EPA/600/R-08/033 

March 2008 

Development and Evaluation of 

Novel Dose-Response Models 

for Use in Microbial Risk 

Assessment 

National Center for Environmental Assessment 

Office of Research and Development 

U.S. Environmental Protection Agency 

Cincinnati, OH 45268 
NOTICE

 The U.S. Environmental Protection Agency through its Office of Research and 

Development partially funded and collaborated on the research described here under 

contract no. CI0790101N with the Intergovernmental Personnel Act Agreement between 

U.S. EPA and Miami University and contract no. DW89921719-01 with the Interagency 

Agreement between the U.S. EPA and the Department of Energy (ORISE). It has been 

subjected to the Agency’s peer and administrative review and has been approved for 

publication as an EPA document. Mention of trade names or commercial products does 

not constitute endorsement or recommendation for use. 

ABSTRACT

 Dose-response relationships relating population infection and illness responses 

to drinking water dose are important to support development of drinking water and 

wastewater reuse policy. Illness endpoints, in particular, are of primary importance in 

economic benefits assessment and management of risk. Dose-response relationships 

for both endpoints, therefore, are needed for assessing the full extent of disease burden 

attributable to pathogens in drinking water and to evaluate the need for new regulation. 

The purpose of this document is to present the predictive Bayesian framework as an 

alternative to the current methods for expressing the risk of infection and illness 

resulting from exposure to pathogens in drinking water. Secondarily, an alternative nonPoisson approach for characterizing the exposure distribution at the tap is offered in the 

context of the dose-response function. Together, these new methods may provide a 

more realistic and rigorous depiction of the impact of water-borne pathogens on drinking 

water consumers. 

Preferred citation: 

U.S. EPA. 2008. Development and Evaluation of Novel Dose-Response Models for Use in Microbial 

Risk Assessment. U.S. Environmental Protection Agency, National Center for Environmental 

Assessment, Cincinnati, OH. EPA/600/R-08/033. 

ii
TABLE OF CONTENTS

Page

LIST OF ABBREVIATIONS.............................................................................................iv 

PREFACE ....................................................................................................................... v

AUTHORS, CONTRIBUTORS AND REVIEWERS .........................................................vi 

1. INTRODUCTION: SCOPE, PURPOSE AND OBJECTIVES ................................ 1 

 1.1. THE PREDICTIVE BAYESIAN FRAMEWORK FOR MICROBIAL 

RISK ASSESSMENT................................................................................. 1 

 1.2. A NON-POISSON DISTRIBUTION AND RELATED DOSERESPONSE FUNCTION FOR PATHOGENS IN TREATED 

DRINKING WATER ................................................................................... 4 

 1.3. THE DISCRETE-SCALING DOSE-RESPONSE FUNCTION ................... 4 

2. METHODS............................................................................................................ 6 

 2.1. PREDICTIVE BAYESIAN METHODS........................................................ 6 

 2.2. DISCRETE-SCALING DISTRIBUTION...................................................... 7 

 2.3. BETA DISCRETE-SCALING DISTRIBUTION DOSERESPONSE FUNCTION (beta-DSD) ........................................................ 7 

3. SUMMARY OF RESULTS.................................................................................... 8 

 3.1. PREDICTIVE BAYESIAN DOSE-RESPONSE ASSESSMENT 

FOR APPRAISING ABSOLUTE HEALTH RISK FROM 

AVAILABLE INFORMATION ................................................................... 10 

 3.2. PREDICTIVE POPULATION DOSE-RESPONSE ASSESSMENT 

FOR CRYPTOSPORIDIUM PARVUM: INFECTION ENDPOINT ............ 11 

 3.3. PREDICTIVE BAYESIAN MICROBIAL DOSE-RESPONSE 

ASSESSMENT BASED ON SUGGESTED SELFORGANIZATION IN PRIMARY ILLNESS RESPONSE: 

CRYPTOSPORIDIUM PARVUM ............................................................. 11 

 3.4. A NEW THEORETICAL DISCRETE “SCALING” PROBABILITY 

DISTRIBUTION WITH VERIFICATION FOR MICROBIAL 

COUNTS IN WATER ............................................................................... 12 

 3.5. A DOSE-RESPONSE FUNCTION FOR DISCRETE-SCALING 

DISTRIBUTED PATHOGENS SUCH AS FOUND IN DRINKING 

WATER.................................................................................................... 12 

4. DISCUSSION AND CONCLUSIONS ................................................................. 14 

5. RESEARCH NEEDS .......................................................................................... 19 

6. REFERENCES................................................................................................... 21 

iii
LIST OF ABBREVIATIONS

CCL Contaminant Candidate List 

DSD discrete-scaling distribution 

PLN Poisson lognormal 

SOC self-organized criticality 

U.S. EPA U.S. Environmental Protection Agency 

iv
PREFACE

 This document was prepared by the National Center for Environmental 

Assessment for the Office of Ground Water and Drinking Water. The document is a 

summary and synthesis of five publication manuscripts authored by Jeff Swartout in 

collaboration with Jim Englehardt of the University of Miami. The document contains a 

description of dose-response modeling methods designed to provide a robust approach 

under uncertainty for predicting human population risk from exposure to pathogens in 

drinking water. There was no literature search specific to this document, only in 

conjunction with the publication manuscripts, three of which have been published in the 

peer-reviewed literature and two that have been submitted for publication as of May 7, 

2007. 

v
vi

AUTHORS, CONTRIBUTORS AND REVIEWERS

AUTHORS

James D. Englehardt, Ph.D., P.E. 

ORISE Fellow 

University of Miami 

Jeff Swartout 

U.S. Environmental Protection Agency 

Office of Research and Development 

National Center for Environment Assessment 

Cincinnati, OH 45268 

CONTRIBUTORS

Chad Loewenstine 

U.S. Environmental Protection Agency 

Office of Research and Development 

National Center for Environment Assessment 

Cincinnati, OH 45268 

REVIEWERS

Glenn Suter II, Ph.D. 

U.S. Environmental Protection Agency 

Office of Research and Development 

National Center for Environment Assessment 

Cincinnati, OH 45268 

Nicholas J. Ashbolt, Ph.D. 

U.S. Environmental Protection Agency 

Office of Research and Development 

National Exposure Research Laboratory 

Cincinnati, OH 45268 
1. INTRODUCTION: SCOPE, PURPOSE AND OBJECTIVES 

Dose-response relationships relating population infection and illness responses 

to exposure to pathogens in drinking water are needed for the quantitative assessment 

of pathogen risks used in developing drinking water and wastewater reuse policy. 

Infection response is of importance in assessing secondary disease transmission and 

population-based pathogen risks (so-called dynamic risk models; Eisenberg et al., 

2005). Illness endpoints have typically been used to indicate the health burden, of 

primary importance in economic benefits assessment and management of risk. Doseresponse relationships for both endpoints, therefore, are needed for assessing the full 

extent of disease burden attributable to pathogens in drinking water and to evaluate the 

need for new regulation. 

The purpose of this document is to describe a body of literature on a predictive 

(unconditional) Bayesian framework as an alternative to the currently used approach 

(variations on ILSI, 2000) to express the risk of infection and illness resulting from 

exposure to pathogens in drinking water. Secondarily, an alternative to the Poisson 

distribution for characterizing pathogens in tap water is also described. Together, these 

new methods may provide a more realistic and rigorous depiction of the impact of 

water-borne pathogens on drinking water consumers. 

The objectives of this work are to: 

(1) Present the unconditional (predictive) Bayesian forms of infection and illness 

dose-response functions as an alternative to confidence bounds for the 

assessment of drinking water pathogen risks; 

(2) Derive a (discrete-scaling) form for the distribution of microbe counts in drinking 

water at the point of entry to the distribution system, based on a theoretical 

treatment process failure structure; and 

(3) Derive general infection and illness dose-response functions for pathogens in 

drinking water and other media based on the new discrete-scaling count 

distribution. 

1.1. THE PREDICTIVE BAYESIAN FRAMEWORK FOR MICROBIAL RISK 

ASSESSMENT 

Pathogen dose-response data are fundamentally sparse and noisy due to 

limitations in the numbers of human subjects that can be tested, and a range of 

uncertainties resulting from defining infection and illness and pathogen strain variations 

(e.g., Teunis et al., 2002). To address limitations in empirical verification, confidence1
based approaches such as likelihood profiles, parametric bootstrapping and the 

Benchmark Dose have been used. Traditional confidence-based approaches based on 

frequentist methods, however, are not ideal for predictive modeling, as is needed for 

microbial risk assessment. Confidence bounds are dependent—or conditional—on a 

specific confidence level (e.g., 5% or 10%), such that the relative ranking of two health 

stressors often reverses, given another level of confidence (Englehardt, 2004). In 

addition, at a 95% level, the assessed bound may be far from the assessed mean risk 

and potentially subject to large errors. Therefore, confidence bounds are not generally 

comparable among health stressors. 

Bayesian approaches have been replacing frequentist methods within predictive 

models, partially to overcome such limitations and partially from recognition that 

probability judgements are fundamentally subjective. In essence, Bayesian methods 

utilize more information in a more transparent manner than do frequentist methods. 

Bayesian methods are often confidence-based, but generally characterized by wider 

confidence bounds than the corresponding frequentist approach. In contrast, the 

predictive Bayesian method yields an unconditional result—one that is not dependent 

on a specific choice of confidence bounds. In the predictive Bayesian approach, the 

uncertainty represented in the confidence limits is integrated into the output, resulting in 

a more robust prediction. Figure 1 illustrates the results of an unconditional predictive 

Bayesian dose-response function compared to one (beta-Poisson) based on frequentist 

confidence limits. As the figure shows, the unconditional beta-Poisson response is 

slightly higher than the frequentist response determined by the direct fit of the betaPoisson dose-response function to the data. For perfect information, the two curves 

would be the same. Otherwise, the unconditional response is more health protective 

(conservative), but still represents the expected risk, with uncertainty incorporated 

directly. Risk management decisions are often “quantified” largely on the expected 

value and are only “qualified” by the confidence bounds. That is, the key quantitative 

risk estimates and corresponding benefits are based on the central tendency, not the 

extremes. The unconditional risk estimate provided by the predictive Bayesian 

approach allows for a more conservative estimate of the central tendency when 

information is not perfect. 

2
cumulative response

dose

10^-3 10^-1 10^1 10^3 10^5 10^7 10^9

10^0 10^-1 10^-2 10^-3 10^-4 10^-5 10^-6

Response data

Frequentist (beta-Poisson) fit

95% upper confidence bound

Unconditional (pred. Bayesian) DRF

FIGURE 1 

Comparison of an Unconditional Predictive-Bayesian Dose-Response Function 

(Englehardt, 2004) with Frequentist Confidence Bound Approach 

An analogous unconditional risk can be assessed using frequentist methods, 

though efficient computational methods have been lacking and the approach has not 

found use. More important is that the frequentist approach precludes the use of any 

information other than numerical dose-response data, such as data on genetic 

prevalence that is rapidly finding application in dose-response assessment, and 

epidemiological information, which can substantially increase the knowledge base for a 

dose-response assessment. Equally important, the predictive Bayesian approach 

computed using Markov chain Monte Carlo methods developed over the last decade, 

gives the frequentist answer directly, if no non-traditional dose-response information is 

used in the analysis. 

3
1.2. A NON-POISSON DISTRIBUTION AND RELATED DOSE-RESPONSE 

FUNCTION FOR PATHOGENS IN TREATED DRINKING WATER 

The second objective of this work was to show how the distribution of pathogens 

in finished drinking water is likely to deviate from the Poisson distribution, and illustrate 

how that deviation will effect the dose-response function for consumers. The form of 

the function relating dose of pathogens in drinking water to illness response depends 

not only on the numerous characteristics of gastrointestinal pathogenesis, but also on 

the distribution of pathogens in the water. The pathogen distribution in finished drinking 

water over time is highly disperse, unlike the well-mixed uniformly-distributed (Poisson) 

laboratory sample used in controlled dose-response studies. For the low concentrations 

of pathogens typically found in drinking water, where the expected exposure at the tap 

is, at most, one organism at a time, a more disperse distribution (i.e., non-Poisson) will 

result in a reduction of the risk of infection in the population, compared to a Poisson 

distribution of pathogens, for any given long-term mean concentration. The discretescaling distribution (DSD; Englehardt et al., 2008b) was developed to account for this 

greater dispersion of microbes expected in finished drinking water. The DSD was 

based on a theoretical consideration of dependent multiplicative failure processes in 

drinking water treatment plants, as opposed to the conventional assumption of 

independent processes, resulting in lognormal means. The DSD provided the best fit to 

several finished and raw water data sets, the common characteristics being a large 

number of zeroes (non-detects) and low counts for positive samples. Conversely, data 

sets characterized by few or no zeroes and high positive sample counts was fitted better 

by the Poisson-lognormal distribution. Englehardt et al. (2008b) recommend the use of 

the DSD for any set of water samples where the number of zero-count samples is 

greater than the number of any other unique count samples. That is, as the DSD is a 

strictly decreasing probability mass function, the highest probability for any single count 

value resides at zero. 

1.3. THE DISCRETE-SCALING DOSE-RESPONSE FUNCTION 

 In combining the results meeting the first two objectives, a new dose-response 

function was derived mathematically based on the new DSD count distribution. For the 

corresponding dose-response assessment of infection or illness risk, the DSD was 

integrated into a dose-response function by replacing the Poisson exposure component 

of the beta-Poisson function with the DSD. The beta-DSD dose-response function 

should be used whenever the DSD is the preferred distribution fit to water pathogen 

count data for a given site assessment. For over-dispersed (non-uniform) pathogen 

4
distributions, compared to the DSD, the beta-Poisson dose-response function could 

overestimate the risk substantially, depending on the magnitude of the measured 

pathogen concentrations. 

5
2. METHODS 

The mathematical methods presented in the five associated papers (described in 

Section 3) are summarized here in two parts. Section 2.1 describes the predictive 

Bayesian approach, which is used in conjunction with both the beta-Poisson and betadiscrete dose-response functions. Section 2.2 describes the discrete-scaling 

distribution for modeling the occurrence of pathogens in finished drinking water. The 

DSD, in turn, is the basis for the exposure component of the beta-discrete doseresponse function. 

2.1. PREDICTIVE BAYESIAN METHODS

To address the nature of low-dose response assessment, an unconditional 

probability of illness, or other response, can be obtained as the conditional doseresponse function (sampling distribution) multiplied by the distribution of uncertainty in 

model parameters, integrated over the full range of variability and uncertainty in the 

parameters (Englehardt, 2004). The result is a distribution for the quantity of interest 

that is quantitatively narrower as more information becomes available, consistent with 

principles of Shannon entropy (Shannon, 1948). Unconditional Bayesian distributions 

have been termed predictive distributions (Aitchison and Dunsmore, 1975). A predictive 

Bayesian probability distribution integrates all available information, as well as the full 

range of possible confidence levels. In addition, because the predictive distribution is 

sensitive to the assumed parametric sampling distribution, the method is powerful for 

the exploitation of information on the expected form of the distribution obtained by 

theoretical derivation. In the general case, the distribution of variability and uncertainty 

can be a Bayesian posterior, incorporating non-traditional information (e.g., genetic or 

epidemiological) as well as numerical dose-response data. The approach produces an 

unconditional probability distribution that is wider and more conservative when less 

information is available. The unconditional distribution is sensitive to, and exploitive of, 

knowledge of the form of the dose-response function. 

For both the beta-Poisson (Englehardt and Swartout, 2006) and beta-discrete 

illness functions (Englehardt et al., 2008a), knowledge of the form of the function was 

obtained from simple models of the pathogenic process in the gut. The models were 

based on the principle of self-organized criticality (SOC) (Bak et al., 1988). The original 

model of species extinction proposed by Bak and coworker represents the process by 

which natural selection can drive evolutionary self-organization in stochastic systems, 

such that general patterns of outcomes emerge from specific, complex, small-scale 

6
interactions. Although there is no direct practical utility of such models applied to the 

management of microbial risks, the approach was useful for investigation of integrative 

dose-response characteristics that may emerge from complex, adaptive host-pathogen 

interactions (Englehardt and Swartout, 2006; Englehardt et al., 2008a). 

2.2. DISCRETE-SCALING DISTRIBUTION 

The DSD was developed for assessing the occurrence-distribution of pathogens 

in finished drinking water, specifically for the case of dependent multiplicative-error 

processes in water treatment plants, where the failure of one process is, at least, 

partially dependent on the failure of the immediately preceding process (Englehardt, 

1995; Englehardt et al., 2008b). The DSD can be generalized to cases of partiallydependent and independent causes. That is, the result of Frisch and Sornette (1997), 

showing the Weibull tail for incident sizes arising as the product of independent 

exponential cause sizes, was generalized by Englehardt et al. (2008b) to the case of 

discrete geometric cause sizes. The DSD is meant to be used for fitting to microbial 

count data from drinking water samples taken over time. Englehardt et al. (2008b) 

provided fits for the DSD to several example data sets. 

2.3. BETA DISCRETE-SCALING DISTRIBUTION DOSE-RESPONSE FUNCTION 

(beta-DSD) 

 The beta-DSD integrates the beta distribution for inter-individual variability in 

response with the DSD as the exposure component. The beta-DSD is characterized by 

three parameters. Two of the parameters are identical to the beta parameters of the 

beta-Poisson, representing the distribution of response variability in the human 

population. The third parameter is the characteristic exponent, η, from the DSD function 

itself. The functional form is somewhat complex and is not reproduced here. The 

reader is referred to Englehardt et al. (2008a) for the technical details. The beta-DSD is 

linear at very low doses (dilutions of single organisms), but approaches linearity more 

slowly than the beta-Poisson because of the higher probability of exposure to more than 

one organism at a time. However, as explained previously, the beta-DSD predicts lower 

risk at low doses prior to becoming linear with respect to the beta-Poisson. Conversely, 

the beta-DSD predicts higher risk than the beta-Poisson at high doses. 

7
3. SUMMARY OF RESULTS 

 The principal results of this work are as follows. 

1. Englehardt (2004) and Englehardt and Swartout (2004, 2006) demonstrate that 

the relative importance of two health stressors may depend on the chosen level 

of confidence. In other words, because the confidence-bound approach is based 

on an isolated, somewhat arbitrary level of confidence, to the exclusion of the 

remaining distribution of parameter uncertainty, stressors may vary with respect 

to their relative importance. 

2. A predictive Bayesian pathogen dose-response function that expresses the 

unconditional probability of infection was developed (Englehardt, 2004). The 

function is rigorously conservative and consistent among health stressors. The 

function is general for Poisson-distributed pathogens as found in laboratory 

samples. An unconditional assessment was presented based on available data 

for rotavirus. 

3. An unconditional dose-response assessment, based on available infection data 

for C. parvum, was developed and published (Englehardt and Swartout, 2004). 

4. A new general pathogen dose-response function for the illness endpoint was 

derived based on a model of self-organized pathogenesis in the human 

gastrointestinal tract and published (Englehardt and Swartout, 2006). The 

conditional function derived is a generalization of the beta-Poisson model, and 

was presented as a predictive Bayesian, unconditional model and demonstrated 

for C. parvum. 

5. The assumption of Poisson-distributed pathogens underlying current doseresponse assessments was shown to be inappropriate for environmental 

samples such as drinking water and source water (Englehardt et al., 2008a,b). A 

new discrete scaling distribution (DSD) for pathogen counts in drinking water was 

derived from theoretical principles and shown to describe well the distribution of 

pathogens in water samples characterized by a high number of zero-count 

observations (Englehardt et al., 2008b). The DSD is expected to accurately 

model distributions of pathogens in any media, including food and contaminated 

buildings, in which low pathogen counts are observed. The existing (Poisson 

lognormal) distribution was preferred for pathogen counts in environmental 

samples where higher counts (few zero counts) are observed. 

6. A new dose-response function based on the new count distribution was derived 

for pathogens in environmental samples such as drinking water, for the infection 

and illness endpoints, not accounting for secondary transmission of infection, as 

shown in Figure 2 (Englehardt et al., 2008b). The predictive Bayesian version of 

the conditional function derived was presented and demonstrated for C. parvum. 

8
-10 10 -5 10 0 10 5 10

-10 10

-8 10

-6 10

-4 10

-2 10

0 10

ytilobabie prviatulumC

Dose (mean viable DSD oocysts per exposure)

Poisson exposure probability

Unconditional illness response, N=1

Unconditional illness response, N=0.5

Unconditional illness response, N=0.33

Unconditional illness response, N=0.25

Unconditional illness response, N=0.2

FIGURE 2 

Predictive Bayesian Beta-Discrete Scaling Illness Dose-Response Function for 

C. parvum (all isolates) for the Inverse Number of Unit Processes η = 1, 0.5, 0.33, 0.25, 

0.2 Corresponding to Treatment Plant Sophistication Ranging from Basic (η = 1) to 

Advanced (η < 0.2) 

 

In the five papers summarized in the remainder of this chapter, unconditional 

infection/illness dose-response functions for both Poisson and discrete scalingdistributed pathogens are proposed for use in risk characterization of pathogen 

exposure in drinking water. In the first two papers (§3.1), the predictive Bayesian doseresponse assessment approach is proposed for the infection endpoint and 

demonstrated using human dose-response data for rotavirus and C. parvum. In the 

third paper (§3.2), the generalized beta-Poisson is proposed for evaluating illness 

response in laboratory data, and demonstrated for C. parvum. In the fourth paper 

(§3.3), the discrete scaling distribution (DSD) is derived for assessing exposure to 

pathogens in drinking water, and verified versus simulated long-term drinking water data 

and validated versus field data. In the fifth paper (§3.4), the beta-discrete scaling doseresponse function is derived for drinking water dose-response assessment and 

demonstrated for Cryptosporidium. 

9
3.1. PREDICTIVE BAYESIAN DOSE-RESPONSE ASSESSMENT FOR 

APPRAISING ABSOLUTE HEALTH RISK FROM AVAILABLE 

INFORMATION (Englehardt, 2004) 

Englehardt (2004) presents the predictive Bayesian approach for use in microbial 

dose-response assessment as an improvement over other methods. In contrast with 

other methods of handling limited information, Bayesian methods exploit available 

subjective and related information (unlike resampling plans such as the bootstrap, and 

likelihood methods such as the Benchmark Dose), as well as numeric data (unlike 

fuzzy-logic and unconditional frequentist methods). The approach allows quantitative 

assessment of probabilities (unlike interval bounding methods). Bayesian methods, in 

general, are becoming widely used to obtain distributions of confidence around the 

parameters of a function of interest. 

In contrast to confidence bounds, a predictive Bayesian probability distribution is 

an unconditional one, integrated over the entire distribution of parameter uncertainty. 

That is, the answer takes into account all possible values of the parameters, effectively 

incorporating all confidence levels. The result can be thought of as the expected value 

under uncertainty. A primary property, and one of the principal advantages, of the 

predictive Bayesian distribution (dose-response function) is that it becomes 

quantitatively narrower as more information is obtained. This means that higher risks 

(relative to the true unknown risk) are predicted when information is limited. In addition, 

because the predictive distribution is sensitive to the assumed parametric sampling 

distribution (i.e., the dose-response model), the method is powerful for the exploitation 

of information on the expected form of the distribution obtained by theoretical derivation. 

As any number of unrelated empirical models will probably fit any given (high-dose) 

data set well, a strong theoretical basis for the conditional model is critical to any doseresponse assessment, and the predictive Bayesian approach leverages this information. 

 Englehardt (2004) demonstrated the predictive Bayesian approach for human 

infection response data for orally-ingested rotavirus (Ward et al., 1986). The 

unconditional predictive Bayesian assessed risk was lower than the theoretical 

maximum risk (exposure risk), but higher than the risk based on fitting the conditional 

dose-response function to the data. Simulating additional data caused the 

unconditional risk to drop closer to the conditional result. That is, assessed risk was 

lower with increasing information availability, as expected, based on information theory. 

The exercise demonstrates the simplicity of evaluating the value of additional 

information. The change in risk and the corresponding monetary benefit can be 

estimated directly from the predictive Bayesian dose-response plot. 

10
3.2. PREDICTIVE POPULATION DOSE-RESPONSE ASSESSMENT FOR 

CRYPTOSPORIDIUM PARVUM: INFECTION ENDPOINT (Englehardt and 

Swartout, 2004) 

Englehardt and Swartout (2004) presented a predictive human population doseresponse assessment for C. parvum for the infection endpoint, demonstrating a 

hierarchical predictive approach for microbial dose-response assessment. Available 

data on the infectivity of three isolates of Cryptosporidium parvum were adjusted for 

sensitive and resistant subpopulations not proportionately represented in the data by 

bootstrap analysis. The diverse mean infectivities of the isolates were used to obtain a 

predictive distribution for population infectivity, used in turn to obtain the predictive 

population dose-response function. 

3.3. PREDICTIVE BAYESIAN MICROBIAL DOSE-RESPONSE ASSESSMENT 

BASED ON SUGGESTED SELF-ORGANIZATION IN PRIMARY ILLNESS 

RESPONSE: CRYPTOSPORIDIUM PARVUM (Englehardt and Swartout, 

2006) 

 This paper describes the first probabilistic derivation of a general model of 

pathogen dose-response and is based on conceptual models of the process of 

gastroenteric pathogenesis. A self-organized, critical (SOC) model of host-pathogen 

interactions was postulated, such that once an infection is established in the GI tract, 

the wellness (i.e., fitness) of the least well GI tract segment will increase (host winning) 

or decrease (pathogen winning) randomly in a time step. The basic model was a linear 

30-segment representation of the GI tract, the only parameters of which were those of 

the infection dose-response relationship (beta-Poisson) and a simulated clinical 

diagnosis of illness (e.g., diarrhea). Infection in the basic model was established 

according to the beta-Poisson relationship. Illness is determined to be over when the 

wellness value of all segments in the GI tract are above a self-organized critical value 

(0.678) that emerged from the simulation runs. Characteristics of the resulting 

probability of illness are determined entirely by self-organization—that is, by selfselection of the least well segment and its neighbors for random wellness revision in 

each time step. Simulations involved testing of one million “hosts” at multiple doses. 

The results of the simulation (probability of illness with respect to dose) was best 

described by a beta-Poisson distribution multiplied by a constant (<1). The constant 

fractional multiplier is mathematically equivalent to the susceptible proportion of the 

population, or one minus the immune fraction. Distributions of severity at a dose were 

characterized by power law distributions ranging over several orders of magnitude. 

That is, the severity of illness in this model behaved as an “incident,” the size of which is 

11
proportional to the product of the preceding cause sizes according to early complex 

systems theory (Chow, 1954; Lomnitz, 1964; Benjamin and Cornell, 1970; Englehardt, 

2002). This finding may have significance in the future development of more realistic 

morbidity models incorporating variable severities, such as attempted for by the 

disability adjusted life year metric for Campylobacter jejuni illnesses (Havelaar et al., 

2000). 

3.4. A NEW THEORETICAL DISCRETE “SCALING” PROBABILITY 

DISTRIBUTION WITH VERIFICATION FOR MICROBIAL COUNTS IN WATER 

(Englehardt et al., 2008b) 

The form of the distribution of pathogen counts in drinking water and other low 

mean-count media is currently unknown, partly because most counts are zero. 

However, the mean of the count distribution is directly proportional to the dose of 

pathogens and, as the outcome of a complex system, the count distribution potentially 

“scales” over several orders of magnitude. Therefore, the long-term dose may be 

governed by rare, high-count events not generally represented in available short-term 

plant monitoring data, and the distributional form is needed to help assess this dose. 

The form of the count distribution is also needed in dose-response assessment, to 

extrapolate health response from high doses to low. In this paper a new discrete 

scaling distribution (DSD) was derived for assessing microbial counts in finished 

drinking water. The exponential parameter of the new distribution corresponds in 

concept to a real-valued inverse number of causes of pathogen counts. Scaling of the 

distribution is a consequence of interaction (dependence) among cause sizes. The 

DSD was shown to fit well to low mean-count data (containing many zeros) such as 

drinking water. Conversely, the Poisson-lognormal (PLN), a commonly used distribution 

(Haas et al., 1999; Masago et al., 2004) was more efficient at fitting high mean-count 

samples such as source water. 

3.5. A DOSE-RESPONSE FUNCTION FOR DISCRETE-SCALING DISTRIBUTED 

PATHOGENS SUCH AS FOUND IN DRINKING WATER (Englehardt et al., 

2008a) 

This paper presents the derivation of a generalized dose-response function 

based on the new DSD count distribution, for the illness and infection endpoints, 

analogous to the generalized beta-Poisson function derived for Poisson-distributed 

organisms in Englehardt and Swartout (2006). The shape of the conditional function is 

established for the illness endpoint by means of simple, self-organized, critical models 

of gastro-enteric pathogenesis. The unconditional predictive Bayesian form of the 

12
derived beta-discrete scaling dose-response function is then plotted using posterior 

distributions computed previously from available data for Cryptosporidium, for infection 

and illness assuming various numbers of conceptual unit processes. This range of 

values of the treatment parameter is typical of the number of unit treatment processes in 

drinking water treatment plants, and of microbe count distributions fitted to empirical 

data previously. 

 In contrast to the beta-Poisson, the beta-discrete scaling function is not linear at 

doses above 10−5. Therefore, linear low-dose extrapolation assuming a dose-response 

slope of unity, as assumed for Poisson-distributed organisms, may not be appropriate. 

In particular, long-term average drinking water concentrations accounting for plant upset 

events may be above the linear range (Englehardt et al., 2008b). In such cases, the 

implementation of additional treatment barriers at a plant would result in a lower-mean, 

higher-skew distribution of pathogen counts in the treated water, and a relaxation of the 

dose associated with an acceptable response level. On the other hand, at the level of 

10-7 illnesses/exposure originally envisioned in the U.S. Environmental Protection 

Agency (U.S. EPA) Surface Water Treatment Rule, the allowable C. parvum

concentration would be on the order of 10-6 oocysts/L for any treatment plant. These 

results do not consider secondary transmission of illness or exposure-driven immunity, 

both subjects of current research. 

13
4. DISCUSSION AND CONCLUSIONS 

 The primary theme in this body of work is the utility of the predictive Bayesian 

framework for the evaluation of pathogen dose-response functions in a data-poor 

environment. Proponents of Bayesian methods, in general, argue that a Bayesian 

approach is a virtual requirement when trying to assess probabilities of future outcomes, 

rather than the frequencies of past events (Bernardo and Smith, 1994; Berry, 1996; 

Carlin and Louis, 2001). Bayesian methods provide a rigorous framework for commonsense interpretation of statistical conclusions. Frequentist approaches can only place 

confidence limits on a result that depends on specific conditions, leading to inferences 

that might be made in repeated practice. Bayesian proponents point out that most 

people interpret, erroneously, frequentist results in the Bayesian sense, anyway 

(Gelman et al., 2004). Bayesian methods generally provide much greater flexibility than 

do frequentist methods, allowing them to cope with very complex, data-limited 

problems. 

The predictive Bayesian approach is different from “traditional” Bayesian 

methods in that the result is unconditional. In more standard Bayesian analyses, prior 

distributions are assigned to individual parameters of the model and the outputs are 

expressed similarly, as updated “posterior” distributions of the parameters based on a 

consideration of the empirical data. If no non-numerical dose-response information is 

available or desired for use, the prior can be made non-informative, resulting in the 

traditional frequentist assessment. The mean or mode of the posterior is then taken 

alone and used as a Bayesian estimate of the parameters of the conditional doseresponse function. The predictive Bayesian approach, in contrast, involves integrating 

the dose-response function over the entire range of the posterior distribution, effectively 

averaging the dose-response function (not the posterior) over the range of uncertainty, 

to produce a dose-response function that is not conditional upon the uncertain 

parameters. Because the predictive Bayesian integrates across all values of the 

parameters, interpretation of the “answer” does not depend on consideration of some 

relatively arbitrary confidence limit. With respect to dose-response functions, the 

predictive Bayesian output is a single risk curve—devoid of confidence bounds—that 

can be described as the “believed” risk, but is more rigorously termed the unconditional 

(with respect to parameter uncertainty) probability of response, representing the 

expected value of the dose-response function under uncertainty. This unconditional 

probability of response is more conservative (higher) when information is scarce than 

when there is a surfeit of data. As shown in Section 1.1 (Figure 1), the predictive 

14
Bayesian result is more conservative than the corresponding frequentist expected 

value. With “perfect” information, the predictive Bayesian result converges on the 

frequentist solution, which is why interpretation of the latter is more difficult. These 

characteristics make the predictive Bayesian approach an ideal choice for representing 

the expected outcomes of competing risk management options. 

 The predictive Bayesian approach has been demonstrated for both the infection 

and illness endpoints for rotavirus and Cryptosporidium parvum, under several different 

assumptions about the nature of the pathogen and the host and the exposure 

distribution. The rotavirus assessment (Englehardt, 2004) is the simpler case involving 

the infectivity of a single strain administered to a single population of susceptible adult 

hosts, primarily as an introduction to the use of the method for pathogen dose response. 

The assessment of C. parvum infection was the more complex application. Both 

pathogen and host variability were assessed in multiple dimensions. The assessment 

was unique in that host response was modeled for the entire human population, 

including sensitive and resistant sub-populations, as well as for susceptible adults. The 

human population response was evaluated separately across three separate isolates of 

C. parvum and integrated into a final unconditional population response curve by means 

of the predictive Bayesian method. The result was shown to be more health-protective 

than frequentist methods, but not extremely so (Englehardt and Swartout, 2004). A 

much more conservative result was obtained when all the C. parvum human response 

data are pooled, assuming no difference among the isolates (Englehardt and Swartout, 

2006; Englehardt et al., 2008b). Under this assumption, the cross-isolate variability was 

subsumed within the inter-individual host-pathogen variability, resulting in a stretching 

out of the dose-response curve. The unconditional unit infectivity estimate increased by 

about an order of magnitude compared to the separate strain assumption for the 

individual isolates. In addition, as the morbidity ratio was very high in the study 

subjects, and given slightly greater noise in the illness data compared to infection, the 

unconditional probability of illness was virtually the same as for infection for the single 

strain scenario. Simply fitting the beta-Poisson dose-response function to the same 

data results in a slightly lower probability of infection at low dose (0.45 vs. 0.50), but a 

significantly lower probability of illness (0.38 vs. 0.50). The difference reflects the 

incorporation of uncertainty by the predictive Bayesian method into the unconditional 

result. 

 The self-organized-criticality (SOC) model used in both illness modeling papers 

(Englehardt and Swartout, 2006; Englehardt et al., 2008a) provides a unique and 

innovative approach by suggesting a general form of the illness dose-response 

15
relationship. The behavior of the illness severity measure as an “incident,” similar to 

other phenomena in nature and engineered systems, may have significance in the 

future development of more realistic morbidity models incorporating variable severities. 

That is, insights into the distribution of various outcomes in a comprehensive model of 

continuously-variable severities, from infection to diarrhea to death, is of critical 

importance. The realization of such a model would greatly reduce the response 

classification error inherently present in the current categorical response approaches. 

The discrete-scaling distribution developed in Englehardt et al. (2008b) is based 

on the consideration of water treatment processes having failure rates that are, at least, 

partially (positively) dependent. In this sense, failure rates have been likened to cause 

sizes in a complex system incident-size distribution analysis. Other published water 

treatment process simulations have considered the failure rates to be independent 

(Haas and Trussell, 1998; Masago et al., 2004; Signor and Ashbolt, 2005). The primary 

difference in the assumptions is that dependent processes will result in a more skewed 

distribution of pathogens in finished drinking water than for independent processes. 

The implications of this behavior are two-fold: 

First, adding more treatment processes may lower mean occurrence but will 

likely increase dispersion, resulting in a higher probability of larger events. The 

latter could mean a greater risk of outbreaks than would be predicted under the 

assumption of a less-skewed distribution. Longer monitoring records are needed 

to predict long-term exposure. Perhaps of more significance is the potential for 

significant underestimation of the long-term mean pathogen concentration when 

the temporal distribution is highly skewed, where high-count rare events greatly 

influence the mean. Short-term monitoring efforts, even with adequate sample 

volumes, may underestimate the longer-term population risk significantly. 

Second, a more disperse distribution will generally result in a lower aggregate 

risk of infection with a given mean pathogen concentration. The latter is the 

result of an increase in the probability of exposure per exposure event to more 

than one pathogen unit (organism) at low exposure concentrations (<<1 

organism per liter), but a corresponding increase in the number of null exposures 

(failure to ingest any pathogens). Because the increase of unexposed individuals 

results in a proportional decrease in population risk and the increase in risk from 

exposure to more than one unit per event is less than proportional, the net result 

is a decrease in overall population risk for more disperse exposure distributions. 

The difference becomes more pronounced with increasing skewness and 

increasing dose. 

Follow-up work will necessarily include long-term data simulation to investigate the 

impact of distribution parameter assumptions on population risk. 

16
 In a practical sense, the DSD appears to be favored over the PLN for modeling 

exposure to sparse, high-zero, low-count data (i.e., occurrence data characterized by 

mostly non-detects and positive counts covering a relatively narrow range). This 

conclusion is not based so much on the results presented in Englehardt et al. (2008b) 

where a slight preference for the DSD was found, but rather on preliminary results from 

a more detailed analysis of New York City reservoir data and simulations of observation 

records of various lengths (Englehardt et al., 2007). In the latter study, a strong 

preference for the DSD over the PLN is shown by the ability of the model to predict the 

longer-term distribution from short-term records. Additional work is being conducted to 

clarify these relationships. 

The impact of discrete-scaling distributed pathogens in drinking water was 

investigated in Englehardt et al. (2008a), by assuming that exposure at the tap for 

drinking water consumers would be similar to the one found at the treatment plant. The 

effect of the use of the proposed function, as opposed to the beta-Poisson doseresponse function, was shown to be a partial relaxation of the apparent allowable 

concentration of pathogens in drinking water for a fixed level of acceptable risk. This 

difference was more noticeable for microbial count distributions representing the case of 

treatment plants that present a more effective overall barrier to pathogen breakthrough 

(i.e., more treatment processes). However, the difference between treatment plants 

may or may not be appreciable at very low exposure levels depending on the maximum 

acceptable risk level assumed. For example, given the results of this analysis 

(Englehardt et al., 2008a), the concentration corresponding to an acceptable risk level 

of 10-4 illnesses/year would be on the order of 10-6 oocysts/L for any treatment plant, 

assuming a drinking water consumption of one liter per day. This exposure estimate 

does not consider secondary transmission of illness or acquired immunity, both subjects 

of current research. With “perfect” information, that is knowledge of the true population 

response, the dose corresponding to an unconditional risk of 10-4 illnesses per year 

might be an order of magnitude higher. The extent to which the predictive Bayesian 

underestimates that dose (i.e., overestimates risk) reflects the degree to which the data 

are imperfect. Frequentist methods typically reflect only particular types of experimental 

and physical variability, without addressing the often-dominant uncertainties rooted in 

information limitations.

 With respect to Cryptosporidium parvum, used as an example in most of the 

papers cited above, the single most significant factor is pathogen variability. Treating 

the C. parvum isolates as distinct strains results in an unconditional unit risk of infection 

of about 0.05 (Englehardt and Swartout, 2004). If the isolates, instead, are assumed to 

17
be random samples of the same strain, the unconditional unit risk of infection is an order 

of magnitude higher (Englehardt and Swartout, 2006; Englehardt et al., 2008b). 

Although part of the increased risk in the latter scenario was due to the addition of new 

data, most was a result of the single strain assumption. In contrast, human population 

variability does not appear to be as big a factor. Taken individually, most of the 

C. parvum isolate response data sets are fit adequately by the exponential model, 

implying a lack of inter-individual variability in the human hosts. The simulation of 

sensitive and resistant subpopulations had little impact on the overall population risk 

(Englehardt and Swartout, 2004). Changing the underlying pathogen distribution to a 

more disperse one resulted in significant impact in the extreme case at higher 

exposures but, in most cases, did not change the risk substantially at expected 

exposure levels in drinking water (<10-4 oocysts per liter). It would be premature to 

generalize these conclusions, however, as C. parvum may be somewhat unique with 

respect to other microorganisms. Follow-up work in evaluating the nature of the 

response differences across the C. parvum isolates is indicated. 

The methods developed in this body of work are designed to provide scientificallydefensible and rapid (essentially on-demand) dose-response assessments with the 

following caveats: (a) currently, only primary response to an ingested dose (e.g., in 

drinking water) can be assessed, (b) the temporal effects of acquired immunity on 

population response are not well-understood and (c) the applicability of laboratory 

human dose-response data to actual response to environmental strains of 

pathogens is not well-understood. First, to understand the total disease burden in 

the population, modeling of the secondary transmission of disease and doseresponse for other sequellae (other disease endpoints from the one pathogen 

infection) in the population is necessary. Second, perhaps using the same model, 

the long-term effect of acquired immunity on the primary response needs to be 

evaluated. Third, questions remain as to whether strains of pathogens cultured in 

the lab over various time periods accurately represent the infectivity of the strains 

existing in the environment. 

18
5. RESEARCH NEEDS 

 The following list of research needs is not necessarily comprehensive or in order 

of priority. The highest priority tasks will depend on the specific needs of the U.S. EPA 

program offices as overarching regulatory processes develop. The first two research 

needs, however, address specific regulatory programs within the Office of Water and 

may have higher priority. 

1. Assess the relative virulence and the effect of laboratory culturing on strain 

virulence for all C. parvum isolates and determine their prevalence in the 

environment. Fully assess the relationship among C. parvum and C. hominis

isolates (one strain or many?) by means of statistical modeling. Published 

assessments of the infectivity and morbidity of C. parvum vary by more than an 

order of magnitude. The choice of parameter values will have a significant 

impact on the 6-year review of the Long-Term 2 Surface Water Treatment Rule 

(due in 2012). 

2. Develop unconditional dose-response functions for CCL organisms based on 

available information. There are a few organisms on the proposed CCL3 for 

which some data exist. 

3. Model the effect of long-term source-water variability on finished-water pathogen 

distributions. The current version of the DSD does not take into account, 

explicitly, source-water variability. Many pathogens exhibit seasonal variability in 

occurrence. Preliminary analysis of the New York City source water supply 

suggests that other temporal cycles may exist. This research will have direct 

impact on human exposure estimation. 

 

4. Evaluate the monitoring strategy necessary for predicting the long-term 

distribution of pathogens in drinking water (length of record, frequency of 

measurements, sample volume). The DSD fits to some data sets suggest highly 

disperse distributions, characterized mostly by zero or low-count samples and 

long-term mean values dependent on large rare events, suggesting longer 

monitoring periods, more frequent sampling and larger sample volumes. 

Alternatively, quantification of pathogens in raw waters and their removal (largely 

using surrogates) may provide more accurate assessment of their distributions in 

treated waters (Signor et al., 2007). 

5. Evaluate the effect of secondary transmission on the shape of the dose-response 

function to estimate the total disease burden from primary drinking water 

exposure. 

6. Evaluate the effect of exposure-induced immunity in the population to assess the 

impact of prior exposure on primary response to pathogens in drinking water. 

19
7. Demonstrate the incorporation of information other than numerical doseresponse data in the predictive Bayesian posterior, such as genetic prevalence 

data and epidemiological information. This work allows for a fuller exploitation of 

existing data in a rigorous framework, with an expectation of reduced uncertainty 

in the prediction. 

 

8. Further evaluate the behavior of the predictive Bayesian with respect to change 

in shape relative to amount of data, using simulated data, to evaluate the value of 

information. 

20
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