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.     

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.     

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.     

Harnessing the Theoretical Foundations of the Exponential and Beta‐Poisson Dose‐Response Models to Quantify Parameter Uncertainty Using Markov Chain Monte Carlo

Dose‐response models are the essential link between exposure assessment and computed risk values in quantitative microbial risk assessment, yet the uncertainty that is inherent to computed risks because the dose‐response model parameters are estimated using limited epidemiological data is rarely quantified. Second‐order risk characterization approaches incorporating uncertainty in dose‐response model parameters can provide more complete information to decisionmakers by separating variability and uncertainty to quantify the uncertainty in computed risks. Therefore, the objective of this work is to develop procedures to sample from posterior distributions describing uncertainty in the parameters of exponential and beta‐Poisson dose‐response models using Bayes's theorem and Markov Chain Monte Carlo (in OpenBUGS). The theoretical origins of the beta‐Poisson dose‐response model are used to identify a decomposed version of the model that enables Bayesian analysis without the need to evaluate Kummer confluent hypergeometric functions. Herein, it is also established that the beta distribution in the beta‐Poisson dose‐response model cannot address variation among individual pathogens, criteria to validate use of the conventional approximation to the beta‐Poisson model are proposed, and simple algorithms to evaluate actual beta‐Poisson probabilities of infection are investigated. The developed MCMC procedures are applied to analysis of a case study data set, and it is demonstrated that an important region of the posterior distribution of the beta‐Poisson dose‐response model parameters is attributable to the absence of low‐dose data. This region includes beta‐Poisson models for which the conventional approximation is especially invalid and in which many beta distributions have an extreme shape with questionable plausibility.     

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.     

The Beta Poisson Dose-Response Model Is Not a Single-Hit Model

The choice of a dose-response model is decisive for the outcome of quantitative risk assessment. Single-hit models have played a prominent role in dose-response assessment for pathogenic microorganisms, since their introduction. Hit theory models are based on a few simple concepts that are attractive for their clarity and plausibility. These models, in particular the Beta Poisson model, are used for extrapolation of experimental dose-response data to low doses, as are often present in drinking water or food products. Unfortunately, the Beta Poisson model, as it is used throughout the microbial risk literature, is an approximation whose validity is not widely known. The exact functional relation is numerically complex, especially for use in optimization or uncertainty analysis. Here it is shown that although the discrepancy between the Beta Poisson formula and the exact function is not very large for many data sets, the differences are greatest at low doses--the region of interest for many risk applications. Errors may become very large, however, in the results of uncertainty analysis, or when the data contain little low-dose information. One striking property of the exact single-hit model is that it has a maximum risk curve, limiting the upper confidence level of the dose-response relation. This is due to the fact that the risk cannot exceed the probability of exposure, a property that is not retained in the Beta Poisson approximation. This maximum possible response curve is important for uncertainty analysis, and for risk assessment of pathogens with unknown properties.     

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.     

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.     

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.     

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.     

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.     

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.     

Application of the Hyper‐Poisson Generalized Linear Model for Analyzing Motor Vehicle Crashes

The hyper‐Poisson distribution can handle both over‐ and underdispersion, and its generalized linear model formulation allows the dispersion of the distribution to be observation‐specific and dependent on model covariates. This study's objective is to examine the potential applicability of a newly proposed generalized linear model framework for the hyper‐Poisson distribution in analyzing motor vehicle crash count data. The hyper‐Poisson generalized linear model was first fitted to intersection crash data from Toronto, characterized by overdispersion, and then to crash data from railway‐highway crossings in Korea, characterized by underdispersion. The results of this study are promising. When fitted to the Toronto data set, the goodness‐of‐fit measures indicated that the hyper‐Poisson model with a variable dispersion parameter provided a statistical fit as good as the traditional negative binomial model. The hyper‐Poisson model was also successful in handling the underdispersed data from Korea; the model performed as well as the gamma probability model and the Conway‐Maxwell‐Poisson model previously developed for the same data set. The advantages of the hyper‐Poisson model studied in this article are noteworthy. Unlike the negative binomial model, which has difficulties in handling underdispersed data, the hyper‐Poisson model can handle both over‐ and underdispersed crash data. Although not a major issue for the Conway‐Maxwell‐Poisson model, the effect of each variable on the expected mean of crashes is easily interpretable in the case of this new model.     

Development of a Dose‐Response Model for SARS Coronavirus

In order to develop a dose‐response model for SARS coronavirus (SARS‐CoV), the pooled data sets for infection of transgenic mice susceptible to SARS‐CoV and infection of mice with murine hepatitis virus strain 1, which may be a clinically relevant model of SARS, were fit to beta‐Poisson and exponential models with the maximum likelihood method. The exponential model (k= 4.1 × l0²) could describe the dose‐response relationship of the pooled data sets. The beta‐Poisson model did not provide a statistically significant improvement in fit. With the exponential model, the infectivity of SARS‐CoV was calculated and compared with those of other coronaviruses. The does of SARS‐CoV corresponding to 10% and 50% responses (illness) were estimated at 43 and 280 PFU, respectively. Its estimated infectivity was comparable to that of HCoV‐229E, known as an agent of human common cold, and also similar to those of some animal coronaviruses belonging to the same genetic group. Moreover, the exponential model was applied to the analysis of the epidemiological data of SARS outbreak that occurred at an apartment complex in Hong Kong in 2003. The estimated dose of SARS‐CoV for apartment residents during the outbreak, which was back‐calculated from the reported number of cases, ranged from 16 to 160 PFU/person, depending on the floor. The exponential model developed here is the sole dose‐response model for SARS‐CoV at the present and would enable us to understand the possibility for reemergence of SARS.     

Extension of the Application of Conway‐Maxwell‐Poisson Models: Analyzing Traffic Crash Data Exhibiting Underdispersion

The objective of this article is to evaluate the performance of the COM‐Poisson GLM for analyzing crash data exhibiting underdispersion (when conditional on the mean). The COM‐Poisson distribution, originally developed in 1962, has recently been reintroduced by statisticians for analyzing count data subjected to either over‐ or underdispersion. Over the last year, the COM‐Poisson GLM has been evaluated in the context of crash data analysis and it has been shown that the model performs as well as the Poisson‐gamma model for crash data exhibiting overdispersion. To accomplish the objective of this study, several COM‐Poisson models were estimated using crash data collected at 162 railway‐highway crossings in South Korea between 1998 and 2002. This data set has been shown to exhibit underdispersion when models linking crash data to various explanatory variables are estimated. The modeling results were compared to those produced from the Poisson and gamma probability models documented in a previous published study. The results of this research show that the COM‐Poisson GLM can handle crash data when the modeling output shows signs of underdispersion. Finally, they also show that the model proposed in this study provides better statistical performance than the gamma probability and the traditional Poisson models, at least for this data set.     

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.     

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.     

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.     

Classic Dose‐Response and Time Postinoculation Models for Leptospira

Leptospirosis is a preeminent zoonotic disease concentrated in tropical areas, and prevalent in both industrialized and rural settings. Dose‐response models were generated from 22 data sets reported in 10 different studies. All of the selected studies used rodent subjects, primarily hamsters, with the predominant endpoint as mortality with the challenge strain administered intraperitoneally. Dose‐response models based on a single evaluation postinfection displayed median lethal dose (LD₅₀) estimates that ranged between 1 and 10⁷ leptospirae depending upon the strain's virulence and the period elapsed since the initial exposure inoculation. Twelve of the 22 data sets measured the number of affected subjects daily over an extended period, so dose‐response models with time‐dependent parameters were estimated. Pooling between data sets produced seven common dose‐response models and one time‐dependent model. These pooled common models had data sets with different test subject hosts, and between disparate leptospiral strains tested on identical hosts. Comparative modeling was done with parallel tests to test the effects of a single different variable of either strain or test host and quantify the difference by calculating a dose multiplication factor. Statistical pooling implies that the mechanistic processes of leptospirosis can be represented by the same dose‐response model for different experimental infection tests even though they may involve different host species, routes, and leptospiral strains, although the cause of this pathophysiological phenomenon has not yet been identified.