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Characterization of drinking water treatment for virus risk assessment
water research 43 (2009) 395–404
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Characterization of drinking water treatment for virus risk assessment P.F.M. Teunisa,b,*, S.A. Rutjesa, T. Westrellc, A.M. de Roda Husmana a
National Institute for Public Health and the Environment, RIVM, Bilthoven, The Netherlands Emory University, Hubert Department of Global Health, Rollins School of Public Health, Atlanta, GA, USA c European Centre for Disease Prevention and Control (ECDC), Stockholm, Sweden b
article info
abstract
Article history:
Removal or inactivation of viruses in drinking water treatment processes can be quantified
Received 21 August 2008
by measuring the concentrations of viruses or virus indicators in water before and after
Received in revised form
treatment. Virus reduction is then calculated from the ratio of these concentrations. Most
20 October 2008
often only the average reduction is reported. That is not sufficient when treatment effi-
Accepted 22 October 2008
ciency must be characterized in quantitative risk assessment.
Published online 8 November 2008
We present three simple models allowing statistical analysis of series of counts before and after treatment: distribution of the ratio of concentrations, and distribution of the proba-
Keywords:
bility of passage for unpaired and paired water samples.
Drinking water treatment
Performance of these models is demonstrated for several processes (long and short term
Virus
storage, coagulation/filtration, coagulation/sedimentation, slow sand filtration, membrane
Log reduction
filtration, and ozone disinfection) using microbial indicator data from full-scale treatment
Heterogeneity
processes.
Uncertainty
All three models allow estimation of the variation in (log) reduction as well as its uncer-
Quantitative risk assessment
tainty; the results can be easily used in risk assessment. Although they have different characteristics and are present in vastly different concentrations, different viruses and/or bacteriophages appear to show similar reductions in a particular treatment process, allowing generalization of the reduction for each process type across virus groups. The processes characterized in this paper may be used as reference for waterborne virus risk assessment, to check against location specific data, and in case no such data are available, to use as defaults. ª 2008 Elsevier Ltd. All rights reserved.
1.
Introduction
Pristine sources for drinking water production have become preciously scarce, as a result of ever increasing contamination of the environment due to population growth and economic growth. At the same time, demand for potable water continues to increase. Instruments for safeguarding the
microbial quality of drinking water are necessary. All over the world, quantitative risk assessment is now utilized by policy makers and production companies to project expected health risks of existing or planned catchment, treatment, and distribution systems (WHO, 2004). Quantitative risk assessment for waterborne pathogens often identifies water treatment as the critical stage, with
* Corresponding author. National Institute of Public Health and the Environment, RIVM, P.O. Box 1, NL-3720 BA Bilthoven, the Netherlands. Tel.: þ31 30 274 2937; fax: þ31 30 274 4456. E-mail address: [email protected] (P.F.M. Teunis). 0043-1354/$ – see front matter ª 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2008.10.049
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respect to nominal risk as well as uncertainty (Teunis et al., 1997). Limited data exist on microbial reduction by drinking water treatment processes, and even less is known quantitatively about virus reduction. Human pathogenic viruses may be transmitted via drinking water causing diseases such as meningitis, gastroenteritis, or hepatitis (Amvrosieva et al., 2006). These enteroviruses, noroviruses and other viruses may be present in the source waters for drinking water production (Gilgen et al., 1997; Lodder and de Roda Husman, 2005). Treatment processes may reduce virus concentrations to different levels depending on properties of the virus and properties of the treatment process. Reduction is often estimated indirectly, using nominal or expected performance of processes under idealized conditions, for indicator organisms or non-microbial particles. Water utilities usually express the performance of their treatment plants with a single metric, the log reduction or decimal reduction. This number is calculated from microbial counts by taking the difference in 10-base logarithms of their (arithmetic) averages before and after treatment. Such a procedure is attractive because it is simple and produces an unbiased result (Teunis et al., 1999). However, hardly any real treatment process can be trusted to always perform the same, regardless of conditions like turbidity, temperature, or pH which may vary considerably even at a single site. Extreme cases may occur, where processes reach nominal (designed) performance almost always, but may break down to extremely poor performance in rare occasions. Such a pattern of variation can be shown to have a profound influence on the distribution of risk in an exposed population (Teunis et al., 1997; Teunis and Havelaar, 1999; Teunis et al., 2003). Therefore, characterization of pathogen reduction in treatment processes must include a description of the variation in (log) reduction as a probability distribution, for instance. Extending the methods in (Teunis et al., 1999) we present three methods for characterizing the log reduction of drinking water treatment: (1) estimation of the ratio of concentrations in treated and untreated water, (2) estimation of the probability of passage with unpaired samples, and (3) estimation of that same probability assuming the samples must be paired. These three methods are compared by applying them to several different full-scale processes, using virus data: indicator or human pathogenic viruses.
2.
Data
Only data from full-scale processes, part of existing drinking water production plants, have been used in this survey. Most data have been collected in the course of routine surveillance, to monitor the performance of treatment processes. Long term storage in three chain-linked reservoirs with an average retention time of 5 months. Counts of F-specific bacteriophages, enteroviruses and reoviruses in samples of river water at the entrance of a cascade of storage reservoirs and in water leaving the reservoirs have been taken at 12 successive monthly intervals (de Roda Husman and Ketelaars, 2004) (Appendix, Table 4). Short term storage of treated water with an average retention time of 2 weeks prior to distribution (Teunis et al., 1999).
Counts of enteroviruses and reoviruses before and after storage in 10 monthly samples are available (Appendix, Table 5). Slow sand filtration. A full-scale seeding study (Schijven et al., 2008) using MS2 bacteriophages under varying conditions, including a defective ‘‘Schmutzdecke’’ (experiment 1), data in Appendix Table 6. Coagulation/filtration from two separate treatment plants (locations a and b), both using coagulation followed by filtration (de Roda Husman et al., 2005). F-specific phages have been counted before and after treatment at both sites, in unbalanced samples (6 samples from treated water, 3 from untreated water at one site, 5 from treated water, 3 from untreated water at the other site, see Appendix, Table 7). Coagulation/sedimentation with counts of enteroviruses and reoviruses in 10 monthly samples before and after treatment (Teunis et al., 1999) (Appendix, Table 8). Membrane filtration counts of somatic coliphages and Fspecific phages in 6 samples before and after a membrane filtration process (see Appendix, Table 9). Ozone disinfection. Two sets of samples from disinfection processes using Ozone (locations 1 and 2) are available. Counts of somatic coliphages and F-specific phages from all these samples are available for analysis (Appendix, Tables 10 and 11). Viruses present in water samples of up to 1000 liters were concentrated by negative membrane filtration using a conventional absorption–elution method, followed by ultrafiltration (Rutjes et al., 2005). These ultrafiltered concentrates were analysed for the presence of bacteriophages and human pathogenic viruses. Somatic and F-specific bacteriophages have been enumerated according to ISO 10705-2 (2000) and ISO 10705-1 (1995) standards, respectively (Havelaar et al., 1995). Enteroviruses and reoviruses have been enumerated according to standard NEN-EN 14486 (2005) (Lodder and de Roda Husman, 2005).
3.
Statistical methods
3.1.
Ratio of pathogen concentrations
For the first class of models – estimating concentration ratio – variations in concentration in treated and untreated water are estimated separately, and the distribution of their ratio is calculated, not making any additional assumptions about the process. As a ratio may have any value mathematically, such a procedure does not exclude an apparent increase in pathogen concentration, instead of reduction.
3.1.1.
Poisson counts in a Gamma mixture
First we provide the negative binomial distribution as a Poisson–Gamma mixture, explicitly using the counted number of micro-organisms and the sample volume and the parameters of the Gamma distribution for their concentration. In a volume V from a well mixed suspension, with Poisson distributed counts the distribution of counts n is
pðnjcVÞ ¼
ðcVÞn ecV n!
(1)
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If the concentration c varies between samples, with Gamma pdf f ðcjr; lÞ ¼
lr r1 c=l c e GðrÞ
(2)
the marginal distribution for the counts can be written as gðn; Vjr; lÞ ¼
Vn lr Gðn þ rÞ Gðn þ rÞ ðlVÞn ¼ n!GðrÞ ð1=l þ VÞnþr GðrÞn! ð1 þ lVÞnþr
(3)
If we substitute 1/(1 þ lV) ¼ p, this may be rewritten into the familiar equation for the negative binomial distribution.
3.1.2.
Distribution of the ratio of two Gamma variables
If we have two sets of samples, one from water entering a treatment process, the other set from the treated water, we can use Eq. (3) to estimate Gamma parameters for the distributions of concentrations in raw and treated waters. When we have the two concentrations as independent Gamma distributed random variables, c1 and c2, with probability densities f1(c1jr1,l1) and f2(c2jr2,l2) the distribution of their ratio u ¼ c1/c2 is r1 1 l2 l2 u Gðr1 þ r2 Þ l1 l1 (4) hðujr1 ; l1 ; r2 ; l2 Þ ¼ ðr1 þr2 Þ Gðr1 ÞGðr2 Þ l2 1þ u l1 a Type 2 Beta distribution, also known as the F-distribution(Stuart and Ord, 1987). Here, r1 and r2 are the shape parameters of the two fitted negative binomial distributions, and the scale factor is the ratio of the scale parameters l2/l1. A transformation l1 w u¼ (5) l2 1w converts h(wjr1,r2) into a (Type 1) Beta distribution.
3.2.
Binomial models of pathogen removal
The second class of models – binomial probability of passage – has been reported before (Teunis et al., 1999), and is based on the idea that any single microbial particle has a certain probability of passing the barrier posed by the treatment process. If particles pass the process independently, in other words: if the probability of passage does not depend on the concentration of particles, this is a binomial process. We have described two alternatives which may be seen as extremes: paired and unpaired samples. With paired samples any pair of samples taken simultaneously is assumed to represent a single (binomial) reduction, randomly taken from a distribution that describes its variation (among different pairs of samples). For unpaired samples we use all samples to estimate the variation in concentrations before and after treatment, with a binomial probability linking out and input concentrations, completely neglecting the order of the samples.
3.2.1.
Input and output samples unpaired
Suppose, we have a data set consisting of m input samples, with counts n ¼ fn1 ; .; nm g in volumes V ¼ fV1 ; .; Vm g
and l output samples, with counts k ¼ fk1 ; .; kl g in volumes W ¼ fW1 ; .; Wl g If we assume that the distribution of organisms in a given sample is Poisson, with Gamma distributed concentration (among samples), then counts are negative binomially distributed, as shown previously (Section 3.1). The distribution of organisms in any single sample (volume Wi) after passage is also Poisson distributed, with parameter pcWi, where p is the probability of passage, scaling down the concentration to pc. If c is Gamma distributed with parameters l and r, pc is Gamma distributed as well with parameters l/p and r. If p varies randomly among samples, as a Beta distributed variable with parameters (a,b), the marginal distribution of the counts in the treated sample can be written as (Teunis et al., 1999) hðk; Wjl; r; a; bÞ ¼ ðlWÞk 2 F1 ðk
Gðr þ kÞ Gða þ bÞGða þ kÞ GðrÞk! GðaÞGðab þ kÞ
þ r; a þ k; a þ b þ k; lWÞ
(6)
in which 2F1($) is a Hypergeometric function (Abramowitz and Stegun, 1984). The likelihood function is the joint distribution of counts in raw and treated water m Y
Lðl; r; a; bjn; V; k; WÞ ¼
gðni ; Vi jl; rÞ
i¼1
l Y h kj ; Wj l; r; a; b
(7)
j¼1
Unequal numbers of samples before and after treatment (m and l, respectively) are allowed. Maximum likelihood estimates for all parameters (l, r, a, and b) can be found by numerical optimization.
3.2.2.
Input and output samples paired
Suppose, we have a data set consisting of m sample pairs with counts n ¼ fn1 ; .; nm g
in volumes V ¼ fV1 ; .; Vm g
of water before treatment and counts k ¼ fk1 ; .; km g
in volumes W ¼ fW1 ; .; Wm g
after treatment. The distribution of organisms in any single sample is assumed Poisson (before treatment parameter cV, after treatment pcW ). A mixture of the joint Poisson densities before and after treatment with a Gamma distributed concentration (parameters (r,l)) can be written as hðn; V; k; Wjp; l; rÞ ¼
1 l
Vn ðpWÞk lr Gðn þ k þ rÞ nþkþr n!k!GðrÞ þ V þ pW
(8)
With p Beta distributed (variable fraction, parameters (a,b)), the joint density becomes (Teunis et al., 1999)
hðn;V;k;Wjl;r;a;bÞ¼
Vn Wk lr
GðrþnþkÞ GðaþbÞGðaþkÞ
ðVþ1=lÞnþkþr GðrÞn!k! GðaÞGðaþbþkÞ W nþkþr;aþk;aþbþk; F 2 1 ð1=lÞþV
with 2F1($) defined as in Section 3.2.
(9)
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This distribution may be used to calculate a likelihood function Lðl; r; a; bjn; V; k; WÞ ¼
m Y
hðni ; Vi ; ki ; Wi jl; r; a; bÞ
(10)
i¼1
Maximum likelihood estimates for all parameters (l, r, a, and b) can be found by numerical optimization.
3.3. Parameter estimation and prediction of log reduction The likelihood functions corresponding to all of the above models can be used for statistical analysis. Maximum likelihood parameter estimates were obtained by (numerical) optimization of the likelihood function (Mathematica 6.0 Wolfram Research, 2008); likelihood ratio tests (95% confidence level) were used to assess goodness of fit and to test whether models could be fitted to joint data, for instance for different organisms in the same treatment process, or in similar treatment processes (Teunis et al., 1999). For uncertainty analysis we used a Bayesian approach, employing normal (uninformed) priors after parameter transformation (log transformation of r and l, a, and b) and Markov chain Monte Carlo, using adaptive rejection sampling (Metropolis–Hastings algorithm, Gilks et al., 1996). Posterior predictive intervals were obtained from Monte Carlo parameter samples by calculating quantiles of (Beta) probability densities as a function of the argument (ratio, or fractions of concentrations before and after treatment). As we usually are interested in log reduction and not in the fraction (or probability), we transformed its probability density to a log-scale. If f ðxjqÞ is the pdf of the fraction x (parameter vector q), the pdf of y ¼ 10 log x is f ð10y jqÞ10y logð10Þ
(11)
This is the distribution of the log reduction shown in the graphs in Section 4.
(to all tested viruses) in many of the studied processes. Joint models could not be found for the paired binomial model for coagulation/sedimentation, and the ratio distribution for ozone disinfection at locations 1 and 2. Likewise, generalization of model results across all separate slow sand filtration seeding experiments failed, with poor model fit to pooled data, even when experiment 1 (having poor removal compared to the rest) was omitted. Table 2 shows quantiles of the best fit (or posterior mode) distributions, illustrating the variation in performance estimated by the three models. Supplementary information on the uncertainty in estimated quantiles of log reduction for all processes is provided separately (see Appendix Tables 12 and 13). Models for joint data of a process are again designated ‘‘All’’. Figs. 1–5 show graphs of the reduction distributions, with 95% (posterior predictive) intervals. These graphs correspond to the joint estimates (rows marked ‘‘All’’) in Tables 1 and 2. Note that often the ratio model shows a greater range in log reductions, both in variation (the width of the curves) and uncertainty (the widths of the grey intervals). It is also apparent that the ratio model does not restrict the sign of the log reduction to negative values (here, negative numbers indicate reduction in virus concentrations). Compared to the unpaired binomial model, the paired analysis appears to sometimes produce wider uncertainty intervals (Fig. 2f and Fig. 4c). This is not only caused by ‘‘true’’ variation in reduction, but also because some unpaired observations had to be excluded so that these distributions are based on a smaller data set. An overview of estimated quantiles of log reductions (Fig. 6) shows a considerable range, with some processes (coagulation/filtration, membrane filtration, and ozone disinfection) achieving high reductions but with a large range of variation, while other processes (slow sand filtration and coagulation/sedimentation) combine a somewhat smaller log reduction with less variation.
5. 4.
Discussion
Results
Although all three models can provide reduction estimates for any data sets of counted samples in known volumes, their application does not always make sense. For long term storage, paired samples, for instance taken on the same day at in and output of the reservoirs, are not expected to show correlation. The paired binomial model has therefore not been applied to the long term storage data as this seems a rather implausible model for such a process. Table 1 shows maximum likelihood parameter values for the parameters of each of the models for pathogen reduction. For each process type, models were fitted to the data for each pathogen or indicator separately. Then a joint model was fitted, with a single set of parameter values for the reduction distribution (ratio or Beta), leaving all other parameters free (rows marked ‘‘All’’ in Table 1). The F-specific and somatic coliphages data for the two coagulation/filtration processes (a and b) did not indicate significantly different distributions for their reduction. Therefore they were treated as a single data set. It appears that the estimated reduction can be generalized
Water treatment processes may be characterized by their potential for reducing the numbers of pathogens in the processed water. As a design target, this is a theoretical quantity, possibly determined from an idealized process model. We are here concerned not with the potential pathogen reduction, but with the reduction in pathogen concentrations achieved by a process under field conditions. Ideally, we want to estimate the reduction in pathogen concentrations from measurements of pathogen concentrations, in water entering and leaving the process. If this is not feasible (as will often be the case) the second best alternative is the use of microbial indicators similar in morphology and physico-chemical properties to the pathogens they represent (Payment and Franco, 1993). Whether we are studying pathogens or indicators, microorganisms must be enumerated: microbes are discrete particles and their concentration is usually estimated by counting them in a sample of a specific volume (or detecting presence or absence in a dilution series but that is not considered here). Therefore analysis combines two stages: (1) estimating the concentration of micro-organisms from (a series of) counts in
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Table 1 – Optimum (posterior mode) parameters for the ratio (Type 2 Beta) distribution l2/l1, r1, and r2 (Section 3.1) or the binomial probability (Type 1 Beta) distribution a and b. Ratio ^l2 =^l1
Unpaired
b r1
b r2
b a
Paired b b
b a
b b
Storage (long term) F-specific phages Enterovirus Reovirus All
292.9 143.2 18.92 55.43
0.193 0.414 0.359 0.209
0.370 0.638 0.719 0.504
0.413 2.054 0.817 0.554
272.7 431.0 29.76 41.77
Storage (short term) Enterovirus Reovirus All
7.14 20.19 34.88
0.0061 0.233 0.176
0.790 1.278 1.019
0.029 0.388 0.269
66.29 48.31 52.66
0.0095 0.211 0.142
23.27 15.04 16.16
Slow sand filtration Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiments 2–4 All
0.851 149.2 57.29 3.15 103 37.20* 0.080*
20.36 5.800 7.031 5.635 0.894* 0.205*
73.41 73.00 58.60 11.14 24.84* 29.08*
34.88 7.66 103 2.70 103 2.02 104 893.7* 2.185*
22.03 2.18 3.00 0.91 0.782* 0.202*
47.26 4.16 103 1.44 103 3.88 103 832.2* 2.517*
Coagulation/filtration Somatic coliphages F-specific phages All
39.70 21.76 35.60
0.107 0.049 0.088
0.420 0.297 0.355
0.128 0.089 0.150
10.39 9.642 20.93
0.129 0.128 0.149
6.053 2.394 4.695
Coagulation/sedimentation Enterovirus Reovirus All
26.52 20.35 27.90
0.790 1.278 1.019
0.741 2.466 1.161
1.408 2.002 2.131
30.26 73.81 68.93
4.499 2.161 2.212*
83.88 82.24 60.30*
Membrane filtration Somatic coliphages F-specific phages All
1.12 1013 2.19 1013 1.17 1011
2.49 106 4.56 104 7.27 103
0.438 0.132 0.208
0.071 0.041 0.062
55.00 21.63 95.23
0.067 0.029 0.040
59.13 2.258 5.377
Ozone disinfection (location 1) Somatic coliphages 7.97 102 F-specific phages 1.64 102 All 8.28 102
0.0955 0.141 0.0955
0.778 0.727 0.762
0.120 0.089 0.118
5.90 102 79.91 3.10 102
0.0532 0.0283 0.055
33.35 10.94 45.78
Ozone disinfection (location 2) Somatic coliphages 4.98 103 F-specific phages 24.23 All 54.49
0.123 0.095 0.067
0.233 0.276 0.245
0.328 0.195 0.176
2.22 103 9.203 46.52
0.072 0.103 0.086
4.274 2.034 2.975
Ozone disinfection (locations 1 þ 2 pooled) All 63.97*
0.059*
0.347*
0.126
85.97
0.0580
3.250
16.94 4.26 5.83 3.59 0.900* 0.202*
These parameters define distributions describing the variation in (pathogen or indicator) reduction for several drinking water treatment processes. Pooled models (‘‘All’’) are all accepted, except the paired binomial model for coagulation/sedimentation (*) and the ratio distribution model for ozone disinfection at locations 1 and 2 pooled, which are not. Slow sand filtration data cannot be pooled at all.
(replicate) samples, before and after treatment, and (2) estimating the reduction from these concentrations. Water utilities traditionally report log reductions, using series of counts in standard samples from untreated and treated waters. Average log reductions may be calculated, for instance by taking the logarithm of the ratio of counts from pairs of samples and averaging these. As this does assign equal weights to each pair of counts, whether low or high numbers of particles are detected, such a procedure
produces an incorrect, biased estimate. An unbiased estimate is easily calculated by taking the ratio of the sums of counts before and after treatment (Teunis et al., 1999). The ‘‘decimal elimination capacity’’ (van Breemen et al., 1998) is calculated from this ratio, by taking its 10-base logarithm. Strictly speaking, decimal elimination capacity (DEC) is a misnomer since it does not measure the capacity or potential reduction, but the actual or realized average reduction.
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Table 2 – Quantiles of the 10 log reduction in various treatment processes, estimated by the three different models. Ratio
Paired
b 0:50 Q
b 0:95 Q
b 0:05 Q
b 0:50 Q
b 0:95 Q
8.46 5.03 4.71 7.49
3.24 2.50 1.83 2.67
0.11 0.59 0.00 0.12
5.71 3.06 3.09 4.06
3.24 2.40 1.76 2.18
2.21 1.95 1.07 1.32
214.53 7.03 8.94
50.16 2.72 3.26
4.34 0.94 1.09
46.80 5.17 6.73
12.41 2.55 2.99
2.78 1.48 1.62
Storage (long term) F-specific phages Enterovirus Reovirus All Storage (short term) Enterovirus Reovirus All
Unpaired
b 0:05 Q
Slow sand filtration Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiments 2–4 All
0.68 3.65 3.02 4.21 4.43* 6.88*
Coagulation/filtration Somatic coliphages F-specific phages All Coagulation/sedimentation Enterovirus Reovirus All Membrane filtration Somatic coliphages F-specific phages All Ozone disinfection: location somatic coliphages F-specific phages All
1
Ozone disinfection: location somatic coliphages F-specific phages All
2
0.49 3.30 2.70 3.81 3.19* 2.00*
0.32 3.01 2.43 3.44 2.50* 0.33*
13.08 26.77 15.44
3.71 6.31 4.07
0.10 þ0.21 þ0.18
2.92 2.66 2.77
1.38 1.65 1.52
11.80 7.55 9.38
b 0:50 Q
b 0:95 Q
138.72 7.53 10.56
33.33 2.77 3.52
3.95 1.15 1.31
0.49 3.29 2.69 3.79 3.18* 1.93*
0.36 2.98 2.42 3.45 2.51* 0.36*
11.34 15.72 10.19
3.55 4.55 3.52
1.15 1.26 1.40
11.05 10.67 9.58
þ0.21 0.85 0.35
2.31 2.32 2.23
1.45 1.65 1.59
0.94 1.21 1.16
1.72 2.29 2.14*
5.90 5.20 5.59
3.56 2.60 0.77
20.16 33.42 23.36
6.17 8.93 7.10
2.12 2.03 2.44
21.50 45.28 33.47
6.51 10.86 8.46
2.19 1.23 1.42
16.34 11.25 16.35
5.87 4.13 5.88
2.49 1.49 2.49
13.83 16.76 13.70
5.49 5.51 5.25
2.94 2.19 2.66
26.22 47.24 25.57
7.41 11.90 7.37
2.06 2.01 2.18
12.89 13.90 19.52
4.77 3.36 4.76
0.03 þ1.24 þ0.89
7.46 7.80 9.25
4.38 2.67 3.56
3.18 0.96 1.70
18.87 13.03 15.67
4.99 3.34 4.11
0.99 0.51 0.75
5.96*
0.45*
12.47
4.53
2.08
23.11
5.87
0.96
Ozone disinfection: locations 1 þ 2 pooled All 22.98*
0.65 3.70 3.03 4.25 4.41* 6.90*
b 0:05 Q
0.64 3.98 3.25 5.04 4.62* 6.95*
0.50 3.35 2.73 3.81 3.23* 2.00*
0.38 2.92 2.36 3.14 2.51* 0.42*
3.29 2.86 2.85
0.92 0.52 0.76
1.32 1.66 1.52*
0.00 1.23 1.10*
Symbol: (*) Model fit not acceptable (Table 1).
The low concentrations in post-treatment samples frequently cause zero counts, even for indicator organisms in large volume samples. Calculation of log reductions then becomes problematic, requiring ad hoc solutions like adding 1 to all counts. Even in case all post-treatment samples should be zero, the methods presented here still allow estimation of a minimum log reduction. Some of the distributions reported here approach such a case where only the minimum reduction is well defined (Fig. 4b, c and Fig. 5b, c). For estimation of the average infection risk we need the (arithmetic) average reduction, and not the average log
reduction, to estimate the average drinking water concentration (Haas, 1996). However, point estimates of risk are obsolete and should be avoided. For instance, if we want to know whether a source of drinking water complies with an annual infection risk of 1 in 10,000 a point estimate of risk is not much use. Any meaningful test of compliance includes a measure of confidence, indicating for instance with what level of probability the estimated risk is 1 in 10,000 or less. This requires knowledge of the variation (and uncertainty) at all stages in the risk chain (Teunis et al., 1997). Many of the (log) reduction distributions shown here appear to be skewed to the left, displaying a tail towards
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Long term storage
a
0.30
b
Ratio
0.8
Unpaired
0.6
0.20
f(y)
f(y)
0.25 0.15
0.4
0.10
0.2
0.05 -8
-6
-4
-2
-8
0
-6
-4
-2
0
y (10log reduction)
y (10log reduction)
Short term storage
d
e
Unpaired
0.3
0.6
0.2
0.4
Paired
0.4 0.3 0.2
0.2
0.1
0.6 0.5
f(y)
Ratio
f(y)
f(y)
c
0.4
0.1 -8
-6
-4
-2
0
-8
y (10log reduction)
-6
-4
-2
-8
0
-6
-4
-2
0
y (10log reduction)
y (10log reduction)
Fig. 1 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Top (long term storage): virus reduction by storage based on counted numbers in 13 monthly samples of F-specific bacteriophages, enteroviruses, and reoviruses (see Appendix, Table 4). Bottom (short term storage): virus reduction by storage of treated (see Table 5) river water based on counted numbers in 10 weekly samples of enteroviruses and reoviruses.
higher (log) reductions. This suggests that in these treatment processes, viruses may be removed very efficiently. However, most distributions also show that small log reductions are also likely.
Variation in the performance of drinking water treatment processes is an interesting finding, indicating that periods of normal operation (within design limits) may be interrupted by short periods of degraded performance (Teunis et al., 2003;
Slow sand filtration: experiment 1
a
Ratio
5
c
Unpaired
7 5
3
4
1
f(y)
4
3 2
2
-6
-4
-2
0
1 -8
y (10log reduction)
3 2
1 -8
Paired
6
4
f(y)
f(y)
5
b
-6
-4
-2
0
-8
y (10log reduction)
-6
-4
-2
0
y (10log reduction)
Slow sand filtration: experiment 4
e
Ratio
2.5 2.0
1.5
f(y)
f(y)
2.0
f
Unpaired
1.0 0.5
1.5
f(y)
d
1.0 0.5
-8
-6
-4
-2
y (10log reduction)
0
-8
-6
-4
-2
y (10log reduction)
0
1.4 1.2 1.0 0.8 0.6 0.4 0.2
Paired
-8
-6
-4
-2
0
y (10log reduction)
Fig. 2 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Bacteriophage removal by slow sand filtration (data in Appendix, Table 6). Two examples, of a failing (top) and properly operating process (bottom).
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Coagulation/filtration
b
Ratio
c
Unpaired
0.4
f(y)
0.15
f(y)
0.5
0.10 0.05
0.2
-6
-4
-2
0
-8
y (10log reduction)
Paired
0.2 0.1
0.1 -8
0.4 0.3
0.3
f(y)
a
-6
-4
-2
-8
0
y (10log reduction)
-6
-4
-2
0
y (10log reduction)
Coagulation/sedimentation
d 0.4
f(y)
f(y)
0.6
0.2 -8
-6
-4
-2
3.5 3.0 2.5 2.0 1.5 1.0 0.5
y (10log reduction)
Paired 1.5 1.0 0.5
-8
0
f
Unpaired
f(y)
e
Ratio
-6
-4
-2
0.0
0
y (10log reduction)
-8
-6
-4
-2
0
y (10log reduction)
Fig. 3 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Top (coagulation/ filtration): bacteriophage removal by coagulation/filtration treatment (data in Appendix, Table 7). Bottom (coagulation/ sedimentation): virus removal by coagulation/sedimentation treatment (data in Appendix, Table 8).
Westrell et al., 2003). Such bimodal performance, if it occurs, may have a profound effect on long term infection risks (Teunis and Havelaar, 1999). The objective of the present study was to characterize variable reduction in drinking water treatment with simple models, useful for straightforward quantitative risk assessment (Vose, 1996). Process models of water treatment, for instance describing coagulant trapping of virus particles, or colloid adsorption processes in soil, are eminently useful for understanding the mechanisms of reduction and guiding the design of processes. In particular when covariables controlling the efficient operation of the process are known a model that incorporates mechanistic aspects of the removal or inactivation processes may predict changes in virus reduction corresponding to those covariables. However, such models are not well equipped for predicting removal in a real-world situation. In such, usually ill-defined conditions, we have only partial knowledge: of the process parameters as well as the validity of the implicit assumptions about the structure of the
a
b
Ratio
c
Unpaired
0.20
f(y)
0.3
f(y)
Membrane filtration 0.25
0.2 0.1
0.15
f(y)
0.4
process model. There statistical models are useful, since in risk assessment we are primarily interested in heterogeneity and uncertainty: how does the (log) reduction vary, when repeatedly observed and, given the available observations, how uncertain is our estimate of the variable (log) reduction? When comparing reductions estimated by each of the used models, it should be noted that going from the ratio model via the binomial unpaired to the binomial paired model of virus reduction, model assumptions are increasingly restrictive: the ratio model does not restrain removal in any way; the binomial unpaired model assumes removal, an increase in virus concentration is not accounted for, and the binomial paired model additionally assumes strong correlation between sample pairs. Preferably, a process should be characterized by a set of quantiles of its reduction distribution, as shown in Table 2. Remarkably, most studied processes show similar reduction among the tested viruses, as judged by likelihood ratio test.
0.10 0.05
-8
-6
-4
-2
y (10log reduction)
0
-8
-6
-4
-2
y (10log reduction)
0
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
Paired
-8
-6
-4
-2
y (10log reduction)
Fig. 4 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Reduction of bacteriophages by membrane filtration (data in Appendix, Table 9).
0
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b
Ozone disinfection
c
0.14 Ratio
0.5
0.12 0.10 0.08 0.06 0.04 0.02 0.00
0.4
0.20
0.3
0.15
0.25
Unpaired
f(y)
f(y)
f(y)
a
0.2
-6
-4
-2
0
0.10 0.05
0.1 -8
Paired
0.0
-8
y (10log reduction)
-6
-4
-2
0
0.00
y (10log reduction)
-8
-6
-4
-2
0
y (10log reduction)
Fig. 5 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Distribution of inactivation of somatic coliphages and F-specific phages by ozone disinfection (data in Appendix Tables, 10 and 11).
Physico-chemical properties of some of the viruses may be similar, some of the processes may not depend strongly on those properties, but we must also keep in mind that we have small data sets, often with many zero counts in treated waters, therefore statistical power is low, sometimes not allowing much more than establishing a minimum log reduction. In cases where there are sufficient observations, the estimated median reductions for the three models are also often not very much different. From these results, it is obvious that in many cases the average (log) reduction provides little information on the performance of a treatment process. For waterborne virus risk assessment the variation in reduction should be characterized by determining quantiles or, preferably, by defining its probability distribution. The present study provides such characteristics for a choice of important drinking water treatment processes. These results may be used for reference in waterborne virus risk assessment, either to compare with location specific data or in the absence of location specific information, as default distributions. In view of the importance of the variation in reduction for risk estimates, it would be useful to define some measure of
reliability for treatment processes, possibly based on the variance of the distribution of their (log) reduction. Especially when there is not a single process but a chain of several processes (some in parallel, some in series), knowledge of their variation would allow to not only predict the increase in average reduction by multiple barriers, but, more importantly, the gain in reliability. Such an approach to multiple barriers would allow the assessment of reliability in a risk framework (Teunis et al., 2003; Haas and Trussell, 1998).
6.
Conclusions
We have shown that it is possible to infer distributions for the (log) reduction based on limited observations. We have also demonstrated that, given the three applied models, different viruses (or virus indicators) are reduced similarly in the same treatment process. Although the parametric models developed here may appear complicated, the resulting statistics are very simple indeed, and application of the basic (Beta) distributions in risk calculations is straightforward.
0
-5
-10
-15
-20
LTS
STS
SSF1
SSF2
SSF3
SSF4
C/F
C/S
MF
Ozone
Fig. 6 – Summary. Quantiles of estimated log reduction of the three models (concentration ratio, binomial unpaired, and binomial paired, respectively) for long term storage (LTS); short term storage (STS); slow sand filtration (SSF1–SSF4); coagulation–filtration (C/F); coagulation–sedimentation (C/S); membrane filtration (MF); and ozone disinfection (Oz). Boxes indicate 90% range of log reduction (with median), whiskers indicate uncertainty as upper 95% level of Q0.95 and lower 5% level of Q0.05.
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The examples shown here provide information on virus reduction in several categories of drinking water treatment, and may be used as a reference against which new data on similar processes may be compared. Parameter estimates as Monte Carlo samples may be requested (by e-mail) from the corresponding author.
Acknowledgements We thank Dr Henk Ketelaars (WBB) for making available data on virus and indicators in long term storage reservoirs. Ria de Bruyn, Willemijn Lodder, Harold van den Berg and Jack Schijven (all RIVM) have provided expert assistance with enumeration of viruses and bacteriophages.
Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.watres.2008.10.049.
references
Abramowitz, M., Stegun, I.A., 1984. Pocketbook of Mathematical Functions. Verlag Harri Deutsch, Thun Frankfurt/Main. Amvrosieva, T.V., Paklonskaya, N.V., Biazruchka, A.A., Kazinetz, O.N., Bohush, Z.F., Fisenko, E.G., 2006. Enteroviral infection outbreak in the Republic of Belarus: principal characteristics and phylogenetic analysis of etiological agents. Central European Journal of Public Health 14 (2), 67–73. de Roda Husman, A.M., Ketelaars, H., 2004. Humane virussen in de maas. Report 330 200 001/2004, RIVM, Bilthoven, The Netherlands (in Dutch). de Roda Husman, A.M., Penders, E.J.M., Krom, A.P., Bakker, G.L., Hooogenboezem, W., 2005. Viruses in the Rhine and source waters for drinking water production. Report isbn: 90-6683-000-x, RIWA. Gilgen, M., Germann, D., Luthy, J., Hubner, P., 1997. Three-step isolation method for sensitive detection of enterovirus, rotavirus, hepatitis A virus, and small round structured viruses in water samples. International Journal of Food Microbiology 37 (2–3), 189–199. Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (Eds.), 1996. Markov Chain Monte Carlo in Practice. Chapman and Hall, London. Haas, C.N., 1996. How to average microbial densities to characterize risk. Water Research 30 (4), 1036–1038. Haas, C.N., Trussell, R.R., 1998. Frameworks for assessing reliability of multiple, independent barriers in potable water reuse. Water Science and Technology 38 (6), 1–8. Havelaar, A.H., van Olphen, M., Schijven, Y.C., 1995. Removal and inactivation of viruses by drinking water treatment processes
under full scale conditions. Water Science and Technology 31 (5–6), 55–62. Lodder, W.J., de Roda Husman, A.M., 2005. Presence of noroviruses and other enteric viruses in sewage and surface waters in the Netherlands. Applied and Environmental Microbiology 71 (3), 1453–1461. Payment, P., Franco, E., 1993. Clostridium perfringens and somatic coliphages as indicators of the efficiency of drinking water treatment for viruses and protozoan cysts. Applied and Environmental Microbiology 59 (8), 2418–2424. Rutjes, S.A., Italiaander, R., van den Berg, H.H., Lodder, W.J., de Roda Husman, A.M., 2005. Isolation and detection of enterovirus RNA from large-volume water samples by using the Nuclisens miniMAG system and real-time nucleic acid sequence-based amplification. Applied and Environmental Microbiology 71 (7), 3734–3740. Schijven, J.F., Colin, M., Dullemont, Y., Hijnen, W.A.M., MagicKnezev, A., Oorthuizen, W., Rutjes, S.A., de Roda Husman, A. M., 2008. Verwijdering van micro-organismen door langzame zandfiltratie (removal of microorganisms by slow sand filtration). Technical Report 330204001, RIVM; Waternet; Kiwa Research; het Waterlaboratorium; Duinwaterbedrijf ZuidHolland, Bilthoven (in Dutch). Stuart, A., Ord, J.K., 1987. Kendall’s Advanced Theory of Statistics 1: Distribution Theory, fifth ed. Charles Griffin and Co., London. Teunis, P.F.M., Davison, A., Deere, D., 2003. Short term fluctuations in drinking water quality and their significance for public health. WHO Report, World Health Organization, Geneva. Prepared for revision of Guidelines for Drinking Water Quality. Teunis, P.F.M., Evers, E.G., Slob, W., 1999. Analysis of variable fractions resulting from microbial counts. Quantitative Microbiology 1, 63–88. Teunis, P.F.M., Havelaar, A.H., 1999. Cryptosporidium in drinking water: evaluation of the ILSI/RSI quantitative risk assessment framework. Technical Report 284 550 006, RIVM, Bilthoven, The Netherlands. Teunis, P.F.M., Medema, G.J., Kruidenier, L., Havelaar, A.H., 1997. Assessment of the risk of infection by Cryptosporidium or Giardia in drinking water from a surface water source. Water Research 31 (6), 1333–1346. van Breemen, L.W.C.A., Ketelaars, H.A.M., Hoogenboezem, W., Medema, G.J., 1998. Storage reservoirs – a first barrier for pathogenic micro-organisms in the Netherlands. Water Science and Technology 37 (2), 253–260. Vose, D., 1996. Quantitative Risk Analysis. A guide to Monte Carlo Modelling. John Wiley & Sons, Chichester. Westrell, T., Bergstedt, O., Stenstro¨m, T.-A., Ashbolt, N.J., 2003. A theoretical approach to assess microbial risks due to failures in drinking water systems. International Journal of Environmental Health 13, 181–197. WHO, 2004. Guidelines for Drinking Water Quality. World Health Organization, Geneva. http://www.who.int/water_sanitation_ health/dwq. Wolfram Research, 2008. Mathematica Version 6. Champaign, Illinois, USA; url: http://www.wolfram.com.
Available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/watres
Characterization of drinking water treatment for virus risk assessment P.F.M. Teunisa,b,*, S.A. Rutjesa, T. Westrellc, A.M. de Roda Husmana a
National Institute for Public Health and the Environment, RIVM, Bilthoven, The Netherlands Emory University, Hubert Department of Global Health, Rollins School of Public Health, Atlanta, GA, USA c European Centre for Disease Prevention and Control (ECDC), Stockholm, Sweden b
article info
abstract
Article history:
Removal or inactivation of viruses in drinking water treatment processes can be quantified
Received 21 August 2008
by measuring the concentrations of viruses or virus indicators in water before and after
Received in revised form
treatment. Virus reduction is then calculated from the ratio of these concentrations. Most
20 October 2008
often only the average reduction is reported. That is not sufficient when treatment effi-
Accepted 22 October 2008
ciency must be characterized in quantitative risk assessment.
Published online 8 November 2008
We present three simple models allowing statistical analysis of series of counts before and after treatment: distribution of the ratio of concentrations, and distribution of the proba-
Keywords:
bility of passage for unpaired and paired water samples.
Drinking water treatment
Performance of these models is demonstrated for several processes (long and short term
Virus
storage, coagulation/filtration, coagulation/sedimentation, slow sand filtration, membrane
Log reduction
filtration, and ozone disinfection) using microbial indicator data from full-scale treatment
Heterogeneity
processes.
Uncertainty
All three models allow estimation of the variation in (log) reduction as well as its uncer-
Quantitative risk assessment
tainty; the results can be easily used in risk assessment. Although they have different characteristics and are present in vastly different concentrations, different viruses and/or bacteriophages appear to show similar reductions in a particular treatment process, allowing generalization of the reduction for each process type across virus groups. The processes characterized in this paper may be used as reference for waterborne virus risk assessment, to check against location specific data, and in case no such data are available, to use as defaults. ª 2008 Elsevier Ltd. All rights reserved.
1.
Introduction
Pristine sources for drinking water production have become preciously scarce, as a result of ever increasing contamination of the environment due to population growth and economic growth. At the same time, demand for potable water continues to increase. Instruments for safeguarding the
microbial quality of drinking water are necessary. All over the world, quantitative risk assessment is now utilized by policy makers and production companies to project expected health risks of existing or planned catchment, treatment, and distribution systems (WHO, 2004). Quantitative risk assessment for waterborne pathogens often identifies water treatment as the critical stage, with
* Corresponding author. National Institute of Public Health and the Environment, RIVM, P.O. Box 1, NL-3720 BA Bilthoven, the Netherlands. Tel.: þ31 30 274 2937; fax: þ31 30 274 4456. E-mail address: [email protected] (P.F.M. Teunis). 0043-1354/$ – see front matter ª 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2008.10.049
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respect to nominal risk as well as uncertainty (Teunis et al., 1997). Limited data exist on microbial reduction by drinking water treatment processes, and even less is known quantitatively about virus reduction. Human pathogenic viruses may be transmitted via drinking water causing diseases such as meningitis, gastroenteritis, or hepatitis (Amvrosieva et al., 2006). These enteroviruses, noroviruses and other viruses may be present in the source waters for drinking water production (Gilgen et al., 1997; Lodder and de Roda Husman, 2005). Treatment processes may reduce virus concentrations to different levels depending on properties of the virus and properties of the treatment process. Reduction is often estimated indirectly, using nominal or expected performance of processes under idealized conditions, for indicator organisms or non-microbial particles. Water utilities usually express the performance of their treatment plants with a single metric, the log reduction or decimal reduction. This number is calculated from microbial counts by taking the difference in 10-base logarithms of their (arithmetic) averages before and after treatment. Such a procedure is attractive because it is simple and produces an unbiased result (Teunis et al., 1999). However, hardly any real treatment process can be trusted to always perform the same, regardless of conditions like turbidity, temperature, or pH which may vary considerably even at a single site. Extreme cases may occur, where processes reach nominal (designed) performance almost always, but may break down to extremely poor performance in rare occasions. Such a pattern of variation can be shown to have a profound influence on the distribution of risk in an exposed population (Teunis et al., 1997; Teunis and Havelaar, 1999; Teunis et al., 2003). Therefore, characterization of pathogen reduction in treatment processes must include a description of the variation in (log) reduction as a probability distribution, for instance. Extending the methods in (Teunis et al., 1999) we present three methods for characterizing the log reduction of drinking water treatment: (1) estimation of the ratio of concentrations in treated and untreated water, (2) estimation of the probability of passage with unpaired samples, and (3) estimation of that same probability assuming the samples must be paired. These three methods are compared by applying them to several different full-scale processes, using virus data: indicator or human pathogenic viruses.
2.
Data
Only data from full-scale processes, part of existing drinking water production plants, have been used in this survey. Most data have been collected in the course of routine surveillance, to monitor the performance of treatment processes. Long term storage in three chain-linked reservoirs with an average retention time of 5 months. Counts of F-specific bacteriophages, enteroviruses and reoviruses in samples of river water at the entrance of a cascade of storage reservoirs and in water leaving the reservoirs have been taken at 12 successive monthly intervals (de Roda Husman and Ketelaars, 2004) (Appendix, Table 4). Short term storage of treated water with an average retention time of 2 weeks prior to distribution (Teunis et al., 1999).
Counts of enteroviruses and reoviruses before and after storage in 10 monthly samples are available (Appendix, Table 5). Slow sand filtration. A full-scale seeding study (Schijven et al., 2008) using MS2 bacteriophages under varying conditions, including a defective ‘‘Schmutzdecke’’ (experiment 1), data in Appendix Table 6. Coagulation/filtration from two separate treatment plants (locations a and b), both using coagulation followed by filtration (de Roda Husman et al., 2005). F-specific phages have been counted before and after treatment at both sites, in unbalanced samples (6 samples from treated water, 3 from untreated water at one site, 5 from treated water, 3 from untreated water at the other site, see Appendix, Table 7). Coagulation/sedimentation with counts of enteroviruses and reoviruses in 10 monthly samples before and after treatment (Teunis et al., 1999) (Appendix, Table 8). Membrane filtration counts of somatic coliphages and Fspecific phages in 6 samples before and after a membrane filtration process (see Appendix, Table 9). Ozone disinfection. Two sets of samples from disinfection processes using Ozone (locations 1 and 2) are available. Counts of somatic coliphages and F-specific phages from all these samples are available for analysis (Appendix, Tables 10 and 11). Viruses present in water samples of up to 1000 liters were concentrated by negative membrane filtration using a conventional absorption–elution method, followed by ultrafiltration (Rutjes et al., 2005). These ultrafiltered concentrates were analysed for the presence of bacteriophages and human pathogenic viruses. Somatic and F-specific bacteriophages have been enumerated according to ISO 10705-2 (2000) and ISO 10705-1 (1995) standards, respectively (Havelaar et al., 1995). Enteroviruses and reoviruses have been enumerated according to standard NEN-EN 14486 (2005) (Lodder and de Roda Husman, 2005).
3.
Statistical methods
3.1.
Ratio of pathogen concentrations
For the first class of models – estimating concentration ratio – variations in concentration in treated and untreated water are estimated separately, and the distribution of their ratio is calculated, not making any additional assumptions about the process. As a ratio may have any value mathematically, such a procedure does not exclude an apparent increase in pathogen concentration, instead of reduction.
3.1.1.
Poisson counts in a Gamma mixture
First we provide the negative binomial distribution as a Poisson–Gamma mixture, explicitly using the counted number of micro-organisms and the sample volume and the parameters of the Gamma distribution for their concentration. In a volume V from a well mixed suspension, with Poisson distributed counts the distribution of counts n is
pðnjcVÞ ¼
ðcVÞn ecV n!
(1)
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If the concentration c varies between samples, with Gamma pdf f ðcjr; lÞ ¼
lr r1 c=l c e GðrÞ
(2)
the marginal distribution for the counts can be written as gðn; Vjr; lÞ ¼
Vn lr Gðn þ rÞ Gðn þ rÞ ðlVÞn ¼ n!GðrÞ ð1=l þ VÞnþr GðrÞn! ð1 þ lVÞnþr
(3)
If we substitute 1/(1 þ lV) ¼ p, this may be rewritten into the familiar equation for the negative binomial distribution.
3.1.2.
Distribution of the ratio of two Gamma variables
If we have two sets of samples, one from water entering a treatment process, the other set from the treated water, we can use Eq. (3) to estimate Gamma parameters for the distributions of concentrations in raw and treated waters. When we have the two concentrations as independent Gamma distributed random variables, c1 and c2, with probability densities f1(c1jr1,l1) and f2(c2jr2,l2) the distribution of their ratio u ¼ c1/c2 is r1 1 l2 l2 u Gðr1 þ r2 Þ l1 l1 (4) hðujr1 ; l1 ; r2 ; l2 Þ ¼ ðr1 þr2 Þ Gðr1 ÞGðr2 Þ l2 1þ u l1 a Type 2 Beta distribution, also known as the F-distribution(Stuart and Ord, 1987). Here, r1 and r2 are the shape parameters of the two fitted negative binomial distributions, and the scale factor is the ratio of the scale parameters l2/l1. A transformation l1 w u¼ (5) l2 1w converts h(wjr1,r2) into a (Type 1) Beta distribution.
3.2.
Binomial models of pathogen removal
The second class of models – binomial probability of passage – has been reported before (Teunis et al., 1999), and is based on the idea that any single microbial particle has a certain probability of passing the barrier posed by the treatment process. If particles pass the process independently, in other words: if the probability of passage does not depend on the concentration of particles, this is a binomial process. We have described two alternatives which may be seen as extremes: paired and unpaired samples. With paired samples any pair of samples taken simultaneously is assumed to represent a single (binomial) reduction, randomly taken from a distribution that describes its variation (among different pairs of samples). For unpaired samples we use all samples to estimate the variation in concentrations before and after treatment, with a binomial probability linking out and input concentrations, completely neglecting the order of the samples.
3.2.1.
Input and output samples unpaired
Suppose, we have a data set consisting of m input samples, with counts n ¼ fn1 ; .; nm g in volumes V ¼ fV1 ; .; Vm g
and l output samples, with counts k ¼ fk1 ; .; kl g in volumes W ¼ fW1 ; .; Wl g If we assume that the distribution of organisms in a given sample is Poisson, with Gamma distributed concentration (among samples), then counts are negative binomially distributed, as shown previously (Section 3.1). The distribution of organisms in any single sample (volume Wi) after passage is also Poisson distributed, with parameter pcWi, where p is the probability of passage, scaling down the concentration to pc. If c is Gamma distributed with parameters l and r, pc is Gamma distributed as well with parameters l/p and r. If p varies randomly among samples, as a Beta distributed variable with parameters (a,b), the marginal distribution of the counts in the treated sample can be written as (Teunis et al., 1999) hðk; Wjl; r; a; bÞ ¼ ðlWÞk 2 F1 ðk
Gðr þ kÞ Gða þ bÞGða þ kÞ GðrÞk! GðaÞGðab þ kÞ
þ r; a þ k; a þ b þ k; lWÞ
(6)
in which 2F1($) is a Hypergeometric function (Abramowitz and Stegun, 1984). The likelihood function is the joint distribution of counts in raw and treated water m Y
Lðl; r; a; bjn; V; k; WÞ ¼
gðni ; Vi jl; rÞ
i¼1
l Y h kj ; Wj l; r; a; b
(7)
j¼1
Unequal numbers of samples before and after treatment (m and l, respectively) are allowed. Maximum likelihood estimates for all parameters (l, r, a, and b) can be found by numerical optimization.
3.2.2.
Input and output samples paired
Suppose, we have a data set consisting of m sample pairs with counts n ¼ fn1 ; .; nm g
in volumes V ¼ fV1 ; .; Vm g
of water before treatment and counts k ¼ fk1 ; .; km g
in volumes W ¼ fW1 ; .; Wm g
after treatment. The distribution of organisms in any single sample is assumed Poisson (before treatment parameter cV, after treatment pcW ). A mixture of the joint Poisson densities before and after treatment with a Gamma distributed concentration (parameters (r,l)) can be written as hðn; V; k; Wjp; l; rÞ ¼
1 l
Vn ðpWÞk lr Gðn þ k þ rÞ nþkþr n!k!GðrÞ þ V þ pW
(8)
With p Beta distributed (variable fraction, parameters (a,b)), the joint density becomes (Teunis et al., 1999)
hðn;V;k;Wjl;r;a;bÞ¼
Vn Wk lr
GðrþnþkÞ GðaþbÞGðaþkÞ
ðVþ1=lÞnþkþr GðrÞn!k! GðaÞGðaþbþkÞ W nþkþr;aþk;aþbþk; F 2 1 ð1=lÞþV
with 2F1($) defined as in Section 3.2.
(9)
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This distribution may be used to calculate a likelihood function Lðl; r; a; bjn; V; k; WÞ ¼
m Y
hðni ; Vi ; ki ; Wi jl; r; a; bÞ
(10)
i¼1
Maximum likelihood estimates for all parameters (l, r, a, and b) can be found by numerical optimization.
3.3. Parameter estimation and prediction of log reduction The likelihood functions corresponding to all of the above models can be used for statistical analysis. Maximum likelihood parameter estimates were obtained by (numerical) optimization of the likelihood function (Mathematica 6.0 Wolfram Research, 2008); likelihood ratio tests (95% confidence level) were used to assess goodness of fit and to test whether models could be fitted to joint data, for instance for different organisms in the same treatment process, or in similar treatment processes (Teunis et al., 1999). For uncertainty analysis we used a Bayesian approach, employing normal (uninformed) priors after parameter transformation (log transformation of r and l, a, and b) and Markov chain Monte Carlo, using adaptive rejection sampling (Metropolis–Hastings algorithm, Gilks et al., 1996). Posterior predictive intervals were obtained from Monte Carlo parameter samples by calculating quantiles of (Beta) probability densities as a function of the argument (ratio, or fractions of concentrations before and after treatment). As we usually are interested in log reduction and not in the fraction (or probability), we transformed its probability density to a log-scale. If f ðxjqÞ is the pdf of the fraction x (parameter vector q), the pdf of y ¼ 10 log x is f ð10y jqÞ10y logð10Þ
(11)
This is the distribution of the log reduction shown in the graphs in Section 4.
(to all tested viruses) in many of the studied processes. Joint models could not be found for the paired binomial model for coagulation/sedimentation, and the ratio distribution for ozone disinfection at locations 1 and 2. Likewise, generalization of model results across all separate slow sand filtration seeding experiments failed, with poor model fit to pooled data, even when experiment 1 (having poor removal compared to the rest) was omitted. Table 2 shows quantiles of the best fit (or posterior mode) distributions, illustrating the variation in performance estimated by the three models. Supplementary information on the uncertainty in estimated quantiles of log reduction for all processes is provided separately (see Appendix Tables 12 and 13). Models for joint data of a process are again designated ‘‘All’’. Figs. 1–5 show graphs of the reduction distributions, with 95% (posterior predictive) intervals. These graphs correspond to the joint estimates (rows marked ‘‘All’’) in Tables 1 and 2. Note that often the ratio model shows a greater range in log reductions, both in variation (the width of the curves) and uncertainty (the widths of the grey intervals). It is also apparent that the ratio model does not restrict the sign of the log reduction to negative values (here, negative numbers indicate reduction in virus concentrations). Compared to the unpaired binomial model, the paired analysis appears to sometimes produce wider uncertainty intervals (Fig. 2f and Fig. 4c). This is not only caused by ‘‘true’’ variation in reduction, but also because some unpaired observations had to be excluded so that these distributions are based on a smaller data set. An overview of estimated quantiles of log reductions (Fig. 6) shows a considerable range, with some processes (coagulation/filtration, membrane filtration, and ozone disinfection) achieving high reductions but with a large range of variation, while other processes (slow sand filtration and coagulation/sedimentation) combine a somewhat smaller log reduction with less variation.
5. 4.
Discussion
Results
Although all three models can provide reduction estimates for any data sets of counted samples in known volumes, their application does not always make sense. For long term storage, paired samples, for instance taken on the same day at in and output of the reservoirs, are not expected to show correlation. The paired binomial model has therefore not been applied to the long term storage data as this seems a rather implausible model for such a process. Table 1 shows maximum likelihood parameter values for the parameters of each of the models for pathogen reduction. For each process type, models were fitted to the data for each pathogen or indicator separately. Then a joint model was fitted, with a single set of parameter values for the reduction distribution (ratio or Beta), leaving all other parameters free (rows marked ‘‘All’’ in Table 1). The F-specific and somatic coliphages data for the two coagulation/filtration processes (a and b) did not indicate significantly different distributions for their reduction. Therefore they were treated as a single data set. It appears that the estimated reduction can be generalized
Water treatment processes may be characterized by their potential for reducing the numbers of pathogens in the processed water. As a design target, this is a theoretical quantity, possibly determined from an idealized process model. We are here concerned not with the potential pathogen reduction, but with the reduction in pathogen concentrations achieved by a process under field conditions. Ideally, we want to estimate the reduction in pathogen concentrations from measurements of pathogen concentrations, in water entering and leaving the process. If this is not feasible (as will often be the case) the second best alternative is the use of microbial indicators similar in morphology and physico-chemical properties to the pathogens they represent (Payment and Franco, 1993). Whether we are studying pathogens or indicators, microorganisms must be enumerated: microbes are discrete particles and their concentration is usually estimated by counting them in a sample of a specific volume (or detecting presence or absence in a dilution series but that is not considered here). Therefore analysis combines two stages: (1) estimating the concentration of micro-organisms from (a series of) counts in
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Table 1 – Optimum (posterior mode) parameters for the ratio (Type 2 Beta) distribution l2/l1, r1, and r2 (Section 3.1) or the binomial probability (Type 1 Beta) distribution a and b. Ratio ^l2 =^l1
Unpaired
b r1
b r2
b a
Paired b b
b a
b b
Storage (long term) F-specific phages Enterovirus Reovirus All
292.9 143.2 18.92 55.43
0.193 0.414 0.359 0.209
0.370 0.638 0.719 0.504
0.413 2.054 0.817 0.554
272.7 431.0 29.76 41.77
Storage (short term) Enterovirus Reovirus All
7.14 20.19 34.88
0.0061 0.233 0.176
0.790 1.278 1.019
0.029 0.388 0.269
66.29 48.31 52.66
0.0095 0.211 0.142
23.27 15.04 16.16
Slow sand filtration Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiments 2–4 All
0.851 149.2 57.29 3.15 103 37.20* 0.080*
20.36 5.800 7.031 5.635 0.894* 0.205*
73.41 73.00 58.60 11.14 24.84* 29.08*
34.88 7.66 103 2.70 103 2.02 104 893.7* 2.185*
22.03 2.18 3.00 0.91 0.782* 0.202*
47.26 4.16 103 1.44 103 3.88 103 832.2* 2.517*
Coagulation/filtration Somatic coliphages F-specific phages All
39.70 21.76 35.60
0.107 0.049 0.088
0.420 0.297 0.355
0.128 0.089 0.150
10.39 9.642 20.93
0.129 0.128 0.149
6.053 2.394 4.695
Coagulation/sedimentation Enterovirus Reovirus All
26.52 20.35 27.90
0.790 1.278 1.019
0.741 2.466 1.161
1.408 2.002 2.131
30.26 73.81 68.93
4.499 2.161 2.212*
83.88 82.24 60.30*
Membrane filtration Somatic coliphages F-specific phages All
1.12 1013 2.19 1013 1.17 1011
2.49 106 4.56 104 7.27 103
0.438 0.132 0.208
0.071 0.041 0.062
55.00 21.63 95.23
0.067 0.029 0.040
59.13 2.258 5.377
Ozone disinfection (location 1) Somatic coliphages 7.97 102 F-specific phages 1.64 102 All 8.28 102
0.0955 0.141 0.0955
0.778 0.727 0.762
0.120 0.089 0.118
5.90 102 79.91 3.10 102
0.0532 0.0283 0.055
33.35 10.94 45.78
Ozone disinfection (location 2) Somatic coliphages 4.98 103 F-specific phages 24.23 All 54.49
0.123 0.095 0.067
0.233 0.276 0.245
0.328 0.195 0.176
2.22 103 9.203 46.52
0.072 0.103 0.086
4.274 2.034 2.975
Ozone disinfection (locations 1 þ 2 pooled) All 63.97*
0.059*
0.347*
0.126
85.97
0.0580
3.250
16.94 4.26 5.83 3.59 0.900* 0.202*
These parameters define distributions describing the variation in (pathogen or indicator) reduction for several drinking water treatment processes. Pooled models (‘‘All’’) are all accepted, except the paired binomial model for coagulation/sedimentation (*) and the ratio distribution model for ozone disinfection at locations 1 and 2 pooled, which are not. Slow sand filtration data cannot be pooled at all.
(replicate) samples, before and after treatment, and (2) estimating the reduction from these concentrations. Water utilities traditionally report log reductions, using series of counts in standard samples from untreated and treated waters. Average log reductions may be calculated, for instance by taking the logarithm of the ratio of counts from pairs of samples and averaging these. As this does assign equal weights to each pair of counts, whether low or high numbers of particles are detected, such a procedure
produces an incorrect, biased estimate. An unbiased estimate is easily calculated by taking the ratio of the sums of counts before and after treatment (Teunis et al., 1999). The ‘‘decimal elimination capacity’’ (van Breemen et al., 1998) is calculated from this ratio, by taking its 10-base logarithm. Strictly speaking, decimal elimination capacity (DEC) is a misnomer since it does not measure the capacity or potential reduction, but the actual or realized average reduction.
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Table 2 – Quantiles of the 10 log reduction in various treatment processes, estimated by the three different models. Ratio
Paired
b 0:50 Q
b 0:95 Q
b 0:05 Q
b 0:50 Q
b 0:95 Q
8.46 5.03 4.71 7.49
3.24 2.50 1.83 2.67
0.11 0.59 0.00 0.12
5.71 3.06 3.09 4.06
3.24 2.40 1.76 2.18
2.21 1.95 1.07 1.32
214.53 7.03 8.94
50.16 2.72 3.26
4.34 0.94 1.09
46.80 5.17 6.73
12.41 2.55 2.99
2.78 1.48 1.62
Storage (long term) F-specific phages Enterovirus Reovirus All Storage (short term) Enterovirus Reovirus All
Unpaired
b 0:05 Q
Slow sand filtration Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiments 2–4 All
0.68 3.65 3.02 4.21 4.43* 6.88*
Coagulation/filtration Somatic coliphages F-specific phages All Coagulation/sedimentation Enterovirus Reovirus All Membrane filtration Somatic coliphages F-specific phages All Ozone disinfection: location somatic coliphages F-specific phages All
1
Ozone disinfection: location somatic coliphages F-specific phages All
2
0.49 3.30 2.70 3.81 3.19* 2.00*
0.32 3.01 2.43 3.44 2.50* 0.33*
13.08 26.77 15.44
3.71 6.31 4.07
0.10 þ0.21 þ0.18
2.92 2.66 2.77
1.38 1.65 1.52
11.80 7.55 9.38
b 0:50 Q
b 0:95 Q
138.72 7.53 10.56
33.33 2.77 3.52
3.95 1.15 1.31
0.49 3.29 2.69 3.79 3.18* 1.93*
0.36 2.98 2.42 3.45 2.51* 0.36*
11.34 15.72 10.19
3.55 4.55 3.52
1.15 1.26 1.40
11.05 10.67 9.58
þ0.21 0.85 0.35
2.31 2.32 2.23
1.45 1.65 1.59
0.94 1.21 1.16
1.72 2.29 2.14*
5.90 5.20 5.59
3.56 2.60 0.77
20.16 33.42 23.36
6.17 8.93 7.10
2.12 2.03 2.44
21.50 45.28 33.47
6.51 10.86 8.46
2.19 1.23 1.42
16.34 11.25 16.35
5.87 4.13 5.88
2.49 1.49 2.49
13.83 16.76 13.70
5.49 5.51 5.25
2.94 2.19 2.66
26.22 47.24 25.57
7.41 11.90 7.37
2.06 2.01 2.18
12.89 13.90 19.52
4.77 3.36 4.76
0.03 þ1.24 þ0.89
7.46 7.80 9.25
4.38 2.67 3.56
3.18 0.96 1.70
18.87 13.03 15.67
4.99 3.34 4.11
0.99 0.51 0.75
5.96*
0.45*
12.47
4.53
2.08
23.11
5.87
0.96
Ozone disinfection: locations 1 þ 2 pooled All 22.98*
0.65 3.70 3.03 4.25 4.41* 6.90*
b 0:05 Q
0.64 3.98 3.25 5.04 4.62* 6.95*
0.50 3.35 2.73 3.81 3.23* 2.00*
0.38 2.92 2.36 3.14 2.51* 0.42*
3.29 2.86 2.85
0.92 0.52 0.76
1.32 1.66 1.52*
0.00 1.23 1.10*
Symbol: (*) Model fit not acceptable (Table 1).
The low concentrations in post-treatment samples frequently cause zero counts, even for indicator organisms in large volume samples. Calculation of log reductions then becomes problematic, requiring ad hoc solutions like adding 1 to all counts. Even in case all post-treatment samples should be zero, the methods presented here still allow estimation of a minimum log reduction. Some of the distributions reported here approach such a case where only the minimum reduction is well defined (Fig. 4b, c and Fig. 5b, c). For estimation of the average infection risk we need the (arithmetic) average reduction, and not the average log
reduction, to estimate the average drinking water concentration (Haas, 1996). However, point estimates of risk are obsolete and should be avoided. For instance, if we want to know whether a source of drinking water complies with an annual infection risk of 1 in 10,000 a point estimate of risk is not much use. Any meaningful test of compliance includes a measure of confidence, indicating for instance with what level of probability the estimated risk is 1 in 10,000 or less. This requires knowledge of the variation (and uncertainty) at all stages in the risk chain (Teunis et al., 1997). Many of the (log) reduction distributions shown here appear to be skewed to the left, displaying a tail towards
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Long term storage
a
0.30
b
Ratio
0.8
Unpaired
0.6
0.20
f(y)
f(y)
0.25 0.15
0.4
0.10
0.2
0.05 -8
-6
-4
-2
-8
0
-6
-4
-2
0
y (10log reduction)
y (10log reduction)
Short term storage
d
e
Unpaired
0.3
0.6
0.2
0.4
Paired
0.4 0.3 0.2
0.2
0.1
0.6 0.5
f(y)
Ratio
f(y)
f(y)
c
0.4
0.1 -8
-6
-4
-2
0
-8
y (10log reduction)
-6
-4
-2
-8
0
-6
-4
-2
0
y (10log reduction)
y (10log reduction)
Fig. 1 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Top (long term storage): virus reduction by storage based on counted numbers in 13 monthly samples of F-specific bacteriophages, enteroviruses, and reoviruses (see Appendix, Table 4). Bottom (short term storage): virus reduction by storage of treated (see Table 5) river water based on counted numbers in 10 weekly samples of enteroviruses and reoviruses.
higher (log) reductions. This suggests that in these treatment processes, viruses may be removed very efficiently. However, most distributions also show that small log reductions are also likely.
Variation in the performance of drinking water treatment processes is an interesting finding, indicating that periods of normal operation (within design limits) may be interrupted by short periods of degraded performance (Teunis et al., 2003;
Slow sand filtration: experiment 1
a
Ratio
5
c
Unpaired
7 5
3
4
1
f(y)
4
3 2
2
-6
-4
-2
0
1 -8
y (10log reduction)
3 2
1 -8
Paired
6
4
f(y)
f(y)
5
b
-6
-4
-2
0
-8
y (10log reduction)
-6
-4
-2
0
y (10log reduction)
Slow sand filtration: experiment 4
e
Ratio
2.5 2.0
1.5
f(y)
f(y)
2.0
f
Unpaired
1.0 0.5
1.5
f(y)
d
1.0 0.5
-8
-6
-4
-2
y (10log reduction)
0
-8
-6
-4
-2
y (10log reduction)
0
1.4 1.2 1.0 0.8 0.6 0.4 0.2
Paired
-8
-6
-4
-2
0
y (10log reduction)
Fig. 2 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Bacteriophage removal by slow sand filtration (data in Appendix, Table 6). Two examples, of a failing (top) and properly operating process (bottom).
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Coagulation/filtration
b
Ratio
c
Unpaired
0.4
f(y)
0.15
f(y)
0.5
0.10 0.05
0.2
-6
-4
-2
0
-8
y (10log reduction)
Paired
0.2 0.1
0.1 -8
0.4 0.3
0.3
f(y)
a
-6
-4
-2
-8
0
y (10log reduction)
-6
-4
-2
0
y (10log reduction)
Coagulation/sedimentation
d 0.4
f(y)
f(y)
0.6
0.2 -8
-6
-4
-2
3.5 3.0 2.5 2.0 1.5 1.0 0.5
y (10log reduction)
Paired 1.5 1.0 0.5
-8
0
f
Unpaired
f(y)
e
Ratio
-6
-4
-2
0.0
0
y (10log reduction)
-8
-6
-4
-2
0
y (10log reduction)
Fig. 3 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Top (coagulation/ filtration): bacteriophage removal by coagulation/filtration treatment (data in Appendix, Table 7). Bottom (coagulation/ sedimentation): virus removal by coagulation/sedimentation treatment (data in Appendix, Table 8).
Westrell et al., 2003). Such bimodal performance, if it occurs, may have a profound effect on long term infection risks (Teunis and Havelaar, 1999). The objective of the present study was to characterize variable reduction in drinking water treatment with simple models, useful for straightforward quantitative risk assessment (Vose, 1996). Process models of water treatment, for instance describing coagulant trapping of virus particles, or colloid adsorption processes in soil, are eminently useful for understanding the mechanisms of reduction and guiding the design of processes. In particular when covariables controlling the efficient operation of the process are known a model that incorporates mechanistic aspects of the removal or inactivation processes may predict changes in virus reduction corresponding to those covariables. However, such models are not well equipped for predicting removal in a real-world situation. In such, usually ill-defined conditions, we have only partial knowledge: of the process parameters as well as the validity of the implicit assumptions about the structure of the
a
b
Ratio
c
Unpaired
0.20
f(y)
0.3
f(y)
Membrane filtration 0.25
0.2 0.1
0.15
f(y)
0.4
process model. There statistical models are useful, since in risk assessment we are primarily interested in heterogeneity and uncertainty: how does the (log) reduction vary, when repeatedly observed and, given the available observations, how uncertain is our estimate of the variable (log) reduction? When comparing reductions estimated by each of the used models, it should be noted that going from the ratio model via the binomial unpaired to the binomial paired model of virus reduction, model assumptions are increasingly restrictive: the ratio model does not restrain removal in any way; the binomial unpaired model assumes removal, an increase in virus concentration is not accounted for, and the binomial paired model additionally assumes strong correlation between sample pairs. Preferably, a process should be characterized by a set of quantiles of its reduction distribution, as shown in Table 2. Remarkably, most studied processes show similar reduction among the tested viruses, as judged by likelihood ratio test.
0.10 0.05
-8
-6
-4
-2
y (10log reduction)
0
-8
-6
-4
-2
y (10log reduction)
0
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
Paired
-8
-6
-4
-2
y (10log reduction)
Fig. 4 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Reduction of bacteriophages by membrane filtration (data in Appendix, Table 9).
0
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b
Ozone disinfection
c
0.14 Ratio
0.5
0.12 0.10 0.08 0.06 0.04 0.02 0.00
0.4
0.20
0.3
0.15
0.25
Unpaired
f(y)
f(y)
f(y)
a
0.2
-6
-4
-2
0
0.10 0.05
0.1 -8
Paired
0.0
-8
y (10log reduction)
-6
-4
-2
0
0.00
y (10log reduction)
-8
-6
-4
-2
0
y (10log reduction)
Fig. 5 – Best fit and 95% predictive intervals of probability density functions ( f( y)) of virus log reduction y. Distribution of inactivation of somatic coliphages and F-specific phages by ozone disinfection (data in Appendix Tables, 10 and 11).
Physico-chemical properties of some of the viruses may be similar, some of the processes may not depend strongly on those properties, but we must also keep in mind that we have small data sets, often with many zero counts in treated waters, therefore statistical power is low, sometimes not allowing much more than establishing a minimum log reduction. In cases where there are sufficient observations, the estimated median reductions for the three models are also often not very much different. From these results, it is obvious that in many cases the average (log) reduction provides little information on the performance of a treatment process. For waterborne virus risk assessment the variation in reduction should be characterized by determining quantiles or, preferably, by defining its probability distribution. The present study provides such characteristics for a choice of important drinking water treatment processes. These results may be used for reference in waterborne virus risk assessment, either to compare with location specific data or in the absence of location specific information, as default distributions. In view of the importance of the variation in reduction for risk estimates, it would be useful to define some measure of
reliability for treatment processes, possibly based on the variance of the distribution of their (log) reduction. Especially when there is not a single process but a chain of several processes (some in parallel, some in series), knowledge of their variation would allow to not only predict the increase in average reduction by multiple barriers, but, more importantly, the gain in reliability. Such an approach to multiple barriers would allow the assessment of reliability in a risk framework (Teunis et al., 2003; Haas and Trussell, 1998).
6.
Conclusions
We have shown that it is possible to infer distributions for the (log) reduction based on limited observations. We have also demonstrated that, given the three applied models, different viruses (or virus indicators) are reduced similarly in the same treatment process. Although the parametric models developed here may appear complicated, the resulting statistics are very simple indeed, and application of the basic (Beta) distributions in risk calculations is straightforward.
0
-5
-10
-15
-20
LTS
STS
SSF1
SSF2
SSF3
SSF4
C/F
C/S
MF
Ozone
Fig. 6 – Summary. Quantiles of estimated log reduction of the three models (concentration ratio, binomial unpaired, and binomial paired, respectively) for long term storage (LTS); short term storage (STS); slow sand filtration (SSF1–SSF4); coagulation–filtration (C/F); coagulation–sedimentation (C/S); membrane filtration (MF); and ozone disinfection (Oz). Boxes indicate 90% range of log reduction (with median), whiskers indicate uncertainty as upper 95% level of Q0.95 and lower 5% level of Q0.05.
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The examples shown here provide information on virus reduction in several categories of drinking water treatment, and may be used as a reference against which new data on similar processes may be compared. Parameter estimates as Monte Carlo samples may be requested (by e-mail) from the corresponding author.
Acknowledgements We thank Dr Henk Ketelaars (WBB) for making available data on virus and indicators in long term storage reservoirs. Ria de Bruyn, Willemijn Lodder, Harold van den Berg and Jack Schijven (all RIVM) have provided expert assistance with enumeration of viruses and bacteriophages.
Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.watres.2008.10.049.
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