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Available online at w

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Discharge-based QMRA for estimation of publichealth risks from exposure to stormwater-borne pathogens in recreational waters inthe United States

Graham B. McBride a,*, Rebecca Stott a, Woutrina Miller b, Dustin Bambic c,1,Stefan Wuertz d,e

aNIWA (National Institute of Water and Atmospheric Research), P.O. Box 11-115, Hamilton 3251, New ZealandbDepartment of Pathology, Microbiology, and Immunology, School of Veterinary Medicine, University of California,

Davis, CA 95616, USAcAMEC Earth & Environmental, Nashville, TN 37211, USAdDepartment of Civil & Environmental Engineering, University of California, Davis, CA 95616, USAeSingapore Centre of Environmental Life Sciences Engineering (SCELSE) and School of Civil and

Environmental Engineering, Nanyang Technological University, 60 Nanyang Drive, Singapore

a r t i c l e i n f o

Article history:

Received 19 December 2012

Received in revised form

24 May 2013

Accepted 2 June 2013

Available online 12 June 2013

Keywords:

QMRA

Pathogens

Stormwater

Norovirus

Rotavirus

Health

* Corresponding author. Tel.: þ64 7 856 1726E-mail address: Graham.McBride@niwa.c

1 Present address: Tetra Tech, 712 Melrose0043-1354/$ e see front matter ª 2013 Elsevhttp://dx.doi.org/10.1016/j.watres.2013.06.001

a b s t r a c t

This study is the first to report a quantitative microbial risk assessment (QMRA) on path-

ogens detected in stormwater discharges-of-concern, rather than relying on pathogen

measurements in receiving waters. The pathogen concentrations include seven “Reference

Pathogens” identified by the U.S. EPA: Cryptosporidium, Giardia, Salmonella, Norovirus,

Rotavirus, Enterovirus, and Adenovirus. Data were collected from 12 sites representative of

seven discharge types (including residential, commercial/industrial runoff, agricultural

runoff, combined sewer overflows, and forested land), mainly during wet weather condi-

tions during which times human health risks can be substantially elevated. The risks

calculated herein therefore generally apply to short-term conditions (during and just after

rainfall events) and so the results can be used by water managers to potentially inform the

public, even for waters that comply with current criteria (based as they are on a 30-day

mean risk). Using an example waterbody and mixed source, pathogen concentrations

were used in QMRA models to generate risk profiles for primary and secondary water

contact (or inhalation) by adults and children. A number of critical assumptions and

considerations around the QMRA analysis are highlighted, particularly the harmonization

of the pathogen concentrations measured in discharges during this project with those

measured (using different methods) during the published doseeresponse clinical trials.

Norovirus was the most dominant predicted health risk, though further research on its

doseeresponse for illness (cf. infection) is needed. Even if the example mixed-source

concentrations of pathogens had been reduced 30 times (by inactivation and mixing), the

predicted swimming-associated illness rates e largely driven by Norovirus infections e

can still be appreciable. Rotavirus generally induced the second-highest incidence of risk

among the tested pathogens while risks for the other Reference Pathogens (Giardia,

.o.nz (G.B. McBride).Avenue, Nashville, TN 37211, USA.ier Ltd. All rights reserved.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 7 5283

Cryptosporidium, Adenovirus, Enterovirus and Salmonella) were considerably lower. Sec-

ondary contact or inhalation resulted in considerable reductions in risk compared to pri-

mary contact. Measurements of Norovirus and careful incorporation of its concentrations

into risk models (harmonization) should be a critical consideration for future QMRA efforts.

The discharge-based QMRA approach presented herein is particularly relevant to cases

where pathogens cannot be reliably detected in receiving waters with detection limits

relevant to human health effects.

ª 2013 Elsevier Ltd. All rights reserved.

1. Introduction runoff to receiving rivers/streams and often lower rates of

To date, development of recreational microbiological water

quality criteria has mostly focused on coastal waters (Cabelli

et al., 1982; Cabelli, 1983; Pruss, 1998; Zmirou et al., 2003;

Pond, 2005; Wade et al., 2010; Colford et al., 2012) and lakes

(Dufour, 1984; Wade et al., 2006; Marion et al., 2010), using

epidemiological studies pioneered by Stevenson (1953),

Cabelli (1983) and Dufour (1984). In contrast, very few

studies have included flowing inland waters (Ferley et al.,

1989; Wiedenmann et al., 2006; Kay, 2009). In general all the

aforementioned studies, for either coastal and inland waters,

support the view that risks of mild gastrointestinal (GI) illness

(and sometimes respiratory illness) can be enhanced among

swimmers when the water contains some degree of human

fecal contamination, as presumably indicated by concentra-

tions of fecal indicator bacteria (FIB). In most of these epide-

miological studies the dominant pathogenic material present

has been treated human sewage, often from publicly owned

treatment works (POTWs). Studies by Calderon et al. (1991),

Colford et al. (2007) and Fleisher et al. (2010), along with

some of the beaches included in the studies of Cheung et al.

(1990) and McBride et al. (1998), are notable exceptions in

that the dominant sources of fecal matter were animalsdthe

study of Fleisher et al. (2010) was conducted at a site “without

known sources of sewage”. In these non-sewage studies, evi-

dence for associations between animal-sourced fecal con-

centrations and health risk to recreational users is less clear

cut, with claims in support of a linkage (McBride, 1993) and

others against (Dufour et al., 2012). Nonetheless such doubt

should be treated with some caution, perhaps especially with

regard to livestock (Soller et al., 2010b). For example, there is

evidence in New Zealand that strains of Campylobacter spp.

associated with ovine and bovine animals are commonly

found in recreational waters and are associated with human

health impacts, assessed by Multi-Locus Sequence Typing

(MLST) of human isolates (French et al., 2011; McBride et al.,

2011). Notably, campylobacteriosis is the dominant notifiable

disease reported in that country (www.nzpho.org.nz). Simi-

larly, substantial increases in cryptosporidiosis among rural

dwellers in New Zealand commonly arise during the calving

season (Till and McBride, 2004).

Accordingly, epidemiological data are lacking regarding

the human health impacts of the mixtures of anthroponotic

and zoonotic fecal pathogens expected to be found in flowing

inland waters. Contamination by both can be expected in

many rivers and streams for which recreational water contact

may occur. Furthermore, given the proximity of terrestrial

dilution, one might expect higher levels of zoonotic pathogen

contamination inland than occurs at coastal sites. However,

comprehensive information on the degree of pathogen

contamination in USA inland waters is lackingdespecially as

epidemiological studies seldom include their measurement,

relying instead on fecal indicator bacteria (FIB) as represen-

tative of fecal pollution. Extension of findings from predomi-

nantly coastal waters to flowing inlandwaters could therefore

be viewed with scepticism, especially given the growing body

of evidence suggesting that FIB measured in recreational wa-

ters may be uncoupled from pathogen discharges due to

ubiquity, environmental regrowth/resuscitation, differential

inactivation, and/or persistence in sediments and soils

(Desmarais et al., 2002; Ferguson et al., 2003; Byappanahalli

et al., 2010).

Quantitative microbial risk assessment (QMRA) is a prom-

ising tool for predicting risks associated with water contact

given that it requires specific information on pathogens,

rather than fecal indicator bacteria (Till et al., 2008; Ashbolt

et al., 2010). Examples of that approach have been presented

by Schoen and Ashbolt (2010) who compared health impacts

associated with sewage versus seagull sources, Soller et al.

(2010a) who found that Norovirus tended to dominate the

effective pathogens at recent USA epidemiological study sites,

and Soller et al. (2010b)who usedQMRA to compare impacts of

human versus non-human sources. The latter study inferred

that GI illness risks associatedwith fresh cattle faecesmay not

be substantially different from those associated with human

sources, but that risks associated with fresh gull, chicken, or

pig faeces appear to be substantially lower.

The work reported herein had three objectives: (i) use

QMRA to estimate public health risks associated with storm-

water discharges to recreational waters, (ii) analyze the im-

plications of the findings for QMRA applications, and (iii)

examine the consequences of the selection of appropriate

endpoints for QMRA. In order to meet these objectives, an

extensive microbiological sampling program of runoff/

stormwater sources was conducted throughout the USA,

focusing mostly on wet weather and specific land uses as well

as discharge types (e.g., runoff from catchments categorized

as having land uses that are residential, industrial, agricul-

tural, and so forth). Discharges rather than receiving waters

were sampled because these data should be more applicable

to other watersheds/recreational sites. Data from discharges

were synthesized to represent potential recreational sce-

narios, and only short-term risks are reported, that is,

following discharge events.Wemake generalized calculations

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 75284

of risks associated with exposure directly to the source waters

compared with risks at a recreation site, where the source-

water concentrations have been reduced 30-fold. Actual ex-

posures in a site-specific QMRA would of course be based on

hydrodynamic mixing and inactivation modelsdbeyond the

scope of this paper.

2. Methods

While QMRA requires profiles of pathogen concentrations at

the point of exposure, it can be a more efficient use of re-

sources to characterize pathogen concentrations in the sour-

ces to recreational waters (e.g., runoff from regional

agricultural and residential areas) and to then rely on hydro-

dynamic models to convey the pathogens to the exposure

points. Because many stormwater discharges are intermit-

tent, pathogen concentrations at the point of exposuremay be

below the analytical limits of detection (non-detects) or well

below peak concentrations. Accordingly, the field work for

this study involved sampling of discharges-of-concern that

are not well characterized in the literature, in order to char-

acterize themwith regard to the occurrence and abundance of

high Reference Pathogens and source identifiers. With the

exception of the Southern California sites (see below), moni-

toring was performed during rainfall events.

2.1. Filtration and microbiological methods

At each site 50-L water samples were collected into carboys

and concentrated by ultrafiltration at a location near the

sampling sites. This step concentrated the water sample from

about 50 L to approximately 200mL using a Fresenius filtration

system modified from Hill et al. (2007) and Leskinen and Lim

(2008). Concentrated samples were shipped chilled for over-

night delivery to laboratories at the University of California,

Davis. Field duplicates and field blanks were also collected for

quality assurance and quality control.

Themonitoring toolkit employed a combination of genetic-

, cultivation-, and microscopy-based methodologies for mi-

crobes including Salmonella enterica, Campylobacter jejuni,

Cryptosporidium, Giardia, Adenoviruses, Enteroviruses, Nor-

oviruses and Rotaviruses. Prior to filtration, surrogates were

Table 1 e Methods for detection of pathogenic bacteria, protoz

Microbe type Microbe Molecularapproach

Molec

Pathogenic

Bacteria

Salmonella enterica qPCR Malorny et

Campylobacter jejuni qPCR Nogva et a

Protozoa Cryptosporidium parvum qPCR Fontaine a

Giardia lamblia qPCR Guy et al. (

Virus Adenovirus A, B, C

Adenovirus 40/41

qPCR Leruez-Vill

Rajal et al.

Enterovirus qPCR Fuhrman e

Norovirus (GI and GII) qPCR Wolf et al.

Rotavirus qPCR Gutierrez-A

spiked into environmental samples to estimate the percent

recovery of target organisms and to calculate sample-specific

limits of detection (Rajal et al., 2007a,b). This approach differs

from many others in that filtration recoveries and detection

limits were reported for individual samples, providing

important insights for the interpretation of monitoring re-

sults. Table 1 lists the target organisms and associated

quantification methods.

2.2. Site selection and monitoring

Over the duration of the monitoring period (March-

eSeptember 2011), 25 discharge sites were sampled over a

total of 17 monitoring events, selected to represent different

types of discharges-of concern to recreational waters (Table

2). All but the dry weather urban runoff (URBAN) sites were

sampled for multiple events. The number of sites sampled

during each event was dependent primarily on rain patterns

and where adequate precipitation conditions were present,

noting that the URBAN sites were sampled just once, in dry

weather. The number of monitoring events at the combined

sewer overflow (CSO) site were lower than expected because

of limited access during some storm events. Further infor-

mation on sites, monitoring, land use and soil types is given in

the Supplementary Information (SI). The exact locations of

sites were kept anonymous; instead the site locations are

described as general geographic regions (Mid-Atlantic,

Southeast, or Southern California).

2.2.1. Monitoring conditionsOfficial gages were all located at local major airports in each

zone and were supported by the National Weather Service.

Rain gages supplied 15-min data, helpful in assessing the

weather conditions for monitoring events. Local satellite

radar images were used to track storms and to identify loca-

tions where rain was falling.

2.2.2. Time-of-samplingSamples were collected using large-volume grab sampling

techniques, and the timing of each sampling with respect to

the hydrograph was estimated. Sampling was performed at

peak flow when possible under the assumption that most

fecal material was being transported at this time. However,

oa, and human viruses used in this study.

ular methods Conventionalapproach

Conventional methods

al. (2003) Culture U.S. EPA Method 1682,

Pant and Mittal (2008)

l. (2000) Culture Hijnen et al. (2004),

Wong et al. (2004)

nd Guillot (2003) Microscopy U.S. EPA Method 1623

2004) Microscopy U.S. EPA Method 1623

e et al. (2004),

(2007b)

N/A N/A

t al. (2005) N/A N/A

(2007) N/A N/A

guirre et al. (2008) N/A N/A

Table 2 e Description of discharge types and geographical locations.

Discharge type Acronym No. of sites No. of samples Discharge type characteristics

Residential Stormwater RESID 5 16 Separate municipal stormwater system

draining low-medium density residential

lands: Mid-Atlantic, Southeast, Texas.

Commercial/Light Industrial

Stormwater

COMML 3 13 Separate municipal stormwater system

draining shopping malls, restaurants etc.:

Mid-Atlantic, Southeast.

Agricultural Stormwater AGRIC 2 8 Open channel runoff from an agricultural

area with either row crop agricultural or

livestock grazing: Mid-Atlantic, Southeast.

CSO CSO 1 3 Water collected from within a combined

sewer system prior to discharge: Southeast.

Mixed Use Stormwater MIXED 5 12 Municipal separate stormwater system

draining a variety of land uses (residential,

commercial, agricultural): Texas.

Forested Open Space Stormwater FOREO 1 7 Runoff from a small, forested watershed

with little or no human access (no hiking

trails) and no development. Southeast.

Dry Season Urban Runoff URBAN 8 8 Separate stormwater system draining highly

urbanized residential and commercial areas

(exfiltrating groundwater, irrigation, car

washing, etc.). Samples were collected

during dry weather after several months

without rainfall: Southern California.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 7 5285

the large geographic sampling regions and variability of

rainfall prevented all samples from being collected during

peak flows. During some rainfall events, certain sites received

relatively high rainfall amounts while others remained dry.

Field staff indicatedwhether the volumetric flow rate from the

discharge was rising, at peak, or falling at the time of sam-

pling. See SI for further information.

2.3. QMRA

The following items are described sequentially below: (1)

identifying pathogens and their sources; (2) transport and fate

to and within waterbodies leading to human exposures to the

pathogens; (3) pathogen infectivity (doseeresponse); (4) char-

acterizing the health risks. In so doing, an attempt is made to

characterize fundamental mathematical parameters along

with their variability (such as duration of swimming) and

associated uncertainty (especially with respect to dos-

eeresponse curves). Most parameters are not fixed but instead

are described by statistical distributions that capture the

range and pattern of variation, allowing the use of algorithms

to quantify the level of risk and its variability. These distri-

butions were sampled randomly many times over to build a

risk profile, typically using the Monte Carlo statistical

modeling method (e.g., Haas et al., 1999), in this case using

@RISK software (Palisade Corporation, 2009).

2.3.1. PathogensSeven “Reference Pathogens” were included in this QMRA

study: Salmonella, Cryptosporidium, Giardia, Enterovirus,

Adenovirus, Norovirus and Rotavirus. (Campylobacter was

initially selected also, but our pathogen survey seldom

detected it.) The characteristics of sites to which QMRA has

been applied dictates which of these pathogens (or others)

should be used. For example, Adenoviruses that cause respi-

ratory illness may need to be included where aerosols may be

inhaled (e.g. by water skiers). Salmonella, Cryptosporidium, and

Giardia are included because they are zoonotic and potentially

waterborne, and it should be noted that the type as well as

loading of zoonotic pathogens can be quite variable depending

on factors such as animal host species, animal age, animal

density, distance from the waterway, land management

practices, and season (Cox et al., 2005; Miller et al., 2007). Also

important can be the presence of “supershedder” individuals

in animal herds (French et al., 2005; Chase-Topping et al.,

2008).

2.3.2. ExposuresBecause we report synthesized QMRA results for illustrative

purposes, simple reduction ratios have been adopted (from

source to exposure point), avoiding the need formore complex

contaminant transport and inactivationmodels that would be

required for site-specific assessments.

Few studies are available on swimmers’ ingestion rates

(Schets et al., 2011). In a pilot study, Dufour et al. (2006), used

chloroisocyanurates tracers and found that the average

amount of water swallowed by children and adults during

swimming was 37 mL and 16 mL per event, respectively,

where each event lasted at least 45 min (This was subse-

quently modified in their full study that reported children and

adult rates of 47 mL and 24 mL per event respectivelydEvans

et al., 2006.). One quarter of the swimmers swallowed 85mL or

more, and some swallowed up to 280 mL. Dorevitch et al.

(2010, 2011) used survey methods and chemical testing to

define three modes of contact: low (rowing, boating, fishing,

wading, non-capsizing kayaking and canoeing); Middle

(canoeing and kayaking with occasional capsizing); high

(swimmers). Average ingestions for these three modes (from

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 75286

Dorevitch et al., 2011) were 3.8, 5.8, 10 and mL per event,

respectively, and the authors suggested taking three times

that as a conservative estimate (an “upper confidence limit”).

The duration of each event was generally less than 1 h and

separate ingestion rates between children and adults were not

identified. Haas et al. (1999) used 50 mL ingestion for swim-

ming. Using the ingestion figures from Dufour et al. (2006),

Evans et al. (2006) and Dorevitch et al. (2010, 2011) as a

guide, we have assumed the minimum, mode and maximum

duration of recreational events by adults as 0.25, 0.5 and 2 h,

respectivelydwith associated volumetric intake rates of 10, 50

and 100 mL. The latter rates were increased by 50% for chil-

dren and reduced by 80% for secondary contact or inhalation

(regardless of age). Ingestion per individual per exposure

event is the simple product of the ingestion rate and exposure

duration.

Given the limited number of samples for the pathogens

selected in this study, identification of their appropriate sta-

tistical distributions is challengingdmany distributions

Table 3 e Doseeresponse relationships used in this study.

Pathogen Representative Parameter val

Single-parameter model

Adenovirus Adenovirus 4 r ¼ 0.4172 (2)

Cryptosporidium C. parvum r ¼ 0.05 (14)

Giardia G. lamblia r ¼ 0.02 (35)

Two-parameter model

Enterovirusd Echovirus 12 a, b ¼ 0.401, 2

Norovirus Norwalk virus a, b ¼ 0.04, 0.0

Rotavirus Rotavirus (CJN strain) a, b ¼ 0.2531,

Salmonella Non-typhi Salmonella a, b ¼ 0.33, 13

a ID50 (for infection) values in this column (in parentheses) refer to aver

integer. For the single-parametermodel, Prinf ¼ 1� e�rd and so ID50¼�[n(½

infection given ingestion or inhalation of a single virion. For the two-param

ID50 z b(21/a � 1), where a and b are shape and scale parameters for r’s

applicable for norovirus in which case the ID50 must be obtained by find

Prinf ¼ 1� 1F1(a, aþ b, �d ) ¼½ (Teunis and Havelaar, 2000), where 1F1 is th

Note too that the Adenovirus doseeresponse is for respiratory effects on

others are all for ingestion leading to gastrointestinal effects.

b Based on the proportions of illness exhibited in the relevant trial, regar

2011).

c Mean value of r for the first two models in Exhibit N.20 in U.S. EPA (2005

C. parvum (Teunis, 2009). Soller et al. (2010a) used r ¼ 0.09 (so ID50 z 8) us

strain), as is appropriate for waters impacted by human sources.

d Haas et al. (1999) present a single-parameter model with k (¼1/r) ¼ 78.3

Akin (1981). Thesewere preliminary data for the full trial of 149 volunteers

doses reported by Akin (and by Schiff et al., 1984b) by a factor of 33, thus

different ID50 values for essentially the same trial.

e This QMRA has accounted for a proportion of the exposed population ex

the required secretor phenotype, Teunis et al., 2008). The trial was for No

f This formulation has ID50 ¼ 1003. Other formulations (which exclud

ID50z 20,000 (Haas et al., 1999). Support for a lower ID50 comes from outbr

2008).

would fail to be rejected in a traditional goodness-of-fit test

because the small sample size confers low statistical power on

these tests (McBride, 2005). Instead we used an empirical

“Hockey-stick” distribution, as described in this paper’s

Appendix A, for reasons explained therein. In particular, the

relatively small number of samples collected for each

discharge type poses difficulties for the selection of an

appropriate right-skewed distribution and also this distribu-

tion is “bounded above”, whereasmost standard right-skewed

distributions are not.

2.3.3. DoseeresponseA number of doseeresponse analyses have been reported,

mostly from clinical trials, and can be appliedwithin QMRA. In

general, the doseeresponse data are used to develop a

mathematical function of the likelihood of infection (and

sometimes illness) for given pathogen doses. In this study the

mathematical forms of the infection doseeresponse functions

have been restricted to only those that can be mathematically

ues (ID50)a Prill

b References/comment

50% Couch et al. (1966a,b, 1969),

interpreted by Haas et al. (1999)

50% U.S. EPA (2005) c

45% Rendtorff (1954), Rendtorff and

Holt (1954), interpreted by

Haas et al. (1999)

27.2 (1052) 50% Teunis et al. (1996)

55 (26) 60%e Teunis et al. (2008), with

zero aggregation parameter

0.4265 (6) 35% Ward et al. (1986), interpreted

by Gerba et al. (1996) and Haas

et al. (1999). Note that only

susceptible adults were

recruited for this trial.

9.9 (1003) 100% Rose and Gerba (1991) f

age doses, as administered in a clinical trial, rounded to the nearest

)/rz 0.693/r, where d denotes average dose and r is the probability of

eter approximate (beta-Poisson) model, Prinf z 1� (1 þ d/b)�a and so

beta distribution. Note that the beta-Poisson approximation is not

ing the value of the average dose d satisfying the exact relationship

e confluent hypergeometric function (Abramowitz and Stegun, 1972).

ly (the Couch et al. trial administered Adenovirus 4 by aerosols); the

dless of dose (following the approach of Soller et al., 2010a; Viau et al.,

), consistent with a meta-analysis of the five available trials that used

ing all six available trials (therefore including the “TU502” C. hominis

(so ID50 z 54), using clinical trial data for 60 volunteers presented by

subsequently presented by Schiff et al. (1984a) who alsomultiplied all

accounting for the emergence in the literature of apparently two very

hibiting complete immunity (29% of the clinical trial’s subjects lacked

rwalk virus (G1), assumed to also apply to other G1 and all G2 strains.

e S. Typhimurium and S. Enteritidis, see Coleman et al., 2004) have

eak data related to consumption of contaminated food (Bollaerts et al.,

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 7 5287

derived from fundamental principles (see Teunis et al., 1995;

Haas et al., 1999; McBride, 2005). Other more empirical ap-

proaches may be less generalizable and may not be satisfac-

tory. The development of these restrictedmathematical forms

assumes the following: (i) the clinical trial contains a number

of groups and only the average dose, d, received by each

member of a given group is known (to measure the concen-

tration in each subsample received by each individual is

impractical) and thus the doses received are generally

assumed to follow a Poisson distribution with knownmean d;

(ii) if, with probability r, only one pathogen survives the body’s

defences and reaches a site of the body where infection could

ensue, that is sufficient for that infection to occur (the “single

hit” hypothesis); and (iii) individuals’ susceptibilities and/or a

pathogen’s infectiousness may vary, in which case the single

parameter r is replaced by the more versatile two-parameter

beta distribution. The resulting mathematical functions are

generally well known: either the single-parameter exponen-

tial function (with parameter r) or the two-parameter “beta-

Poisson” function (with parameters a and b). These relation-

ships are described in Table 3 and are depicted in Fig. 1.

It is important to note that if QMRA calculations “expose”

only one person per exposure occasion, then that person be-

comes a representative of all persons exposed on that occa-

siondan “average person”dand so assumption (i) is necessary.

However, for very infectious pathogens (e.g., Adenovirus 4) the

cumulative frequencies of computed infections then become

extremely and implausibly jagged (McBride, 2005). Smooth cu-

mulative frequencies result only if multiple people are exposed

oneachoccasion (and indoing so theoverall infection riskmean

is preserved. In that case assumption (i) is not required (because

their individual computed doses are known) and this consider-

ably simplifies the derivation of doseeresponse curves. This is

Fig. 1 e Doseeresponse curves for infection for the six

Reference Pathogens, also showing the divergent

Norovirus curves when using average doses (1F1 case) or

individual doses (beta-binomial case).

especially important for Norovirus calculations because the

need to evaluate the troublesome confluent hypergeometric

function (seethefirst footnote toTable3)disappears. Inthatcase

infection probabilities are calculated from the beta-binomial

form Prinf ¼ 1 � B(a, b þ i)/B(a, b) (Haas, 2002; McBride, 2005),

where i is an individual’s dose and B is the standard beta func-

tion (Abramowitz and Stegun, 1972). This function is not avail-

able in Excel but it can be calculated from the logarithm of the

gamma function (which is available): B(alpha, beta) ¼EXP(GAMMALN(alpha)þGAMMALN(beta)�GAMMALN(alphaþbeta)). Both functions are displayed in Fig. 1. For the one-

parameter model, abandoning assumption (i) gives rise to the

simple binomial form: Prinf¼ 1� (1� r)i. These simple-binomial

and beta-binomial forms have been used in this work, using

parameter values given in Table 3. All QMRA scenarios reported

are based on exposing 100 people on 1000 independent

occasions.

Considerable care must be taken when using these func-

tions. In particular: (a) The doseeresponse formulation from

trials is always for infection as the endpoint but not always for

consequential illness, so some translation from predicted

cases of infection to cases of illness may be necessary, given

that illness does not always follow from infection. This is a

critical topic because, in commonwith outbreak data analysis

(e.g., Teunis et al., 2005), the epidemiological studies used to

develop recreational water quality criteriameasure the illness

endpoint (not infection); (b) The form of the “dose” used in a

clinical trial needs to be made consistent with the form used

to describe the dose ingested or inhaled. For example, the

relevant Rotavirus clinical trial (Ward et al., 1986) reported

dose as Focus Forming Units (FFU) and there may be multiple

virus particles for each FFU. This is the “harmonization” issue:

if the laboratory method used for pathogen enumeration in

the appropriate clinical trial differs from that used to assay

source-water pathogens, an investigation must be made to

ensure that the results are harmonized, via an adjustment

factor. The approach taken in this study for the seven path-

ogens is summarized in Table 4; (c) Only a limited number of

species or serovars have been examined in clinical trialsdfor

example, only Type 4 Adenovirus data are available, and this

type gives rise to respiratory and conjunctivitis symptoms

(D’Angelo et al., 1979; Mena and Gerba, 2008), whereas the

more commonly monitored Adenovirus 40/41 complex gives

rise to diarrheal illness via ingestion; (d) Trials are performed

on healthy adult volunteers, whereas children, the elderly and

immune-compromised citizens (who likely have different

doseeresponse effects) may be at elevated risk. Additionally,

there is some evidence that among adult groups there can be

differential immunity status (Lake et al., 2011). For example,

adults with regular exposure (e.g., local surfers) may have

acquired a higher immune status than the general adult

population; (e) Uncertainty in the doseeresponse equation

can be captured during the calculation process, in the form of

credible intervals or alternative parameter sets for the dos-

eeresponse curve’s equation.

2.3.4. Risk characterizationFrom repeated random sampling of exposure and pathogen

concentration distributions we obtained a distribution of

doses. Each dose is used in the doseeresponse curve to

Table 4 e Harmonization rules for pathogen detection methods used in the study and those used in clinical trials.

Pathogen This study’s method Clinical trial method Harmonization rule Rationale/Comment

Salmonella qPCR and cultivation CFU: McCullough and

Eisele (1951a,b,c);

Hornick et al. (1970)

Use cultivation results Harmonious with CFU

Cryptosporidium Microscopy IFA cell counts, qPCR:

Okhuysen et al. (1999,

2002); Chappell et al.

(2006)

Microscopy ¼ IFA Assume detects C. parvum and

C. hominis

Giardia Microscopy Microscopy: Rendtorff

(1954); Rendtorff and

Holt (1954)

Microscopy ¼ IFA

Adenovirus 40/41 qPCR (gc) for Adenovirus

A, B, C and 40/41

TCID50 viral particulates

for Adenovirus 4 inhaled

via aerosols; Couch et al.

(1966a,b, 1969)

1 TCID50 ¼ 700 genomes Genome/PFU z 1000 (raw primary

effluentdHe and Jiang, 2005).

1 TCID50 z 0.7 PFU (Dulbecco,

1988).

Enteroviruses qPCR PFU in cell line (Akin, 1981;

Schiff et al., 1984a)

1 PFU ¼ 773 RNA genomes Average genome/PFU based on

results for raw wastewaters and

artificially spiked surface waters

(Jonsson et al., 2009; Puig et al.,

1994)

Norovirus GII qPCR (for GI and GII). qPCR, Teunis et al. (2008) Divide the clinical trial

data by 18.5

Both use RT PCR but on different

genetic target sequences with

differences in critical threshold

standard curves

Rotavirus qPCR for human

rotaviral strains

Focus Forming Units (FFU),

Ward et al. (1986)

Genome: FFU z 1900 Average genome/PFU ¼ 629

(Jonsson et al., 2009; Puig et al.,

1994; de Roda Husman et al., 2009)

and data from

Payne et al. (2006): 1 PFU z 3 FFU.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 75288

calculate an infection probability, which is then translated to

probabilities of illness, as explained in Table 3. This infection

probability value can be read either from the curve itself (not

allowing for any uncertainty in that curve) or from estimated

uncertainty intervals around the curves (in which case a

“second-order” Monte Carlo analysis may be required, e.g.,

Wu and Tsang, 2004). While some of the latter approach has

been implemented in this study, herein we focus on the

former approachdwithout loss of generality. Our Monte Carlo

simulations exposed 100 people at a site on each of 1000

separate occasions, resulting in 105 iterations. The resulting

cumulative frequency distribution is summarized via the

IIRdIndividual’s Illness Riskddefined as the total number of

individual illnesses predicted divided by the total number of

potential exposures (105). For this 100 � 1000 strategy the IIR

(as a percentage) is numerically identical to the mean value of

the short-term predicted illness risk.

Table 5 e Median and maximum values for the Hockey-Stick ddischarge types.a

RESID COMML A

Salmonella (culture), MPN per 10 L (5, 500) (5, 100) (50,

Cryptosporidium (IFA), oocysts/10 L (5, 100) (5, 100) (10,

Giardia (IFA), cysts/10 L (20, 500) (2, 10) (10,

Adenovirus 40/41 (qPCR), genomes/mL (10, 1 � 103) (10, 500) (10,

Enterovirus (qPCR), genomes/mL (10, 100) (50, 500) (1, 1

Norovirus GII (qPCR), genomes/mL (10, 1 � 103) (50, 500) (100

Rotavirus (qPCR), genomes/mL (100, 2 � 103) (50, 5 � 103) (500

a All minima were set to zero.

3. Results

3.1. Risk profiles

The pathogen concentrations that formed the basis of the

QMRA calculations herein represent the first comprehensive

data set on the abundance and occurrence of Reference

Pathogens (as defined by the U.S. EPA) in wet-weather dis-

charges to inland waters in the United States. Measured

concentrations were used to guide the choice of appropriate

medians and maxima for the Hockey-stick distributions

(Table 5). “End-of-pipe" (no dilution and direct stormwater

contact) risk profiles, and IIR values for childrens’ recreational

water ingestion were constructed for the seven source types

and for six of the seven Reference Pathogens (Cryptosporidium

oocysts, Enterovirus, Giardia cysts, Norovirus GII, Rotavirus

istribution of pathogens obtained for seven freshwater

GRIC CSO MIXED FOREO URBAN

500) (1, 100) (100, 3 � 103) (5, 50) (1, 3 � 103)

500) (10, 1 � 103) (2, 10) (100, 5 � 103) (2, 10)

100) (50, 1 � 104) (2, 20) (10, 100) (1, 50)

1 � 103) (100, 1 � 104) (10, 500) (10, 1 � 103) (10, 100)

0) (1, 1 � 103) (10, 100) (10, 100) (50, 500)

, 2 � 104) (20, 2 � 103) (1, 100) (5, 500) (10, 100)

, 5 � 104) (10, 100) (20, 2 � 104) (20, 2 � 103) (100, 1 � 103)

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 7 5289

and Salmonella); see Fig. 2 (for the ingestion pathway for chil-

dren) and Fig. 3 (for the inhalation or secondary contact

pathways by either age group after a 30-fold dilution). These

two cases span a wide range of risks. Adenovirus was not

included in Fig. 2 because of the lack of doseeresponse re-

lationships for ingestion. These graphs show the cumulative

number of cases of illness out of the 100 people exposed on

any random occasion over 1000 independent occasions: as

Fig. 2 e Risk profiles for a child’s ingestion exposure at end-of-

cysts (G), Norovirus GII (N), Rotavirus (R) and Salmonella (S) for t

which is considered for inhalation only. Prominent IIR values (%

explained above, that number numerically corresponds to the

probability of illness given random exposure. For example,

consider the predictedNorovirus illness for childrens’ primary

exposure to agricultural (AGRIC) discharges. Then for 50% of

the time (i.e., 500 of the 1000 occasions) there will be no more

than 22 Norovirus illness cases among the 100 people exposed

to this undiluted wet-weather source, and at no time will

there be more than 35 cases.

pipe for Cryptosporidium oocysts (C), Enterovirus (E), Giardia

he seven discharge types: Note the absence of Adenovirus,

) are shown on the profiles.

Fig. 3 e Risk profiles for secondary exposure (children) and inhalation (any age) after a 30-fold dilution for Cryptosporidium

oocysts (C), Enterovirus (E), Giardia cysts (G), Norovirus GII (N), Rotavirus (R) and Salmonella (S) for the seven discharge types:

Note the inclusion of Adenovirus (for inhalation only). Prominent IIR values (%) are shown on the profiles.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 75290

Fig. 4 presents an example (synthesized) application of

QMRA for evenly-composited mixtures of pathogens found in

this study for four stormwater sources (excluding the dry-

weather urban runoff [URBAN] discharges): residential

(RESID), commercial (COMML), agricultural (AGRIC), and

forested open space (FOREO). These calculations move away

from “end-of-pipe” scenarios, which are likely to be highly

conservative, and instead synthesize a scenario for exposure

to a hypothetical receiving water by allowing a 30:1 dilution of

the source waters, representing the fact that some dilution

and inactivation of pathogens typically occurs between the

source waters and the exposure site. Three groups of swim-

mers were considered (primary contact by adults, primary

contact by children, secondary contact) for ingestion of any or

Fig. 4 e Hypothetical example of recreational risks at a site

impacted by discharge-of-concern, with four types of

stormwater discharges diluted 30:1.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 7 5291

all of the six Reference Pathogens presented in Fig. 2. Fig. 5

shows the same type of analysis, but with Norovirus

excluded (by setting its concentration to zero).

3.2. Sources of risk

Figs. 2e5 show that Norovirus is the most dominant predicted

health risk. Even after source concentrations have been

Fig. 5 e Hypothetical example of recreational risks at a site

impacted by discharge-of-concern, with four types of

stormwater discharges diluted 30:1, with Norovirus

excluded.

reduced 30 times, the predicted short-term Norovirus illness

rates still exceed the current (recently-revised) U.S. EPA

criteria numerical limit for “NGI” (gastrointestinal illness not

necessarily accompanied by fever, mean risk ¼ 3.6%, U.S. EPA

2012)dresults not shown. However that limit refers to a 30-

day mean risk and it is possible that water receiving these

discharges could comply with that mean risk, yet, on occa-

sion, exhibit rather higher short-term risks.

Rotavirus generally induced the second-highest incidence

of risk among the tested pathogens. Risks for the other four

other Reference Pathogens (Giardia, Cryptosporidium, Entero-

virus and Salmonella) are considerably lower. Risks for sec-

ondary contact/inhalation are considerably lower than for

primary contact.

Reduction in pathogen concentrations via mixing and

inactivation does not necessarily cause a significant risk

reduction, particularly with respect to Norovirus for some

source types. This occurswhen concentrations are sufficiently

high such that their reduction does not shift the dos-

eeresponse function response down appreciably (i.e., the

doseeresponse curve is “flat” at higher concentrations, in

which case a moderate decrease in ingested Norovirus con-

centration does not significantly reduce the risk, see Fig. 1).

Appreciable risks are also associated with Adenovirus for

inhalation or secondary exposures, although this result must

be considered to be precautionary because an inhalation

exposure pathway has been assumed for Adenovirus 40/41

(which cause gastrointestinal illness not respiratory illness).

Accounting for uncertainty in doseeresponse (results not

shown here) generally causes a decrease in predicted risks

(although not so in the case of Adenovirus). As such, ac-

counting for uncertainty can lead to a less conservative

(lower) estimate of risks.

4. Discussion

This study is the first worldwide to perform QMRA using

pathogens detected in discharges into flowing inland waters

(termed “discharge-based QMRA”) rather than relying on

measurements in receiving waters themselves. Hence path-

ogens were present at higher concentrations, increasing the

likelihood of their detection compared to the much more

dilute concentrations expected in receiving waters. With the

exception of URBAN sites, data were collected (i) from typical

discharge waters and (ii) during wet weather conditions.

Therefore, the QMRA risks derived here apply to short-term

conditions and not necessarily to longer periodsda full risk

assessment should arguably consider both short-term risks

(e.g., a day after significant rainfall) and long-term risks (over a

month or a whole bathing season). U.S. federal criteria typi-

cally apply over a month (U.S. EPA, 2012) and in New Zealand

over a whole bathing season (MfE/MoH, 2003). Accordingly, if

pathogens are transported to rural receiving waters during

rainfall events, then risks posed to swimmers in those waters

during dry-weather periods should generally be lower than

presented here. However, conditions in highly urbanized

catchments (our URBAN sites) may also exhibit elevated risk

in dry conditions.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 75292

Our results serve to indicate the relative importance of the

Reference Pathogens, and also to illuminate the risks that can

apply during or shortly after rainfall events. Wet weather

sources are typically not well-represented by epidemiological

studies, and pose a significant challenge for public agencies

with requirements to abate stormwater discharges and

comply with recreational water quality criteria. Therefore, in

this regard, QMRA may have an important role in criteria

implementation.

4.1. QMRA procedures

Some caveats must be made concerning QMRA procedures.

First, Adenovirus doseeresponse is for respiratory effects (via

inhalation of aerosols containing Adenovirus 4), so should not

be strictly applied to all adenoviruses. Second, careful

consideration is needed about the appropriate form of the

doseeresponse function. In this work we have paid particular

attention to the appropriate form of the single-parameter and

two-parameter cases. For the two-parameter case (e.g., for

Norovirus or Rotavirus) if we expose multiple individuals on

each exposure occasion, and if we calculate the dose received

by each individual, then the resulting functional form is exact

(the approximate beta-Poisson equation is not required) and is

much simpler to evaluate. Third, results show that children

may deserve special attention in QMRA analyses, even from

the point of view that their water ingestion or inhalation rates

may be higher than is the case for adults (Schets et al., 2011).

For some pathogens at least, children may also be more sus-

ceptible. For example, in a study of children’s consumption of

contaminated milk during farm visits, Teunis et al. (2005) re-

ported stronger responses to Campylobacter comparedwith the

standard infection doseeresponse function derived by

Medema et al. (1996) from clinical trials on healthy adults

(Black et al., 1988). Fourth, when presenting QMRA for use in

assessing human health impacts it will often be advisable to

consider how the uncertainty in doseeresponse curves can be

incorporated into the assessment (results not shown here).

Fifth, although this was a large study, the number of pathogen

concentration data for each discharge type is still rather small

and some of these data contained censored results (less than a

varying detection limit). In the face of that, a precautionary

approach has been taken when assigning the median values

of the pathogen distributions. In particular that resulted in a

higher assigned median rotavirus concentration for the

URBAN category than for a number of other discharge types,

and that has driven a substantial part of rotavirus illness risk

(halving the rotavirus median from 100 to 50 genomes per mL

caused the IIR shown on Fig. 2 to drop from 14.5% to about 9%).

Finally, translating the probability of infection into a proba-

bility of illness needs careful examination, particularly for

Norovirus, as discussed below.

4.2. Norovirus doseeresponse

Based on our QMRA analyses, Norovirus dominated risk pro-

files, driving a large proportion of the risk to recreational

users. Careful attention was given to the incorporation of the

Norovirus doseeresponse curve into our analyses, taking ac-

count of any new information (e.g., Seitz et al., 2011). However,

specific issues arise concerning Norovirus QMRA: aggregation,

illness response, genotype, and analytical method.

(i) Concerning aggregation, the first set of challenges used in

the clinical trial were prepared from a stock suspension in

which the virus particles were highly aggregated, but

administered in relatively low doses (Teunis et al., 2008).

A later part of the trial used an inoculum obtained from a

stool sample from a subject infected in the first part of the

trial. This inoculum was disaggregated but was adminis-

tered only at high doses. Accordingly, there is consider-

able uncertainty in the doseeresponse curve at low doses.

Furthermore the formulae presented by Teunis et al.

(2008) for the aggregation case (in terms of the Gauss

hypergeometric function, 2F1) indicate an ID50 value (for

infection) about two orders-of-magnitude higher than for

the non-aggregation case. Accordingly, Norovirus infec-

tion may be seriously over-estimated were there to be

significant aggregation in these source waters or during

downstream transport. In this study we have followed

others’ practice of ignoring aggregation (Soller et al.,

2010a; Schoen and Ashbolt, 2010; Viau et al., 2011).

(ii) Teunis et al. (2008) report a function for the probability of

illness, given that infection has occurred, predicting

remarkably low values of illness unless dose is very high

(that probability is given by 1 � (1 þ hd )�r, where, for the

non-aggregated case, h¼ 2.55 � 10�3 and r ¼ 0.086). At the

infection ID50 ¼ 26 virions (Table 3) this predicts the

probability of illness to be z0.006. In other words “the

interesting consequence is that low dose exposure may

cause infections with few symptomatic cases, whereas

high doses cause clusters of symptomatic cases” (pers.

comm., Dr P. Teunis, RIVM, The Netherlands). However in

this study we have followed the practice of other authors

(e.g., Schoen and Ashbolt, 2010) and used much higher

values of illness probabilitiesdderived from the trial data

but regardless of dose. Our finding that Norovirus is the

most dominant predicted health risk (as has also been

reported by Sinclair et al., 2009; Soller et al., 2010a) would

be diminished were we to have adopted the Teunis

formulation of the probability of illness.

(iii) TheNorovirus trial was for theNorwalk strain, which falls

into genogroup I. Genogroup II strains also can cause

illness via exposure to contaminated water (Matthews

et al., 2012), so QMRA is forced to consider all strains in

these genogroups to be similarly infectious.

(iv) Several analytical methods have been published for

Norovirus (Kageyama et al., 2003; Wolf et al., 2007, 2010;

Hewitt et al., 2011). For this study, Norovirus concentra-

tions were measured using an assay based on methods

published byWolf et al. (2007). Formost Norovirus assays,

including ours, validation tests have been limited to a

small set of known samples. Although Norovirus (G1 and

G2) is normally assumed to be specific to human fecal

sources, our study detected these viruses in forested open

space with no known human sources. A subset of positive

samples in this study have been sequenced and all were

confirmed to contain Norovirus geneticmaterial. As such,

additional work to validate the performance of Norovirus

assays in environmental waters is warranted.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 7 5293

4.3. Pathogen fate and transport

The synthesized QMRA results herein applied a single dilu-

tion/attenuation factor (30:1) to all pathogens. In practice,

different discharge types will be subject to varying down-

stream transport (e.g., some discharges are near the exposure

site while others are far away) and pathogens will each have

different characteristics for environmental decay/persistence.

In turn, to address these site-specific issues during discharge-

based QMRA, risk assessors will need to (i) consider applying

fate and transport models to account for spatial- and time-

dependent variables and parameters and (ii) carefully apply

available data regarding the decay and persistence ofmodeled

pathogens.

4.4. Using source identifiers for QMRA

While analytical methods are becoming more readily avail-

able and less expensive, monitoring a suite of pathogens re-

quires significant resources. Quantitative data regarding the

relative importance of animal versus human sources could be

a surrogate for pathogen data. Recent research shows that

contemporary assays of Bacteroidales can quantify the rela-

tive contributions of those sources (Wang et al., 2010). Work is

underway to relate Bacteroidales concentrations to pathogens

in discharges to assess their reliability in predicting human

pathogens in discharges and supporting QMRA efforts.

5. Conclusions

The approach developed here provides a method to estimate

public health risks emanating from recreational exposure at

sites downstream of multiple discharges. Discharge-based

QMRA is particularly valuable where: (i) receiving water

measurements are limited or unavailable and (ii) when path-

ogen concentrations are often ‘non-detect’ with limits of

detection that are higher than values relevant to public health.

We provide the basic tool set to implement site-specific

QMRA, but site-specific analyses regarding the fate and

transport of pathogens-of-concern will be needed. Conclu-

sions include the following:

� This study provides strong evidence that the emerging

QMRA discipline sheds light on the potential for human

health effects caused by combinations of pathogens from

human and animal sources, particularly during and shortly

after rainfall events.

� Under the assumptions made, the findings confirm that

Norovirus infection (and possibly illness) can be the pre-

dominant risk from exposure. However additional research

is needed regarding the illness response of this Norovirus

(cf. its infection response).

� Care needs to be exercised to ensure that laboratory

methods for dose measurements are harmonized between

(i) the method used in the clinical trial that led to the

development of the doseeresponse curve and (ii) the mod-

ern methods used to assay pathogens in discharges or

receiving waters.

Acknowledgments

Funding for this project was provided by grant PATH2R08 from

Water Environment Research Foundation under the Water-

borne Pathogens and Human Health Research Program. We

also thank all of the organizations and individuals that have

come forward to give their time and effort toward this project

as a way to improve the state of the science in water quality,

fecal pathogen pollution, and risk assessment issues. In

particular, the WERF Pathogen and Human Health Project

Subcommittee, Rhonda Kranz, the U.S. Environmental Pro-

tection Agency, Desmond Till, Christobel Ferguson and J.

Soller provided guidance to this project. Harris County Flood

Control District provided valuable resources enabling the

project team to include an additional geographic region. C.

Owen from AMEC provided critical support with sample

collection and data analysis efforts. Sample collection in

Texas was performed by staff at PBS&J, an Atkins company. D.

Wang, A. Adell, A. Schriewer, N. Chouicha, A. Melli, A. Kundu

and J. Buchino are gratefully acknowledged for their help in

data collection and analysis.

Appendix A. The Hockey-stick distribution

The statistical distributions of pathogen concentrations in

environmental waters and sources can be expected to be

strongly right-skewed. However, because the sample size

(number of data) for any discharge type is small (at most 16, in

the RESID category, see Table 2), choosing the underlying

distribution is problematicaldtraditional “goodness-of-fit”

tests have difficulty in rejecting any right-skewed skewed

distribution (lognormal, gamma, extreme-value,.). That is

because these tests have low statistical power for small

sample sizes. Recognizing this, an alternative approach has

been taken in this study: the empirical “Hockey-stick” distri-

bution (McBride, 2005).

The essential idea is to use two triangular distributions,

each abutting a right-angled trapezium, given estimates of the

minimum,median andmaximumof the resulting distribution

(i.e., the percentile values X0, X50 and X100), as depicted by the

solid line in Fig. A1. The left triangle’s abscissa extends from

the minimum to the median, and so in that range the density

is a linear function of the percentile value. But it would be folly

to use a single triangular distribution for the right tail: it would

not allow for the right skewness evident in many microbio-

logical datasets (and the median would only be preserved if

the maximum and the minimum were equidistant from the

median). A simple solution is to join the median and

maximum by a “Hockey-stick”dthe line BCD in Fig. A1dto

impart some right-skew (and allowing for themaximumbeing

further away from the median). To determine the required XP

percentile (corresponding to the unknown point C on that

Figure), one further piece of information is needed: the

percentile (P), with an as-yet-unknown value XP, at which the

trapezium and the triangle on its right are joined. That is the

position of the Hockey-stick’s heel on the X-axis, where the

shaft turns into the blade. For example, we could specify that

the heel start at the 95%ile.

wat e r r e s e a r c h 4 7 ( 2 0 1 3 ) 5 2 8 2e5 2 9 75294

To completely specify the distribution, the values of the

probability density ordinates h1 and h2 and the abscissa

percentile XP (see Fig. A1) must be derived, given that X0, X50

and X100 and P will have been stated. These quantities can be

derived from the constraint that the total area under the dis-

tribution should be unity. The algebra includes solving a

resulting quadratic, in the following sequence

h1 ¼ 1X50 � X0

XP ¼ 12

"X50 þ X100 þ 1

h1

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX100 � X50Þ2 þ 1

h21

þ X50ð2� 8qÞ þ X100ð2� 8sÞh1

s #

h2 ¼ 2qX100 � XP

where q ¼ 1 � p and s ¼ p � ½ (with p ¼ P/100, as defined in

Fig. A1). [Note that the h2 equation inMcBride (2005) has an X50

on the denominator, but that should be XP.] The values of X0,

X50, XP, X100, h1 and h2 define the Hockey-stick distribution

from which random samples may be drawn. Linear interpo-

lation between the distribution’s breakpoints is used. Note

that unlike many other pathogen concentration distributions

used in QMRA, the Hockey-stick has an upper bound, obtained

via expert judgment. This can be seen as advantage given that

high proportions of risk can be generated from statistical

sampling of the distribution’s tail.

Fig. A1 e The Hockey-stick empirical distribution.

Appendix B. Supplementary data

Supplementary data related to this article can be found at

http://dx.doi.org/10.1016/j.watres.2013.06.001.

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