Modeling of heterotrophic bacteria counts in a water distribution system

water research 43 (2009) 1075–1087

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Modeling of heterotrophic bacteria counts in a water distribution system Alex Francisquea, Manuel J. Rodriguezb,*, Luis F. Miranda-Morenoc, Rehan Sadiqd, Franc¸ois Proulxe a

Centre de Recherche en Ame´nagement et De´veloppement, Universite´ Laval, 1722 Pavillon Savard, Universite´ Laval, Que´bec City, Qc., Canada G1K 7P4 b´ Ecole Supe´rieure d’Ame´nagement du Territoire, Universite´ Laval, 1628 Pavillon Savard, Universite´ Laval, Que´bec City, Qc., Canada G1K 7P4 c Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montreal, Qc., Canada H3A 2K6 d National Research Council Canada, 1200 Montreal Road, Ottawa, ON, Canada K1A 0R6 e Ville de Que´bec, Service de l’environnement, 210, av. Saint-Sacrement, Que´bec City, Qc., Canada G1N 3X6

article info

abstract

Article history:

Heterotrophic plate count (HPC) constitutes a common indicator for monitoring of

Received 20 June 2008

microbiological water quality in distribution systems (DS). This paper aims to identify

Received in revised form

factors explaining the spatiotemporal distribution of heterotrophic bacteria and model

14 November 2008

their occurrence in the distribution system. The case under study is the DS of Quebec City,

Accepted 22 November 2008

Canada. The study is based on a robust database resulting from a sampling campaign

Published online 3 December 2008

carried out in about 50 DS locations, monitored bi-weekly over a three-year period. Models for explaining and predicting HPC levels were based on both one-level and multi-level

Keywords:

Poisson regression techniques. The latter take into account the nested structure of data,

Distribution systems

the possible spatiotemporal correlation among HPC observations and the fact that

Water quality monitoring

sampling points, months and/or distribution sub-systems may represent clusters. Models

Heterotrophic bacteria

show that the best predictors for spatiotemporal occurrence of HPC in the DS are: free

Multi-level Poisson regression

residual chlorine that has an inverse relation with the HPC levels, water temperature and

Modeling

water ultraviolet absorbance, both having a positive impact on HPC levels. A sensitivity analysis based on the best performing model (two-level model) allowed for the identification of seasonal-based strategies to reduce HPC levels. ª 2008 Elsevier Ltd. All rights reserved.

1.

Introduction

For monitoring of microbiological water quality in distribution systems (DS), heterotrophic plate count (HPC) is used as a typical indicator (Grabow, 1996; Edberg and Allen, 2004; Pavlov et al., 2004). HPC is used as a sentinel in the surveillance of water quality and is generally non-pathogenic bacteria that can be found in DS, in particular when pipes are dirty or when levels of residual disinfectant are insufficient

(Edberg et al., 1997; Rusin et al., 1997; Zhang and DiGiano, 2002; Berry et al., 2006). Growth of bacteria in DS depends on a number of physical, chemical and operational conditions (Zhang and DiGiano, 2002), as well as on seasonal fluctuations (Berry et al., 2006). According to literature the main factors contributing to this are free residual chlorine, the presence of easily assimilated organic carbon (AOC), water temperature, water pH, the nature of the pipes, presence of corrosion and shear in the biofilm–liquid interface (LeChevallier et al., 1990,

* Corresponding author. Tel.: þ1 418 656 2131x8933; fax: þ1 418 656 2018. E-mail address: [email protected] (M.J. Rodriguez). 0043-1354/$ – see front matter ª 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2008.11.030

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1993; Ollos, 1998; Zhang and DiGiano, 2002; Liu et al., 2002; Ndiongue et al., 2005). Some authors also mention the distance from the water treatment plant (i.e., a surrogate for residence time), conductivity and water turbidity (Carter et al., 2000; Momba et al., 2004). High HPC may indicate some failure in the treatment process, especially disinfection, and contamination events within the DS (Allen et al., 2004; Sartory, 2004). The spatial and temporal variability of HPC can indicate, in some cases, a non-specific contaminant intrusion in the DS. In other cases to detect intrusion, other parameters must be considered (such as total/faecal coliforms). In addition, HPC can then used as an early indicator of the deterioration of water quality (Sartory, 2004). In some DS, HPC may reach values greater than 10 000 cfu/ml (Allen et al., 2004). In conditions of high HPC in water, detection of coliforms in culture media with lactose may be inhibited (Geldreich et al., 1972; LeChevallier and McFeters, 1985; Allen et al., 2004; Reasoner, 2004). Largely for this reason, a threshold of 500 cfu/ml is frequently used as an operational criterion when monitoring HPC in North America (USEPA, 1989; Health Canada, 2004). In other countries (e.g., Japan and Germany), a count below 100 cfu/ml for HPC favors acceptable hygienic conditions in the distribution system (Pavlov et al., 2004; Hambsch et al., 2004). Regular monitoring of HPC provides useful information for assessing microbiological drinking water quality. However, surveillance of HPC at high frequency in several locations of the DS, common for other parameters, is expensive and timeconsuming. In addition, to insure control of water quality in the DS, proper understanding of operational and water quality parameters associated with HPC is required. Previous work in this area basically explored individual relationships between HPC and certain parameters (van der Kooij, 1992; Carter et al., 2000; Stine et al., 2005) using generally small databases and, in some cases, developed experimentally. It is important to investigate the simultaneous impact of operational and water quality parameters on HPC occurrence based on a large database collected in full DS and based on robust data analysis methodologies. The study is based on spatiotemporal information on HPC and other water quality and operational parameters generated during a long period in a large DS. The main objectives of this paper are to: i) identify factors empirically explaining the spatiotemporal distribution of HPC in DS and ii) develop and compare models that allow prediction of HPC occurrence.

2.

Methodology

2.1.

Case under study

The case under study is the main DS of Quebec City, Canada, serving nearly 240 000 people. Quebec City is located in a region that experiences major seasonal climate variations. For instance, summer is relatively short, relatively warm (water temperature may reach 25  C) and often wet. On the other hand, winter is very long and very cold (water temperature slightly higher than the freezing point during five months approximately). The treatment plant (TP) of the

DS under study draws raw water from the Saint-Charles River. This water undergoes a treatment process consisting of pre-chlorination followed by coagulation–flocculation– sedimentation, slow sand filtration, ozonation and finally post-chlorination (Rodriguez et al., 2007). Average yearly ozonation dose is about 1.75 mg/l O3, whereas post-chlorination dose is about 1.8 mg/l in average. The average yearly value of total organic carbon (TOC) in the treated water (before distribution) is about 1.5 mg/l. The DS is divided into four (4) sub-systems (DSS) or networks (QC1–QC4), each having different hydraulic characteristics in terms of pipe material, pipe age and pressures, and serving specific sectors of Quebec City. The majority of pipe sections in the DS are made of cast iron, polyvinyl chloride and ductile iron. In all provinces of Canada, surveillance of HPC in drinking water is not mandatory, as it is for faecal and total coliforms (MDDEPQ, 2001). Thus, very little information is available on HPC spatiotemporal variability in DS.

2.2.

Sampling and analytical methods

This study is based on an intensive sampling campaign carried out at several locations of the DS during 2003 (47 locations), 2004 (61 locations) and 2005 (56 locations). In most cases, locations were sampled every two weeks. This database is very rich spatially and temporally compared to those used in other studies. For instance, the database used by Zhang and DiGiano (2002) for the Raleigh DS serving over 250 000 people, contained a maximum of 140 samples. They were collected monthly on 10 sampling points during 14 months. During each campaign, samples were collected to measure HPC, total and faecal coliforms, free residual chlorine, water temperature, ultraviolet absorbance at 254 nm (UV-254 nm), pH, turbidity, true and apparent color and conductivity. UV-254 nm was measured as an indicator of organic matter in water. All samples were collected in commercial and public buildings after flushing water for about 5 min. Water temperature and free residual chlorine were measured in the field. The other parameters were measured at the water quality laboratory. For sampling of microbiological parameters, bottles sterilized with sodium thiosulfate were used. Samples were transported to the laboratory in iceboxes to maintain their temperature at 4  C. Free residual chlorine was measured using the colorimetric method (DPD) with a Hach Pocket colorimeter II. pH was measured based on standard method 4500-HþB (APHA et al., 1998) with an Orion SA720 potentiometer. Turbidity was measured based on method 2130B using a Hach 2100N turbidimeter. Conductivity measurement was based on method 2510B using an Orion 1230 conductivimeter. An Ultrospec Biochrome 1000 spectrophotometer with a 5-cm quartz cell was used to measure UV-254 nm. The same spectrophotometer was used to measure color. Analysis of HPC was based on method 9215B using an R2A culture medium with incubation time of 48 h at 35.0  0.5  C. These incubation conditions are those recommended by the Standard Methods for the Examination of Water and Wastewater (APHA et al., 2005) and other organizations and associations in Europe and North America (Standing Committee of Analysts, 2002; Health Canada, 2006).

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2.3.

Data analysis

The analysis of data resulting from the sampling campaign involves two stages. The first consists of a descriptive analysis based on observation of the HPC distribution according to various parameters. To a certain extent, results obtained are used to guide orientations of the second stage. The second stage consists in modeling the simultaneous impacts of diverse parameters on the levels of HPC using different types of models. These models are based on single-level or multilevel analysis (Cameron and Trivedi, 1998; Myers et al., 2002; Goldstein, 2003).

more than 500 1 to 100

a Distribution of HPC classes (%)

In addition, enumeration at 35  C during a period of 48 h seems the most convenient for identifying potentially pathogenic bacteria growth (Payment, 1999). For faecal coliforms, method 9222D with mFc-Agar culture medium was used (incubation time of 24 h at 44.5  0.2  C). Finally, for total coliforms, method 9222B was used (with m-endo Agar as the culture medium and an incubation time of 24 h at 35.0  0.2  C).

Sampling campaign and measurements resulted in a large database containing more than 3500 measurements of microbiological and physico-chemical parameters. Table 1 presents the distribution of data obtained for different parameters. Results in this table highlight the fact that the system under study distributes water of high quality. In fact, no faecal coliforms were detected during the period under study (likewise, very few cases of total coliforms presence). In addition, turbidity and color levels were very low (only 2.6% of samples with turbidity higher than 1 NTU and only 0.4% with color higher than 15 UAC). The occurrence of HPC was relatively low in comparison to levels reported in other studies (Delahaye et al., 2003; Hambsch et al., 2004), although high variability was observed.

3.1. Relationships between HPC levels and monitoring parameters For analysis purposes, the distribution of HPC was grouped into four categories: (HPC ¼ 0, 1–100; 101–500 and higher than

n =622

n =644

n =1787

n =243

0.1 -0.3

0.3 -0.5

0.5 - 1.0

>1.0

90 80 70 60 50 40 30 20 10 <0.1

Free residual chlorine (mg/L) more than 500 1 to 100

Distribution of HPC classes (%)

Results and discussion

n =299

0

b 3.

100

101 to 500 0

100

101 to 500 0

n =858

n =1103

n =1126

n =508

<4

4.1 - 11

11.1 - 18

>18

90 80 70 60 50 40 30 20 10 0

Temperature (°C) Fig. 1 – a. Effect of free residual chlorine on HPC levels. b. Effect of water temperature on HPC levels.

500 cfu/ml). In Fig. 1a, it is interesting to note that when levels of free residual chlorine were below 0.3 mg/l (a standard target at the exit of the water treatment plant (USEPA, 1989)), HPC more frequently exceeded the threshold of 100 and

Table 1 – Statistical distribution of water quality parameters measured in the DS (2003–2005). Parameter Free residual chlorine (mg/l) Temperature ( C) UV-254 nm (m1) pH Turbidity (NTU) Color (UAC) Faecal coliform (cfu/100 ml) Total coliform (cfu/100 ml) HPC (cfu/ml)

n

Mean value

Standard deviation

Median

Mode

3595 3595 3595 3595 3595 3595 3592 3591 3595

0.50 10.7 2.74 7.60 0.33 4.54 0 0.01 13.8

0.313 6.30 0.888 0.313 0.30 2.58 0 0.705 81.2

0.50 10.0 2.60 7.60 0.30 4 0 0 0

0.60 4.0 2.48 7.50 0.20 4 0 0 0

5th Percentile

Cfu: colony forming units; NTU: nephelometric turbidity unit; UAC: unity of apparent color,
95th Percentile 1.0 20.1 4.20 8.10 0.73 9 0 0 40

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500 cfu/ml. Residual chlorine levels above 0.3 mg/l favored levels of HPC lower than 100 cfu/ml in more than 99% of samples. Moreover, under these conditions, no HPC was detected in more than 70% of samples. These results confirm observations made by other authors concerning the importance of maintaining relatively high levels of residual disinfectant in the DS (Zhang and DiGiano, 2002; Huck and Gagnon, 2004; Ndiongue et al., 2005). Fig. 1b indicates that under winter conditions (water temperature  4  C), HPC was not detected at all in more than 75% of collected samples. When water temperatures were lower than 11  C, the number of samples with levels of HPC higher than 100 cfu/ml was very limited: only 10 samples out of 1961 (<0.6%). It is also noteworthy that the proportion of cases with HPC levels exceeding 100 cfu/ml was much greater (about 5 times higher) for samples collected in warm water (water temperature > 18  C) than for the other samples. Under cold-water conditions, free residual chlorine levels are not a critical parameter to ensure very low HPC in the DS (Fig. 2a). Under these conditions, taking into account the HPC method used, water temperature is too low, in comparison to water temperature during summer, to favor the presence of microorganisms. Fig. 2a also indicates that to ensure very low HPC

in warm water conditions, a free residual chlorine concentration higher than 0.3 mg/l is to be preferred. The figure also suggests that to avoid high levels of HPC in warm waters, a minimum concentration of free residual chlorine of 0.1 mg/l must be maintained. However, it is important to note that the combined analysis on the impact of free residual chlorine and temperature must be conducted with caution because these two parameters are not independent. In fact, due to the higher chlorine demand in warm water conditions, the proportion of cases with low levels of free residual chlorine (below 0.3 mg/l) is much higher when water temperature is above 18  C than it is for lower temperatures. Fig. 2b indicates that higher HPC levels are also associated with higher organic matter in water (measured as UV254 nm). In fact, higher levels of UV-254 nm could indicate the increased presence of biodegradable organic matter (not measured in this study) that promotes bacteria regrowth (LeChevallier et al., 1991; van der Kooij, 1992). In addition, the highest HPC levels were observed in samples with simultaneously high UV-254 nm and low free residual chlorine. However, even when levels of UV-254 nm are high, maintaining free residual chlorine above 0.3 mg/l will favor relatively low levels of HPC (Fig. 2b). The apparent high impact of

n =50

a

160

120

n =91

100 80

0

0.1 - 0.3

n =39

n =507 n =608

n =138

n =63

n =9

0.3 - 0.5

n =40

0.5 - 1.0

>18

) C (°

11.1-18

re

n =273

n =210

n =51 n =113

4.1- 11

tu

n =74

20

n =242 n =430

pe ra

40

<0.1

n =74 n =159

<4

Te m

60

n =258

n =166

Mean of HPC (cfu/ml)

140

≥1.0

Free residual chlorine (mg/L)

b

160

120

n =152

100

n =247 n =31

n =289

n =18

0 <0.1

0.1 - 0.3

0.3 - 0.5

0.5 - 1.0

≥1.0

2-3.2 <2

)

-1

(m

>3.2 n =189

n =1363

n =498 n =57

n =36

n =135

nm

n =7

n =89

4

20

n =344

40

V25

60

U

80 n =140

Mean of HPC (cfu/ml)

140

Free residual chlorine (mg/L) Fig. 2 – a. HPC levels according to free residual chlorine and water temperature. b. HPC levels according to free residual chlorine and UV-254 nm.

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UV-254 nm on HPC may also be due to the fact that the highest levels of UV-254 nm were observed in samples collected during the summer (average value of 2.85 m1 for samples collected from June to September in comparison with average value of 2.69 m1 for the other seasons together). In the summer, chlorine demand is high and the maintenance of adequate free residual chlorine levels is more difficult, in particular in system extremities. Sampling campaign results suggest that higher pH in water led to higher HPC levels. For example, the proportion of samples with HPC levels above 100 cfu/ml was higher when water pH exceeded 8.5 (Table 2a). This result concurs with previous studies indicating a positive correlation between pH and HPC (Carter et al., 2000; Zhang and DiGiano, 2002). This result could be associated principally with the fact that higher pH decreases the efficiency of the chlorine disinfection, leading to more levels of HPC (Carter et al., 2000; Ndiongue et al., 2005) and, secondly, the fact that heterotrophic bacteria develop better at higher pH. The other water quality parameters under study (turbidity, color, conductivity) did not appear to have an observable impact on HPC levels (probably due to their relatively limited variability). As mentioned previously, the distribution system under study was composed of four (4) sub-systems (DSS) with different hydraulic characteristics. Analysis of the data showed that the average HPC count was much lower in one specific sub-system, QC2, in comparison with the others (Table 2b). This result might be explained in part by the fact that sub-system (QC2) is supplied by a reservoir where rechlorination is applied (the three other sub-systems being supplied directly by the treatment plant). In addition, average levels of free residual chlorine during the period under study were higher at QC2 sampling points (0.65 mg/l) in comparison with samples of the other three sub-systems (with an average of 0.46 mg/l). The difficulty of maintaining very low levels of HPC under summer conditions is illustrated in Fig. 3. As mentioned before, in addition to temperature conditions that favor bacteria development, relatively high levels of HPC during this season are also probably associated with the fact that high chlorine demand makes it difficult to maintain adequate levels of free residual chlorine throughout the DS. Moreover, as discussed earlier, summer is also the season during which higher levels of UV-254 nm are observed. Likewise, the difficulty of ensuring very low levels of HPC in summer appears higher in DS extremities (Figs. 3 and 4). In fact, in winter for example, low HPC is ensured irrespective of the DS location.

Table 2a – Distribution (%) of samples per HPC class according to water pH. HPC classes (cfu/ml) 0 1–100 101–500 More than 500

Water pH <7.5 n ¼ 1053

7.5–8.5 n ¼ 2478

>8.5 n ¼ 64

61.1 35.3 2.50 1.14

65.9 32.1 1.45 0.61

60.9 29.7 7.81 1.6

Table 2b – Distribution (%) of samples per HPC class according to distribution sub-system. HPC classes (cfu/ml)

QC1 n ¼ 848

QC2 n ¼ 665

QC3 n ¼ 1292

QC4 n ¼ 790

0 1–100 101–500 More than 500

60.1 36.8 2 1.1

75.3 23.8 0.9 0

60.4 36.3 2.4 0.93

66.2 31.3 1.65 0.9

During the summer, the possible formation of biofilm in DS extremities (where low-diameter pipes favor intimacy between pipe wall and bulk water) could also contribute to high chlorine demand, organic matter release and pH conditions contributing to increased HPC levels.

3.2.

Multivariate modeling of HPC occurrence

The previous analysis showing high variability for levels of HPC suggests that factors potentially influencing their occurrence in the DS could be inter-related. For conditions of high water temperatures, for example, several phenomena may arise simultaneously: development of biofilm in the DS, release of bacteria from the biofilm, increase of organic matter from the biofilm, incremented turbidity, increased chlorine demand and decreased free residual chlorine levels. Consequently, the quantitative analysis of the impact on HPC of each single factor is not sufficient to identify the most critical variables at stake. The use of robust multivariate approaches for data analysis must be considered to identify the simultaneous effects of the above-mentioned variables and to develop models predicting HPC occurrence.

3.2.1.

Model description

Models for explaining and predicting HPC in the DS were developed based on multivariate regression analysis. A regression approach allows identification of the association between HPC and other water quality parameters (Spiegel and Stephens, 2007). Given that HPC is a non-negative count variable, the use of random-effect Poisson models was elected (Hilbe, 2007). Five different models were developed based on assumptions of independence or not about HPC observations. Both single-level and multi-level regression models were considered. The models are the following: Model 1: one-level Poisson model; Model 2: negative binomial (NB) or Poissongamma; Model 3: two-level Poisson with a sampling point-level random effect; Model 4: two-level Poisson with a sampling point-level random effect and a month-fixed effect; and Model 5: three-level Poisson with a sampling point-level and subsystem level random effects. In Models 1 and 2, independence is assumed among observations (the observations are completely independent of each other, which means that no spatiotemporal associations exist among observations). Models 3–5 are multi-level. They can take into account that HPC measures could be correlated on a level of sampling points, months and/or distribution sub-systems. HPC observations at different sampling points are not simple random samples: HPC

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Fig. 3 – Spatial variability of HPC according to season; a) Winter (December, January, February), b) Spring (March, April, May), c) Autumn (September, October, November), d) Summer (June, July, August) (maps show locations for which the seasonal 90th percentile of HPC measured in samples belongs to one or other HPC class).

water research 43 (2009) 1075–1087

HPC 90th Percentile

700 600

methods such as Newton–Raphson are applied to maximize this function.

QC404 QC304 QC105

500 400 300 200 100 0 Winter

1081

Spring

Autumn

Summer

Seasons Fig. 4 – Seasonal distribution of HPC (90th percentile) in three selected sampling points (QC404 located close to the treatment plant, QC304 located in the middle of the DS, and QC105 located at the extremity of the DS).

measures are located in sampling points, which are located in distribution sub-systems. Then HPC levels within the same sampling point would be correlated, as HPC levels within the same distribution sub-system. This is due to the specific characteristics (location, pipe material, pipe age, level of residual disinfectant, water residence time, velocity and flow, etc.) of sampling point and distribution sub-system. In order to perform correct inference, these correlations must be considered. That is why, in this case study, it may not be reasonable to assume that observations in the same sampling point are independent. Multi-level models are then developed to consider the presence of correlation among observations at the different levels (samples, sampling points, distribution sub-systems). Multi-level models allow the introduction of sampling points and sub-systems level random effects into the regression model in order to account for intra-sampling point or intra-subsystem correlation. In this section, we define different HPC model formulations and the way intra-class correlation can be estimated.

3.2.1.1. One-level models. The one-level models suppose the non-existence of spatiotemporal association among observations in accordance with the postulate of independence of observations. Two one-level models, described below, were developed.

3.2.1.1.1. Model 1: one-level Poisson model. Considering that the HPC measure i (number of heterotrophic bacteria) at point p ¼ 1,2,.,n, denoted by Yip is Poisson distributed, a one-level Poisson model for Yip can be represented as,   qip YipwPoisson  (1) ln qip ¼ mip þ 3ip where, qip is the mean number of bacteria, mip ¼ bxip, b ¼ (b0,., bk) is a vector of regression parameters to be estimated from the data and xip ¼ (x1ip,., xkip) is a vector of explanatory variables (in this case, a set of water quality parameters). In addition, 3ip is the model error following a Normal distribution, 3ip w N(0, z). Note that 3ip represents all the unobserved factors or random variations. Usually, numerical maximization

3.2.1.1.2. Model 2: one-level negative binomial model. In the negative binomial model, a Gamma random effect nip ¼ exp(3ip), is introduced in a multiplicative way to mip, which is equivalent to assuming that the mean number of HPC (qip) is randomly distributed (Cameron and Trivedi, 1998). High proportion of zero counts i of HPC may lead to over-dispersion, i.e., E½Yip jmip < Var ½Yip jmip  and the Gamma random effect will take this fact into account (Hilbe, 2007). The model error nip follows a Gamma distribution, nip w Gamma (f, d). By specifying d ¼ f (the ‘‘inverse dispersion parameter’’), we obtain the classical NB model (Lawless, 1987), and then nip w Gamma (f, f). The marginal distribution is obtained by integrating over nip such that:       Z   m Yip mip ; f ¼ f Yip mip ; nip p nip dnip ; (2) where p($) represents the probability density of the Gamma distribution. The interesting point of this model is that the marginal distribution of Yip appears to be the density of the NB distribution, which can be written as: !f !Yip  GY þ f   mip f ip  ; (3) m Yip mip ; f ¼ Yip !GðfÞ f þ mip f þ mip where G($) is the Gamma function.

3.2.1.2. Multi-level models. One of the potential limitations of one-level models is that they ignore the multi-level structure of HPC variability because measures of HPC may be correlated at the level of sampling points or sub-systems. One-level models ignore the possible presence of intra-class correlation at different data levels. This can introduce bias in standard errors and then affect variable selection. To reduce the model error in one-level models, multi-level models were tested/ developed. Multi-level models allow for the consideration of specific characteristics of sampling point and distribution sub-system. These characteristics, as mentioned previously, may lead to correlation between HPC observations within the same sampling point, and/or within the same distribution sub-system. By successively adding one level (sampling point random effect, sub-system random effect, etc.) to the first one-level model above, multi-level models are obtained. 3.2.1.2.1. Model 3: two-level Poisson model with a sampling point-level random effect. Using the same considerations as Model 1, a two-level Poisson model with a sampling point random effect for Yip can be represented as,   YipwPoisson qip  ln qip ¼ mip þ gip þ 3ip

(4)

where, qip and mip are defined as in the Model 1 and, in addition, gp is a sampling point-level random effect, following a Normal distribution, i.e., gp w N(0, s2) and 3ip is the model error following also a Normal distribution, 3ip w N(0, z). Note that 3ip represents all the unobserved factors or random variations not captured by gp.

3.2.1.2.2. Model 4: two-level Poisson model with a sampling point-level random effect and a month-fixed effect. This is the same Model 3 to which a monthly fixed effect was included in

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the regression equation. The effect of each month is represented by a dummy variable.

3.2.1.2.3. Model 5: three-level Poisson model with a sampling point-level and a sub-system level random effects. To test the potential intra-class correlation or nested effect at the level of distribution sub-systems, a sub-system level random effect is introduced as in Eq. (4), that is,   (5) ln qips ¼ mips þ gps þ as þ 3ips ; where, s stands for the distribution sub-system, s ¼ 1,.,4. As in regression Eq. (4), gps and 3ips are Normal distributed. Moreover, the sub-system effect is also assumed to be normally distributed, i.e., as w N(0, s3).

3.2.2.

Intra-class correlation and model fit

To measure intra-class correlation (correlation among observations within a sampling point or sub-system), the correlation coefficient (ICC), denoted by r, was used. This coefficient is computed on the basis of the variance components of models defined previously and ranges from 0 to 1. Note that if all HPC measures are independent of one another, r ¼ 0. On the other hand, if all measures inside each cluster (sampling point or sub-system, for instance) are exactly the same, r ¼ 1, then obviously, a r s 0 implies that the observations are not independent, e.g., ICC > 0 implies that HPC measures in the same sampling point or in the same sub-system are found in similar environments or affected by unobserved sampling point or sub-system factors (location, pipe material, pipe age, water residence time, velocity and flow, etc.). For instance, based on the two-level model defined in Eq. (4), correlation between HPC observations coming from the same point (denoted by rp) is obtained as:   s2 rp ¼ cor Yip; Yi0 p ¼ (6) z þ s2 Based on the three-level model defined in Eq. (5), correlation between HPC observations coming from the same system s but different point p is obtained as rs ¼ cor(Yips,Yi0 p0 s) ¼ s3/(z þ s2 þ s3). Obviously, correlation among observations coming from the same point and system is equal to rps ¼ (s2 þ s3)/(z þ s2 þ s3). The likelihood-ratio (LR) test is computed to select the model that best fits the data. To compare the fit of two models based on the LR test, we first calculate a c2 statistic as the positive difference of L1 and L0, where L1 and L0 are the loglikelihoods of the two compared models. LR has a c2 distribution with degrees of freedom obtained from the difference of the number of parameters to be estimated in the two models. If the difference ðc2 ¼ jL1  L0 jÞ is not statistically significant, this suggests that a more complicated model may not be necessary. For instance, if two-level and three-level models are compared and if the c2 statistic is not significant, this means that a two-level model, which contains fewer parameters, is acceptable to explain HPC occurrence.

3.2.3.

Model results

3.2.3.1. One-level models (Models 1 and 2). Results of models are shown in Table 3. For model parameter estimation,

GLLAMM routine was used and implemented in STATA version 9.2. In Model 1, it was assumed that observations were independent. As mentioned before, 3ip is assumed to be Normal distributed instead of Gamma distributed as in Model 2. Free residual chlorine, temperature and UV-254 nm are statistically significant. The model indicates a negative relationship between HPC and free residual chlorine, and a positive relationship between HPC and temperature and between HPC and UV-254 nm. The directions of these relationships are in accordance with previous studies (LeChevallier et al., 1991; van der Kooij, 1992; Carter et al., 2000; Zhang and DiGiano, 2002; Huck and Gagnon, 2004; Ndiongue et al., 2005) and the descriptive analysis presented in Section 3.1. For the fixed subsystem effect (sub-system introduced to the model as dummy variable), Sub-systems 1 and 3 were significant, contrary to Sub-system 2. Water pH was not significant. Results for Model 2 indicate that all considered variables are statistically significant at the 5% level. The model shows a negative relationship between HPC and free residual chlorine, and a positive relationship between HPC and temperature, UV-254 nm and pH. Considered a reference Sub-system 4, the sub-system fixed effects were also significant in this model. In fact, the HPC of Sub-system 4 is significantly lower than for Sub-systems 1 and 3 and, significantly higher than Sub-system 2, as mentioned previously in Table 2b. Finally, note that the log-likelihood absolute value of Model 1 (6138.78) is lower than the value for Model 2 (6303.41), which means that the Normal distribution assumed for random effect 3ip fits the data better than the Gamma density. Thus, Model 1 seems to perform better than Model 2.

3.2.3.2. Multi-level models (two- and three-level models). Model 3 (Column 4 in Table 3) includes a sampling point-level random effect accounting for intra-point correlation. Free residual chlorine, temperature and UV-254 nm were again significant in the model, which was not the case for pH and sub-system fixed effects. Interestingly, for this outcome we observed that the intra-sampling point correlation coefficient is significantly different from zero, rp > 20%. Moreover, it was noted that the log-likelihood absolute value of this model (6001.08) is significantly lower than for two previous models. These results imply, among other things, that:

- Models 1 and 2 violate the independence postulate of the observations (so, the water samples taken in the same point are correlated between each other); - There is a loss of information concerning the intrasampling point variations higher than 20% (at least 20% of the variability of HPC variable is at the sampling point level); - It is impossible to conclude anything on the variability of HPC on the basis of models 1 and 2. So doing would lead to an ecological error (incorrect inference) consisting of separating the effect of individual measures (in samples) from correlations observed at the aggregated level (sampling point level).

Table 3 – Results of one-level and multi-level multivariate regression models for HPC occurrence. Model 1: one-level Poisson

Coef.

Std. err.

Std. err.

** ** ** ** ** ** **

Model 5: three-level Poisson with a sampling point and a sub-system random effects

Coef.

Coef.

Coef.

Std. err.

** ** ** n.s. n.s. n.s. n.s.

Std. err.

** ** ** n.s. ** ** **

n.a. n.a.

n.a. n.a.

0.23 n.a.

Level 1 Var. (std. err.)

n.a.

n.a.

6.16

(0.29)

**

5.70

(0.22)

**

Level 2: point-level Var. (std. err.)

n.a.

n.a.

1.80

(0.39)

**

1.21

(0.13)

**

Level 3: sub-system level Var. (std. err.) n.a.

n.a.

n.a.

Log-likelihood

6303

6001

Std. err.

Sig.

0.269 0.010 0.045 0.201 n.a. n.a. n.a.

** ** ** n.s. n.a. n.a. n.a.

6.13

(0.28)

**

1.86

(0.32)

**

n.a.

0.35

(0.13)

**

6000

6004

2.925 0.092 0.370 0.036 0.814 0.457 0.754

0.173 0.016 0.035 0.094 0.175 0.209 0.175

Sig.

Intra-class correlation Second level (rp) Third level (rs)

2.770 0.105 0.380 0.006 0.841 0.011 0.632

0.288 0.010 0.065 0.247 0.532 0.676 0.506

Sig.

3.065 0.103 0.480 0.067 0.675 0.035 0.886

2.361 0.142 0.445 0.632 0.634 0.713 0.876

0.162 0.009 0.075 0.172 0.158 0.154 0.154

Sig.

Model 4: two-levela Poisson with a sampling point random effect and a month-fixed effect

0.18 n.a.

2.731 0.104 0.385 0.079 n.a. n.a. n.a.

0.23 0.04

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** ** ** n.s. ** n.s. **

Coef.

Model 3: two-level Poisson with a sampling point random effect

Free residual chlorine Temperature UV-254 nm pH Sub-system 1 Sub-system 2 Sub-system 3

6139

0.216 0.010 0.065 0.205 0.180 0.232 0.168

Sig.

Model 2: one-level Poisson-gamma or NB

** Significant at 5%; * significant at 10%. Coef.: coefficient; Std. Err.: standard error; Sig.: significance; var.: variance; n.a.: not applicable; n.s.: not significant. a Significant months in model 4 (month, Coef., Sig.): February, 0.446, *; March, 0.410, *; April, 0.523, **; August, 0.561, ** and November, 0.482, **.

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Under these circumstances, parameter estimates may be unstable and standard errors biased, leading to the establishment of a relationship between variables that does not really exist. Model 4 (Column 5 in Table 3), is a simple extension of Model 3. In this case, a month-fixed effect is introduced in the equation to account for temporal variations. As with the previous models, free residual chlorine, temperature and UV254 nm are statistically significant at 5% level. pH results are still insignificant, but sub-system effects have a significant effect on HPC. In addition, monthly variations are also significant for some particular months.1 The log-likelihood of this model is comparable with that of Model 3, but the intrasampling point correlation is lower for Model 4. A comparative discussion on these two models is presented later. In Model 5, a second-random effect was considered in order to test for the presence of auto-correlation at the subsystem level. Significant water quality variables were practically identical to those in the other models. Moreover, both sampling point and sub-system intra-class correlations were positive. However, the sub-system level correlation appears to be weak. This result suggests that HPC measures originating from the same sub-system for different points are practically independent. Conversely, measures coming from the same point are greatly influenced by unobserved factors at the sampling point level. Thus, the use of two-level models accounting for a clustering effect at the level of sampling points is suggested for modeling HPC data.

3.2.4.

- Traditional regression models or other one-level models are not appropriate for modeling water quality data in a DS, in particular microbiological counts. They suppose observations independent of others, even at the same sampling point. In addition, statistical inference based on traditional regression models is biased, as ecological error is committed. - The multi-level Poisson modeling approach (in particular Model 3) is suitable for situations where HPC levels are relatively low, as is the case for the distribution system under study. - A relatively high percentage of the variability of the dependent variable (23% in the case of Model 3) is not considered by the traditional regression models or other one-level models. These models cannot take into account that HPC measures could be correlated on level of sampling points (multi-level distribution). - The simultaneous impact of the different independent variables is assessed better with multi-level models than with statistical models for HPC developed in previous work. Indeed, in past work, evaluation of the correlation between HPC and independent variables is generally carried out individually (using the coefficient of correlation). The synergic or antagonistic effect of different independent variables was not really considered in previous work. Very few investigations have reported the development of multivariate models for HPC (e.g., Zhang and DiGiano, 2002) but were based only on traditional regression based-methods.

Model selection

According to the results of Models 3–5, the presence of intrasampling point correlation was detected. This indicates that there is dependence among observations originating from the same sampling point. In other words, the specificity of sampling points has an impact on the variability of the HPC levels in the DS. However, based on results for the three-level model (Model 5), intra-class correlation at the sub-system level is very weak, meaning that the proportion of the total variance of HPC due to the specificity of sub-systems is unimportant even if descriptive results of Table 2b suggested the specificity of Sub-system 2. Thus, the three-level model does not bring about a substantial improvement compared to the two-level models. According to this result and based on the log-likelihood absolute values, Models 3 and 4 appeared to be the most appropriate. To compare Models 3 and 4 in terms of goodness-of-fit and to select the best of them, the likelihood-ratio (LR) test was used. This test is based on the fact that LR has a c2 distribution. From Table 3, we noted that the positive difference between the log-likelihoods of these two models was practically nil, with 11 degrees of freedom (resulting from the difference among model parameters introduced in each model, 18  7 ¼ 11). According to the c2 distribution table, this difference, for 11 degrees of freedom and a ¼ 5% (confidence degree), is inferior to the critical value of c2. Thus, Model 4, with 18 parameters, is no better than Model 3, with only seven parameters. Thus, Model 3 is considered as the simplest and

1

best performing model describing the variability of HPC in the DS under study. In summary, the results of the multi-level modeling of HPC point to the following interesting observations:

Considering December as the month of reference.

4.

Model sensitivity analysis

Models can be used to predict the probability of exceeding specific HPC values according to specific conditions (water quality, operational variables, season, type of sub-system, etc.). A sensitivity analysis on Model 3 (the best model) was carried out to demonstrate its applicability for the identification of seasonal-based strategies to minimize the probability of HPC occurrence in the DS (Fig. 5). In this analysis, modelpredicted probabilities of exceeding 10 and 50 cfu/ml of HPC were considered with three different scenarios of seasonal water temperatures. The maximum seasonal value of UV254 nm was considered in order to better illustrate the simulation. The results presented in Fig. 5 show that for all seasons, to reduce the probabilities of HPC occurrence, higher free residual chlorine levels are required for higher seasonal water temperatures. Variability of required chlorine levels is higher in spring and fall because of the higher variability of water temperature during these two seasons, compared to winter and summer. Results show that according to the water temperature scenarios under consideration, the minimum required levels of free residual chlorine to maintain low probability of HPC occurrence vary according to the season. Due to

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Threshold: 10 cfu/ml, Water temperature: 25th percentile

Threshold: 10 cfu/ml, Water temperature: mean value

th

Threshold: 50 cfu/ml, Water temperature: mean value

th

Threshold: 50 cfu/ml, Water temperature: 75th percentile

Threshold: 50 cfu/ml, Water temperature: 25 percentile Threshold: 10 cfu/ml, Water temperature: 75 percentile

b

1.00 0.90

0.80

0.80

0.70

0.70

0.50

1 1

0.78

0.70

0.63

0.55

0.48

0.40

0.33

0.00

1

0.93

0.00 0.85

0.10

0.00 0.78

0.10 0.70

0.20

0.63

0.20

0.55

0.30

0.25

0.40

0.30

0.48

0.78

0.50

0.18

0.40

0.60

0.10

0.50

0.40

0.93

0.70

0.60

0.33

0.93

0.70

0.25

0.85

0.80

Probability

0.90

0.80

0.18

0.70

1.00

0.90

0.10

0.85

d

1.00

0.00

Probability

0.63

Free residual chlorine (mg/L)

Free residual chlorine (mg/L)

c

0.55

0.00

1

0.93

0.85

0.78

0.70

0.63

0.55

0.00 0.48

0.00 0.40

0.10 0.33

0.10 0.25

0.20

0.18

0.30

0.20

0.10

0.30

0.48

0.40

0.40

0.40

0.33

0.50

0.60

0.25

0.60

0.18

Probability

0.90

0.10

1.00

0.00

Probability

a

Free residual chlorine (mg/L)

Free residual chlorine (mg/L)

Fig. 5 – Predicted probabilities of exceeding HPC thresholds (10 and 50 cfu/ml) according to free residual chlorine levels and water temperature; a) Winter (December, January, February), b) Spring (March, April, May), c) Autumn (September, October, November), d) Summer (June, July, August).

the difference of water temperature levels (more than 14  C on average), the contrast between summer and winter is considerable (Fig. 5a and d). According to these results, minimum required levels of free residual chlorine to maintain below 50% the probability of exceeding 10 cfu/ml of HPC vary approximately from 0.17 to 0.43 mg/l in spring, 0.62 to 0.74 mg/l in summer, 0.24 to 0.54 mg/l in fall and 0.06 to 0.17 mg/l in winter. Model analysis also shows that probabilities of exceeding 50 cfu/ml of HPC are important only in summer. The probability of such events is extremely low in the other seasons, independently of water temperatures. In summer, the maintenance of 0.30 mg/l as a minimum level of free residual chlorine will make the probability of exceeding 50 cfu/ml of HPC almost nil. However, samples collected in summer can expect to have HPC levels higher than 50 cfu/ml if levels of free

residual chlorine in the DS are not above 0.10 and 0.17 mg/l, according to the water temperature. In fact, if values of free residual chlorine are below 0.3 mg/l, the probability of exceeding 50 cfu/ml of HPC increases significantly, with relatively small decreases of free residual chlorine levels. In fact, if concentrations of free residual chlorine values drop below 0.15 mg/l, such probability becomes high, irrespective of water temperatures (chances of exceeding 50 cfu/ml of HPC become higher than 50%).

5.

Conclusions

In this research, the occurrence of HPC was investigated and modeled from information generated during a three-year

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sampling program in a full-scale distribution system. Measured HPC levels were relatively low in comparison with other studies reported in literature. Differences with other studies are associated to the incubation temperature, time and medium as well as microbia occurrence is different from one to another distribution system. In this study, even if low, HPC levels were very variable particularly according to the seasons. Descriptive and statistical analysis of the data validate results of past research concerning parameters that influence HPC occurrence in distribution systems. Development of one-level and multi-level statistical models for HPC occurrence led to the simultaneous consideration of different factors potentially governing HPC levels in the distribution system under study. Model results demonstrated that it is possible to properly describe and predict HPC in distribution systems where bacterial occurrence is relatively low. Modeling results indicate that multi-level models (that have not been applied before in this area) perform better than onelevel models to describe and predict HPC levels. The advantage on the multi-level modeling approach in comparison with conventional statistical approaches is that the nested structure (multiple levels of data associated to point effect, season and sub-system) inherent in HPC data is considered, which is not the case in one-level models. This allows elimination of a certain number of biases, such as the violation of the postulate of independence of observations which drive these models with a single level. The developed models are specific to the DS under study. However, the modeling methodology can be used for any distribution system for which valid information on HPC occurrence (or any indicator of microbial water quality) is generated. As suggested by Speight et al. (2004), valid information refers to the abundance of sampling points and the frequency of samples; this abundance is related to sample size currently used by most utilities for routine monitoring and regulation compliance. Sensitivity analysis demonstrates that models for HPC could be used by utility decision-makers to identify strategies to decrease the probability of HPC occurrence: for example, by identifying the required levels of free residual chlorine or other water quality parameters. Models could also be used by managers to recognize locations where water quality deteriorates more easily (in order to prioritize geographical locations preventive and corrective actions). Finally models could also be used to assess whether the frequency of the sampling and the number of sampling points are sufficient to represent spatiotemporal variations of microbiological water quality in DS. However, the models presented here have certain limitations warranting consideration in future research. These limitations are associated, in particular, with a lack of precise data concerning water residence time estimations at each monitoring point of the DS, the presence of biofilm within the pipes, specific characteristics of the pipelines (materials, age, diameter, etc.) and the lack of consideration of the impact of the conditions of a specific sampling point on the water quality of a downstream sampling point, as recently suggested by DiGiano and Zhang (2004). These aspects must be considered in future research on modeling HPC and other microbiological parameters in distribution systems.

In future research, it will also be important to develop complementary modeling approaches to improve the prediction capabilities of microbiological quality in distribution system. For example, recent reformulations of the Poisson model, the Winsorized and spatial filter specifications (Griffith, 2006) might be combined with multi-level models to better address spatial correlation noted at the sampling point level. This study considers HPC because, despite its limitations, this is the primary parameter for assessment of the general microbiological quality of drinking water (Allen et al., 2004; Sartory, 2004). As suggested by Hammes and Egli (2005) and Hammes et al. (2008), the use of flow-cytometric total bacterial cell counts is an alternative to HPC for diverse reasons (e.g., a method that is easier to reproduce, requires only 20 min in comparison to 1–3 days for the incubation of the plates, the total cell number is counted). Particularly, this allows the detection of all bacteria in a sample including those that are inactive or unculturable. Additional progress in the rapid detection of cellular viability and activity would increase the confidence in and acceptance of such a method. Thus in the future, the modeling approach proposed in this paper could also be applied to model the spatiotemporal variability in distribution systems of the total cell counts obtained by flowcytometry.

Acknowledgments The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), the HydroQuebec Institute in Environment, Development and Society and the Embassy of the Republic of Haiti in Canada for their financial support.

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