to low dose extrapolation. If the dose-response is sufficiently nonlinear at low doses to produce cancer risks near zero, then low-dose risk estimates based on linear extrapolation are likely to overestimate risk and the limits of uncertainty cannot be established.     

Interventions Targeting Deep Tissue Lymph Nodes May Not Effectively Reduce the Risk of Salmonellosis from Ground Pork Consumption: A Quantitative Microbial Risk Assessment

The inclusion of deep tissue lymph nodes (DTLNs) or nonvisceral lymph nodes contaminated with Salmonella in wholesale fresh ground pork (WFGP) production may pose risks to public health. To assess the relative contribution of DTLNs to human salmonellosis occurrence associated with ground pork consumption and to investigate potential critical control points in the slaughter‐to‐table continuum for the control of human salmonellosis in the United States, a quantitative microbial risk assessment (QMRA) model was established. The model predicted an average of 45 cases of salmonellosis (95% CI = [19, 71]) per 100,000 Americans annually due to WFGP consumption. Sensitivity analysis of all stochastic input variables showed that cooking temperature was the most influential parameter for reducing salmonellosis cases associated with WFGP meals, followed by storage temperature and Salmonella concentration on contaminated carcass surface before fabrication. The input variables were grouped to represent three main factors along the slaughter‐to‐table chain influencing Salmonella doses ingested via WFGP meals: DTLN‐related factors, factors at processing other than DTLNs, and consumer‐related factors. The evaluation of the impact of each group of factors by second‐order Monte Carlo simulation showed that DTLN‐related factors had the lowest impact on the risk estimate among the three groups of factors. These findings indicate that interventions to reduce Salmonella contamination in DTLNs or to remove DTLNs from WFGP products may be less critical for reducing human infections attributable to ground pork than improving consumers’ cooking habits or interventions of carcass decontamination at processing.     

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.     

A Generalized QMRA Beta‐Poisson Dose‐Response Model

Quantitative microbial risk assessment (QMRA) is widely accepted for characterizing the microbial risks associated with food, water, and wastewater. Single‐hit dose‐response models are the most commonly used dose‐response models in QMRA. Denoting as the probability of infection at a given mean dose d, a three‐parameter generalized QMRA beta‐Poisson dose‐response model, , is proposed in which the minimum number of organisms required for causing infection, K_min, is not fixed, but a random variable following a geometric distribution with parameter . The single‐hit beta‐Poisson model, , is a special case of the generalized model with K_min = 1 (which implies ). The generalized beta‐Poisson model is based on a conceptual model with greater detail in the dose‐response mechanism. Since a maximum likelihood solution is not easily available, a likelihood‐free approximate Bayesian computation (ABC) algorithm is employed for parameter estimation. By fitting the generalized model to four experimental data sets from the literature, this study reveals that the posterior median estimates produced fall short of meeting the required condition of = 1 for single‐hit assumption. However, three out of four data sets fitted by the generalized models could not achieve an improvement in goodness of fit. These combined results imply that, at least in some cases, a single‐hit assumption for characterizing the dose‐response process may not be appropriate, but that the more complex models may be difficult to support especially if the sample size is small. The three‐parameter generalized model provides a possibility to investigate the mechanism of a dose‐response process in greater detail than is possible under a single‐hit model.     

A Quantitative Microbial Risk Assessment Model for Legionnaires'' Disease: Animal Model Selection and Dose-Response Modeling

Legionnaires' disease (LD), first reported in 1976, is an atypical pneumonia caused by bacteria of the genus Legionella, and most frequently by L. pneumophila (Lp). Subsequent research on exposure to the organism employed various animal models, and with quantitative microbial risk assessment (QMRA) techniques, the animal model data may provide insights on human dose-response for LD. This article focuses on the rationale for selection of the guinea pig model, comparison of the dose-response model results, comparison of projected low-dose responses for guinea pigs, and risk estimates for humans. Based on both in vivo and in vitro comparisons, the guinea pig (Cavia porcellus) dose-response data were selected for modeling human risk. We completed dose-response modeling for the beta-Poisson (approximate and exact), exponential, probit, logistic, and Weibull models for Lp inhalation, mortality, and infection (end point elevated body temperature) in guinea pigs. For mechanistic reasons, including low-dose exposure probability, further work on human risk estimates for LD employed the exponential and beta-Poisson models. With an exposure of 10 colony-forming units (CFU) (retained dose), the QMRA model predicted a mild infection risk of 0.4 (as evaluated by seroprevalence) and a clinical severity LD case (e.g., hospitalization and supportive care) risk of 0.0009. The calculated rates based on estimated human exposures for outbreaks used for the QMRA model validation are within an order of magnitude of the reported LD rates. These validation results suggest the LD QMRA animal model selection, dose-response modeling, and extension to human risk projections were appropriate.