Fractional Poisson—A Simple Dose‐Response Model for Human Norovirus

This study utilizes old and new Norovirus (NoV) human challenge data to model the dose‐response relationship for human NoV infection. The combined data set is used to update estimates from a previously published beta‐Poisson dose‐response model that includes parameters for virus aggregation and for a beta‐distribution that describes variable susceptibility among hosts. The quality of the beta‐Poisson model is examined and a simpler model is proposed. The new model (fractional Poisson) characterizes hosts as either perfectly susceptible or perfectly immune, requiring a single parameter (the fraction of perfectly susceptible hosts) in place of the two‐parameter beta‐distribution. A second parameter is included to account for virus aggregation in the same fashion as it is added to the beta‐Poisson model. Infection probability is simply the product of the probability of nonzero exposure (at least one virus or aggregate is ingested) and the fraction of susceptible hosts. The model is computationally simple and appears to be well suited to the data from the NoV human challenge studies. The model's deviance is similar to that of the beta‐Poisson, but with one parameter, rather than two. As a result, the Akaike information criterion favors the fractional Poisson over the beta‐Poisson model. At low, environmentally relevant exposure levels (<100), estimation error is small for the fractional Poisson model; however, caution is advised because no subjects were challenged at such a low dose. New low‐dose data would be of great value to further clarify the NoV dose‐response relationship and to support improved risk assessment for environmentally relevant exposures.     

Dose‐Response Model of Coxiella burnetii (Q Fever)

Q fever is a zoonotic disease caused by the intracellular gram‐negative bacterium Coxiella burnetii (C. burnetii), which only multiplies within the phagolysosomal vacuoles. Q fever may manifest as acute or chronic disease. The acute form is generally not fatal and manifestes as self‐controlled febrile illness. Chronic Q fever is usually characterized by endocarditis. Many animal models, including humans, have been studied for Q fever infection through various exposure routes. The studies considered different endpoints including death for animal models and clinical signs for human infection. In this article, animal experimental data available in the open literature were fit to suitable dose‐response models using maximum likelihood estimation. Research results for tests of severe combined immunodeficient mice inoculated intraperitoneally (i.p.) with C. burnetii were best estimated with the Beta‐Poisson dose‐response model. Similar inoculation (i.p.) trial outcomes conducted on C57BL/6J mice were best fit by an exponential model, whereas those tests run on C57BL/10ScN mice were optimally represented by a Beta‐Poisson dose‐response model.     

Dose‐Response Model of Rocky Mountain Spotted Fever (RMSF) for Human

Rickettsia rickettsii is the causative agent of Rocky Mountain spotted fever (RMSF) and is the prototype bacterium in the spotted fever group of rickettsiae, which is found in North, Central, and South America. The bacterium is gram negative and an obligate intracellular pathogen. The disease is transmitted to humans and vertebrate host through tick bites; however, some cases of aerosol transmission also have been reported. The disease can be difficult to diagnose in the early stages, and without prompt and appropriate treatment, it can be fatal. This article develops dose‐response models of different routes of exposure for RMSF in primates and humans. The beta‐Poisson model provided the best fit to the dose‐response data of aerosol‐exposed rhesus monkeys, and intradermally inoculated humans (morbidity as end point of response). The average 50% infectious dose among (ID₅₀) exposed human population, N₅₀, is 23 organisms with 95% confidence limits of 1 to 89 organisms. Similarly, ID₁₀ and ID₂₀ are 2.2 and 5.0, respectively. Moreover, the data of aerosol‐exposed rhesus monkeys and intradermally inoculated humans could be pooled. This indicates that the dose‐response models fitted to different data sets are not significantly different and can be described by the same relationship.     

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.     

Dose‐Response Modeling with Summary Data from Developmental Toxicity Studies

Dose‐response analysis of binary developmental data (e.g., implant loss, fetal abnormalities) is best done using individual fetus data (identified to litter) or litter‐specific statistics such as number of offspring per litter and proportion abnormal. However, such data are not often available to risk assessors. Scientific articles usually present only dose‐group summaries for the number or average proportion abnormal and the total number of fetuses. Without litter‐specific data, it is not possible to estimate variances correctly (often characterized as a problem of overdispersion, intralitter correlation, or “litter effect”). However, it is possible to use group summary data when the design effect has been estimated for each dose group. Previous studies have demonstrated useful dose‐response and trend test analyses based on design effect estimates using litter‐specific data from the same study. This simplifies the analysis but does not help when litter‐specific data are unavailable. In the present study, we show that summary data on fetal malformations can be adjusted satisfactorily using estimates of the design effect based on historical data. When adjusted data are then analyzed with models designed for binomial responses, the resulting benchmark doses are similar to those obtained from analyzing litter‐level data with nested dichotomous models.     

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.     

Animal and Human Dose‐Response Models for Brucella Species

Human Brucellosis is one of the most common zoonotic diseases worldwide. Disease transmission often occurs through the handling of domestic livestock, as well as ingestion of unpasteurized milk and cheese, but can have enhanced infectivity if aerosolized. Because there is no human vaccine available, rising concerns about the threat of Brucellosis to human health and its inclusion in the Center for Disease Control's Category B Bioterrorism/Select Agent List make a better understanding of the dose‐response relationship of this microbe necessary. Through an extensive peer‐reviewed literature search, candidate dose‐response data were appraised so as to surpass certain standards for quality. The statistical programming language, “R,” was used to compute the maximum likelihood estimation to fit two models, the exponential and the approximate beta‐Poisson (widely used for quantitative risk assessment) to dose‐response data. Dose‐response models were generated for prevalent species of Brucella: Br. suis, Br. melitensis, and Br. abortus. Dose‐response models were created for aerosolized Br. suis exposure to guinea pigs from pooled studies. A parallel model for guinea pigs inoculated through both aerosol and subcutaneous routes with Br. melitensis showed that the median infectious dose corresponded to a 30 colony‐forming units (CFU) dose of Br. suis, much less than the N₅₀ dose of about 94 CFU for Br. melitensis organisms. When Br. melitensis was tested subcutaneously on mice, the N₅₀ dose was higher, 1,840 CFU. A dose‐response model was constructed from pooled data for mice, rhesus macaques, and humans inoculated through three routes (subcutaneously/aerosol/intradermally) with Br. melitensis.     

Aldrin and Dieldrin: A Reevaluation of the Cancer and Noncancer Dose‐Response Assessments

The dose‐response analyses of cancer and noncancer health effects of aldrin and dieldrin were evaluated using current methodology, including benchmark dose analysis and the current U.S. Environmental Protection Agency (U.S. EPA) guidance on body weight scaling and uncertainty factors. A literature review was performed to determine the most appropriate adverse effect endpoints. Using current methodology and information, the estimated reference dose values were 0.0001 and 0.00008 mg/kg‐day for aldrin and dieldrin, respectively. The estimated cancer slope factors for aldrin and dieldrin were 3.4 and 7.0 (mg/kg‐day)^?1, respectively (i.e., about 5‐ and 2.3‐fold lower risk than the 1987 U.S. EPA assessments). Because aldrin and dieldrin are no longer used as pesticides in the United States, they are presumed to be a low priority for additional review by the U.S. EPA. However, because they are persistent and still detected in environmental samples, quantitative risk assessments based on the best available methods are required. Recent epidemiologic studies do not demonstrate a causal association between aldrin and dieldrin and human cancer risk. The proposed reevaluations suggest that these two compounds pose a lower human health risk than currently reported by the U.S. EPA.     

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.     

Incorporating Time‐Dose‐Response into Legionella Outbreak Models

A novel method was used to incorporate in vivo host–pathogen dynamics into a new robust outbreak model for legionellosis. Dose‐response and time‐dose‐response (TDR) models were generated for Legionella longbeachae exposure to mice via the intratracheal route using a maximum likelihood estimation approach. The best‐fit TDR model was then incorporated into two L. pneumophila outbreak models: an outbreak that occurred at a spa in Japan, and one that occurred in a Melbourne aquarium. The best‐fit TDR from the murine dosing study was the beta‐Poisson with exponential‐reciprocal dependency model, which had a minimized deviance of 32.9. This model was tested against other incubation distributions in the Japan outbreak, and performed consistently well, with reported deviances ranging from 32 to 35. In the case of the Melbourne outbreak, the exponential model with exponential dependency was tested against non‐time‐dependent distributions to explore the performance of the time‐dependent model with the lowest number of parameters. This model reported low minimized deviances around 8 for the Weibull, gamma, and lognormal exposure distribution cases. This work shows that the incorporation of a time factor into outbreak distributions provides models with acceptable fits that can provide insight into the in vivo dynamics of the host‐pathogen system.     

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.     

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.     

Quantal Risk Assessment Database: A Database for Exploring Patterns in Quantal Dose‐Response Data in Risk Assessment and its Application to Develop Priors for Bayesian Dose‐Response Analysis

Quantitative risk assessments for physical, chemical, biological, occupational, or environmental agents rely on scientific studies to support their conclusions. These studies often include relatively few observations, and, as a result, models used to characterize the risk may include large amounts of uncertainty. The motivation, development, and assessment of new methods for risk assessment is facilitated by the availability of a set of experimental studies that span a range of dose‐response patterns that are observed in practice. We describe construction of such a historical database focusing on quantal data in chemical risk assessment, and we employ this database to develop priors in Bayesian analyses. The database is assembled from a variety of existing toxicological data sources and contains 733 separate quantal dose‐response data sets. As an illustration of the database's use, prior distributions for individual model parameters in Bayesian dose‐response analysis are constructed. Results indicate that including prior information based on curated historical data in quantitative risk assessments may help stabilize eventual point estimates, producing dose‐response functions that are more stable and precisely estimated. These in turn produce potency estimates that share the same benefit. We are confident that quantitative risk analysts will find many other applications and issues to explore using this database.     

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.     

Bayesian Development of a Dose‐Response Model for Aspergillus fumigatus and Invasive Aspergillosis

Invasive aspergillosis (IA) is a major cause of mortality in immunocompromized hosts, most often consecutive to the inhalation of spores of Aspergillus. However, the relationship between Aspergillus concentration in the air and probability of IA is not quantitatively known. In this study, this relationship was examined in a murine model of IA. Immunosuppressed Balb/c mice were exposed for 60 minutes at day 0 to an aerosol of A. fumigatus spores (Af293 strain). At day 10, IA was assessed in mice by quantitative culture of the lungs and galactomannan dosage. Fifteen separate nebulizations with varying spore concentrations were performed. Rates of IA ranged from 0% to 100% according to spore concentrations. The dose‐response relationship between probability of infection and spore exposure was approximated using the exponential model and the more flexible beta‐Poisson model. Prior distributions of the parameters of the models were proposed then updated with data in a Bayesian framework. Both models yielded close median dose‐responses of the posterior distributions for the main parameter of the model, but with different dispersions, either when the exposure dose was the concentration in the nebulized suspension or was the estimated quantity of spores inhaled by a mouse during the experiment. The median quantity of inhaled spores that infected 50% of mice was estimated at 1.8 × 10⁴ and 3.2 × 10⁴ viable spores in the exponential and beta‐Poisson models, respectively. This study provides dose‐response parameters for quantitative assessment of the relationship between airborne exposure to the reference A. fumigatus strain and probability of IA in immunocompromized hosts.     

Dose–Response Modeling: Extrapolating From Experimental Data to Real‐World Populations

Dose–response modeling of biological agents has traditionally focused on describing laboratory‐derived experimental data. Limited consideration has been given to understanding those factors that are controlled in a laboratory, but are likely to occur in real‐world scenarios. In this study, a probabilistic framework is developed that extends Brookmeyer's competing‐risks dose–response model to allow for variation in factors such as dose‐dispersion, dose‐deposition, and other within‐host parameters. With data sets drawn from dose–response experiments of inhalational anthrax, plague, and tularemia, we illustrate how for certain cases, there is the potential for overestimation of infection numbers arising from models that consider only the experimental data in isolation.     

Guidelines for Use of the Approximate Beta‐Poisson Dose–Response Model

For dose–response analysis in quantitative microbial risk assessment (QMRA), the exact beta‐Poisson model is a two‐parameter mechanistic dose–response model with parameters and , which involves the Kummer confluent hypergeometric function. Evaluation of a hypergeometric function is a computational challenge. Denoting as the probability of infection at a given mean dose d, the widely used dose–response model is an approximate formula for the exact beta‐Poisson model. Notwithstanding the required conditions and , issues related to the validity and approximation accuracy of this approximate formula have remained largely ignored in practice, partly because these conditions are too general to provide clear guidance. Consequently, this study proposes a probability measure Pr(0 < r < 1 | , ) as a validity measure (r is a random variable that follows a gamma distribution; and are the maximum likelihood estimates of α and β in the approximate model); and the constraint conditions for as a rule of thumb to ensure an accurate approximation (e.g., Pr(0 < r < 1 | , ) >0.99) . This validity measure and rule of thumb were validated by application to all the completed beta‐Poisson models (related to 85 data sets) from the QMRA community portal (QMRA Wiki). The results showed that the higher the probability Pr(0 < r < 1 | , ), the better the approximation. The results further showed that, among the total 85 models examined, 68 models were identified as valid approximate model applications, which all had a near perfect match to the corresponding exact beta‐Poisson model dose–response curve.     

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.     

Dose‐Response Models Incorporating Aerosol Size Dependency for Francisella tularensis

The effect of bioaerosol size was incorporated into predictive dose‐response models for the effects of inhaled aerosols of Francisella tularensis (the causative agent of tularemia) on rhesus monkeys and guinea pigs with bioaerosol diameters ranging between 1.0 and 24 μm. Aerosol‐size‐dependent models were formulated as modification of the exponential and β‐Poisson dose‐response models and model parameters were estimated using maximum likelihood methods and multiple data sets of quantal dose‐response data for which aerosol sizes of inhaled doses were known. Analysis of F. tularensis dose‐response data was best fit by an exponential dose‐response model with a power function including the particle diameter size substituting for the rate parameter k scaling the applied dose. There were differences in the pathogen's aerosol‐size‐dependence equation and models that better represent the observed dose‐response results than the estimate derived from applying the model developed by the International Commission on Radiological Protection (ICRP, 1994) that relies on differential regional lung deposition for human particle exposure.