ISSN 0097-8078, Water Resources, 2006, Vol. 33, No. 6, pp. 637–650. © Pleiades Publishing, Inc., 2006.Original Russian Text © B.M. Dolgonosov, K.A. Korchagin, E.M. Messineva, 2006, published in Vodnye Resursy, 2006, Vol. 33, No. 6, pp. 686–700.

637

INTRODUCTION

The anthropogenic microbiological pollution of nat-ural environment is a serious hazard to drinking watersupply system. The major sources of microbial pollu-tion of water are human and warm-blooded animalexcrements that enter into water bodies with municipalwastewaters, and drains from cattle farms and areaspolluted by manure. Because of this, regular microbio-logical control of source water is carried out at waterstations relative to the list of bacteriological indicesinvolving general and thermotolerant coliform bacteria,sulfite-reducing clostridia, total microbial count,coliphages, and fecal streptococci. As is known, theseindices often experience seasonal variations super-posed with irregular fluctuations alternating with aperi-odic surges with variable magnitude. The normal oper-ation of a water supply system requires the predictionof possible level and duration of bacterial pollution ofthe water source. This problem can be solved based ona model describing the formation of bacterial pollutionof a water source. The inherent effect of random factorson these processes requires the use of stochastic mod-els. This class of models allows probabilistic estimatesof different pollution levels to be obtained.

The time series of bacteriological characteristics arecommonly described in the literature either with the useof simple statistical estimates (seasonal trend, coeffi-cients of correlation, etc.) without the use of models[14, 20, 26] or with a probabilistic description based ondifferent types of distributions [15, 17–19, 21, 22].Thus, indicator organisms (

E. coli

, enterococci, andfecal coliforms) in the water of Lake Kinneret werestudied in [17]. Water samples were taken every 2–4

weeks within five years. The data series were describedbased on the Laplace distribution density

truncated at high values of the argument, and the distri-bution density of extreme values

where

µ

is mean,

a

and

b

are constants,

Z

=

Z

(

N

)

is afunction of the organism population

N

, which can havethe form ln

N

,

e

N

,

N

1/2

,

N

1/3

, and others. Function

N

1/3

was used in [17], whereas function ln

N

was used in [15,18, 19, 21, 22]. The distribution parameters were eval-uated by fitting to empirical data with the use of themethod of moments or maximum likelihood method.With allowance made for data errors and large intervalsbetween measurements, the authors regarded the agree-ment between both distribution and measured values assatisfactory.

The agreement with measurement data testifies onlythat the type of distribution was successfully chosen inthis specific case, but neither the distribution itself, norits parameters convey any essential meaning. This ham-pers the interpretation of results and limits their appli-cation. It would be beneficial to derive the functionalform of the distribution immediately from the equationsof stochastic dynamics of the microbial population. Theaim of this study is to develop such model and apply itto the description of a water source (the Moskva River).

f Z( ) 12b------ Z µ–

b---------------–⎝ ⎠

⎛ ⎞ ,exp=

f Z( ) 1b--- a µ–

b------------ a Z–

b------------+⎝ ⎠

⎛ ⎞exp– ,exp=

WATER QUALITY AND PROTECTION:ENVIRONMENTAL ASPECTS

Model of Fluctuations in Bacteriological Indicesof Water Quality

B. M. Dolgonosov

a

, K. A. Korchagin

a

, and E. M. Messineva

b

a

Water Problems Institute, Russian Academy of Sciences, ul. Gubkina 3, GSP–1, Moscow, 119991 Russia

b

MATI–Moscow State Technological University, ul. Orshanskaya 3, Moscow, 121552 Russia

Received April 4, 2006

Abstract

—The bacteriological indices of water quality, including coliforms, sulfite-reducing clostridia, totalmicrobial count, coliphages, and fecal streptococci were analyzed. A model describing fluctuations in microor-ganism population in the aquatic environment is proposed. A stochastic differential equation was obtained andused to derive a lognormal distribution of organism population. The model was applied to describe time seriesof bacteriological indices of the Moskva River Water Source. Satisfactory agreement was obtained between the-oretical distributions and empirical data in a wide range of index values embracing one to three orders of vari-ations in microorganism population. A forecast procedure was developed and applied to calculate the exceed-ance probabilities of different levels of bacterial pollution of the water source. Time periods with higher bacte-rial pollution were identified.

DOI:

10.1134/S0097807806060054

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To develop a model one should be aware of themeaning of the bacteriological indices involved. Forthis purpose, a general description of these indices isgiven, the pattern of their behavior is identified (in par-ticular, their seasonal variations), and the level of waterbacterial pollution is assessed from the viewpoint of theexisting standards, which is beneficial for the substan-tiation of the importance of the problem. Next, themodel is developed, compared with the available data,and applied for prediction. The input data on the bacte-riological indices of the Moskva River Water Sourceare taken from [2].

The main attention is given to the determination ofthe long-term mean distribution of microorganism pop-ulation. This data is quite sufficient for the solution oflong-term problems, such as designing treatment facil-ities in potable water supply systems, which will func-tion for a long time and have to cope with periodicallyappearing extreme levels of bacterial pollution ofsource water. Averaging over many years fails to allowthe seasonal trend in bacteriological indices to bedescribed explicitly; however, seasonal effects demon-strate themselves in the cases when the values of indi-ces in winter and summer are essentially different.

TIME SERIES CHARACTERISTICS

Coliform bacteria

(coliforms) live and reproducein the lower part of the digestive system of humans andwarm-blooded animals. Some coliforms can bedetected not only in feces, but also in the environment(in waters rich in organic matter, soils, in decayingplant materials, etc.), as well as in drinking water withhigher concentration of organic matter. Coliforms enterinto water, as a rule, with fecal discharges and can per-sist in it for several weeks, though, in the overwhelmingmajority of cases, they do not reproduce [3, 4]. Water iscontrolled in terms of the total population of coliformbacteria and the population of thermotolerant coliform,which can be easily detected.

Coliforms are not hazardous to people (though theycan cause an inflammatory process), but they can dem-onstrate that pathogenic microorganisms may haveentered into water. The total coliform population inwater sources used for centralized drinking and domes-tic water supply must not exceed 1000 units(CFU/100 ml; CFU is colony-forming unit), and that ofthermotolerant coliform must not exceed 100 ml [9].No coliform is allowed to be found in any sample ofdrinking water with a volume of 100 ml [8].

In the Moskva River water, coliforms are repre-sented mostly by thermotolerant species [2]. The timeseries (Fig. 1a) shows that coliform bacteria are fewduring summer low-flow period (commonly from ahundred to a few hundreds units), and their populationrises for a short time (1000 units) only during rainfloods. This is confirmed by data in [23], were bacterialpopulation in river water was found to increase several

times (sometimes, even by an order of magnitude) dur-ing heavy rains. Paradoxically as it may be, the popula-tion of bacteria in winter, notwithstanding the lowwater temperature, is one to one and a half orders ofmagnitude higher than in summer (Fig. 1a).

A similar drop in coliform population in summer–autumn period and its increase in winter–spring byone–three orders of magnitude was mentioned in [26],where bacterial pollution of water in a river estuary wasstudied. The high population of coliforms in winter wasexplained by the release of organic matter at the catch-ment surface because of decay of dead plant materialaccumulated in summer and delivered by surfacewaters into the estuary. An increase in bacterial popula-tion in the estuary water by two to three orders of mag-nitude in the period from January to March was men-tioned in [14], where this effect was accounted for bythe surface water runoff enriched with biogenic sub-stances, as well as by the discharge of wastewaters con-taining fecal pollution.

In addition to indicator microorganisms, water mayalso contain pathogenic microorganisms. As shown in[20], the population of Campylobacter pathogenic bac-teria in winter is also somewhat higher than in summer.Analysis of various pollution sources showed that thismay be due to wastewater discharge from aeration sta-tions, agricultural discharges, polluted watercources, aswell as because of contamination by swimming birds.

The low population of microorganisms in summerand autumn can be due to phytoplankton bloomingwhich brings about an increase in pH (pH is commonly8–9 in summer, and 7.5–8 in winter) and a release ofmetabolites into water

−

these factors suppress the vitalactivity of bacterial flora [24, 25]. The destructive effectof solar radiation, which significantly reduces the bac-terial population, was also mentioned in the literature[12, 13]. The effect of these factors is much weaker inwinter; therefore, the bacterial population increases toa level of several thousands of units.

The highest peaks fall in the periods of voluminoussnow melting, especially during spring floods, whensnowmelt water washes out bacteria from cathcmentsurface. In such periods, the total population ofcoliform bacteria in river water sometimes rises up to6000 units, which is six times greater than the specifiedlimit for water sources (1000 units). Since virtually allcoliforms in the water of the Moskva River watersource are thermotolerant, the restriction imposed ontheir population should be even more strict—no morethan 100 units. Thus, the specified limit for thermotol-erant coliforms in flood periods is exceeded by a factorof 60. The populations’ being much higher than theirrespective limits is a common phenomenon in winterand during floods. This demonstrates the high bacterialpollution of the water source in these periods (Fig. 1a).

Sulfite-reducing clostridia

are anaerobic sporeformers which commonly are present in feces, thoughin concentrations much less than coliforms. Spores of

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MODEL OF FLUCTUATIONS IN BACTERIOLOGICAL INDICES 639

clostridia can persist in water for much longer time thancoliform organisms and are more tolerant to disinfec-tion [3]. Some clostridia are pathogenic [11]. Clostridialiving in water can be not only of fecal origin, but canoriginate from other sources. They are widespread,especially in soil and silty bottom sediments. Clostridiain water of water supply sources are not standardized.Standards for drinking water require the absence ofspores in 20 ml [8].

Data on clostridia in the Moskva River Water Sourceare given in [2]. As can be seen from the time series

(Fig. 1b), clostridia population most of the time variesaround the mean level of 30 units (CFU/20 ml). Highpeaks are met from time to time. The peak of 360 units,recorded on April 11, 2003, coincided with the peak ofspring flood. Small peaks were also recorded duringfloods, however peaks as high as 2003 were notrecorded. Since clostridia are present in bottom sedi-ments, they can enter into water as a result of mechan-ical disturbance of these sediments (for example, in theprocess of cleaning of the river or because of abruptchanges in flow velocity in the bottom layer.

6000

4000

2000

0

Total coliforms, CFU/100 ml

160

120

80

40

0

Sulfite-reducing clostridia, CFU/20 ml

360

(‡)

(b)

Fig. 1.

Time series of bacteriological indices of water quality in the Moskva River Water Source (according to [2]): (a) the line atthe level of 1000 units is the specified limit for water sources in terms of total coliforms, and the line at the level of 100 units is thespecified limit in terms of thermotolerant coliforms; (d) the line at the level of 10 units is the specified limit in terms of coliphages.

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DOLGONOSOV et al.

Total microbial count

(TMC) accounts for the totalpopulation of mesophilous aerobes and facultativeanaerobes [4]. This is an indicator of the general bacte-riological pollution of water, which can indirectly char-acterize the probability of the presence of pathogenicorganisms. This characteristic is not standardized forwater sources. The drinking water standard requires thenumber of bacteria to be no more than 50 units(CFU/ml) [8].

In accordance with measurement data [2], this indexmost of the time is close to 100 units (Fig. 1c). It risesto several hundreds during rains. The largest rise in bac-terial population in river water (up to 2000 units) is

recorded during peak floods, which is due to washingout of soil bacteria by snowmelt water.

Coliphage

is a variety of bacterophage viruses thehosts for which are coliforms. Coliphages are used aswater quality indicators because of their similarity tohuman enteroviruses and the simple procedure ofdetection in water. They are widespread in wastewatersbut relatively rare in fresh excrements of humans andanimals. Coliphages are more viable outside of thehost’s body as compared with bacterial indicators [3].The population of coliphages for sources of centralizeddomestic water supply must not exceed 10 units

250

200

150

100

50

0

Coliphages, PFU/100 ml

(d)

1500

1000

500

0

TMC, CFU/ml

(c)

Fig. 1.

(Contd.)

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MODEL OF FLUCTUATIONS IN BACTERIOLOGICAL INDICES 641

(PFU/100 ml, PFU is plaque-forming unit) [9]. Coliph-ages must not be detected in drinking water [8].

Coliphage population in the Moskva River WaterSource (Fig. 1d) attains its maximum during snowmelting in the periods of winter thaws and in the spring[2]. Coliphage population in summer low-flow periodrarely exceeds 50 units. Peaks of their population dur-ing snow melting can reach 280 units. Figure 1d showsthat the specified limit for water sources is exceededalmost always except for short periods (usually in sum-mer). The population peaks in flood periods can exceedthe specified limit by a factor of up to 28.

Comparison of data on coliforms and coliphages(Figs. 1a and 1d) shows a notable correlation betweenthem (the correlation coefficient of 0.73), which is quiteexplainable for the relationships of the virus–cell type.

Fecal streptococci

commonly are present in humanand animal excrements. They rarely reproduce in con-taminated water [4]. Fecal streptococci are regarded asspecific indicators of fecal water pollution.

In summer low-flow period, fecal streptococci arepresent in small amounts in river water (Fig. 1d), com-monly at a level of about 10 units (CFU/100 ml) [2].During rains, their population increases to several tensand even a hundred of units. The streptococci popula-tion in winter is high—from several hundreds to a thou-sand of units. Peaks of population were recorded inperiods of snow melting; the highest peak was recordedduring flood on April 14, 2003.

DESCRIPTION OF THE MODEL

Let us consider the dynamics of a microbial popula-tion in a water volume carried by river flow. In the ini-tial time moment, this volume contains a certainamount of microorganisms (which have entered into it,for example, with wastewater). Microorganism popula-tion varies over time. It can increase due to either repro-duction (which sometimes can be hampered, as it wasmentioned above) or supply from the catchment area.A decrease in the population can be caused by dying offcells, the elimination by predators, and sedimentationwith suspended particles. Models with different degreeof detail have been developed to describe these pro-cesses [5–7]. Complicated models are practicable in thestudies involving microorganism cultivation under arti-ficial conditions where different characteristics of theprocess can be measured. Simpler models are moreconvenient to use for the description of natural pro-cesses because of the lack of adequate data for evaluat-ing numerous parameters. In this case, we hold to theprinciple of correspondence between the complexity ofthe model and the information content of the sourcedata. Following this principle, we will describe thechanges in bacterial population

N

over time by the sim-ple equation of population dynamics of the Malthusiantype

(1)dN /dt B D–( )N ,=

2000

1500

1000

500

0

Oct

. 1, 2

001

Apr

.19,

200

2

Nov

. 5, 2

002

May

24,

200

3

Dec

. 10,

200

3

June

27,

200

4

Jan.

13,

200

5

Aug

. 1, 2

005

Fecal streptococci, CFU/100 ml

(e)

Fig. 1.

(Contd.)

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DOLGONOSOV et al.

where

B

and

D

are effective parameters accounting fordifferent mechanisms of increase (

B

) and decrease (

D

)in the bacterial population,

t

is time measured from thebeginning of migration.

When the population is large, their mutual influence(competition for the resources and inhibition by metab-olites) should be taken into account. This is made by theintroduction of an additional quadratic term

CN

2

intopopulation dynamics equations. This effect results inthe stabilization of the population at a certain stationarylevel. The conditions of the aquatic environment arefavorable for the reproduction of bacteria, their popula-tion is far less than the stationary level, and the mutualinfluence is insignificant. Therefore, the quadratic termin the population equation can be neglected, as it wasmade in the development of equation (1).

Parameters

B

and

D

depend on the migration pathsand the varying environmental conditions. The temper-ature of the medium, organic matter concentration,light flux, the concentration of inhibitors, the activity ofprotozoa and bacteriophages, and many other charac-teristic can vary along the migration paths. Consideringthat the environment is subject to repeated and irregularvariations, it is natural to assume that the variables

B

and

D

contain random components. Therefore, theresultant rate of increase

r

=

B

–

D

in equation (1) canbe presented as the sum of a systematic and a randomcomponents,

(2)

where

k

is the mean value of

r

for the migration period;the last term in (2) is the random component describedas white noise

ξ

(

t

)

with an intensity of

σ

. From (1) and(2), we have

which also can be written in the form

(3)

where

W

t

is standard Wiener process. Stochastic differ-ential equation (3) describes a random process with amultiplicative noise (since the last term in (3) containsthe product of random variables

N

and

dW

t

). Both itsinterpretations—by Ito and Stratonovich [10]—areequivalent to a shift in constant

k

, which in our case isof no significance, since this constant is evaluatedempirically.

We assume that the main source of bacterial pollu-tion can be represented as an impulse input in the initialtime moment. After that, as the pollution stain moves,the bacterial population can be supplemented by theamounts less than the initial impulse, which is takeninto account parametrically (via

k

and

σ

). The spatialdispersion of microorganisms in the process of motionof the water mass somewhat affects the number of cells,though it is insignificant relative to the processes ofpopulation increase or decrease mentioned above;therefore, this effect is not taken into account in the

r k σξ t( ),+=

dN /dt kN σNξ t( ),+=

dN kNdt σNdWt,+=

model. It is known [16] that in the case of an impulsesource, equation of the type of (3) characterizes a ran-dom variable with a lognormal probability distribution:

(4)

where f is probability density; the initial time momentt = 0 corresponds to the impulse input of bacterial pol-lution into the system. The final time moment t in (4)will be taken as the moment when the water mass con-taining bacteria reaches the check section, i.e., the posi-tion of water intake of the water station, where watersample is taken for bacteriological analysis. The timeinterval from 0 to t thus determined characterizes theduration of the entire migration process of bacteria upto the moment of measurement.

From (4), we can find the cumulative distributionfunction

(5)

where erf is error function,

With α and β known (they can be found by the least-squares method directly from empirical data), we canevaluate the initial parameters kt and σ:

Another method for taking into account fluctuationsin kinetic systems, which is based on the method ofmoments, is given in [7]; however, the relationshipsobtained from it are much more complicated.

Earlier, a stochastic model describing the formationof water chemical composition was constructed in [1].This model yielded a power law for the distribution ofingredient concentrations. Unlike the processes consid-ered there, the environmental conditions of microor-ganism migration can change during the process, so therates of increase and elimination of cells can alsochange. The system has not enough time to reach a sta-tionary state; therefore, we have a lognormal distribu-tion instead of the power distribution obtained in [1].

Note that the notions of population dynamics lead tothe distribution type other than Laplace distribution andthe distribution of extreme values that were mentionedabove.

The obtained form of distribution function (5) isused below for the description of time series of bacte-rial indices taken from [2].

CALCULATION RESULTS

Let us consider the results of calculations by model(5) for individual bacteriological indices (Fig. 2). The

f N( ) 1

N 2πσ2kt-------------------------- N kt–ln( )2

2σ2kt---------------------------–⎝ ⎠

⎛ ⎞ ,exp=

F N( ) f N '( ) N 'd

0

N

∫ 12--- 1 erf α N β–ln( )+[ ],= =

α 2σ2kt( ) 1/2–, β αkt.= =

kt β/α, σ 2αβ( ) 1/2– .= =

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MODEL OF FLUCTUATIONS IN BACTERIOLOGICAL INDICES 643

abscissa is the indices in logarithmic scale, the ordinateis the inverse normal distribution x = Φ–1(y), where

is the standard normal distribution (with zero mean andunit variance). Such coordinate system is convenientfor the construction of lognormal distributions, whichare plotted by straight lines in this case.

Coliforms. The results of approximation by lognor-mal distribution are presented in Fig. 2a. The distribu-tion parameters in all cases were determined by theleast-squares method. Two branches of the distributioncan be distinguished; these correspond to the summerand winter seasons, respectively. The change from onebranch to the other takes place at the coliform popula-tion N = 800 units. Deviations from these regularitiestake place either at very low concentrations (less than20 units) or at very high concentrations (more than6000 units). The populations as low or high as those arevery rare, and the very scanty data make it impossibleto draw a conclusion regarding the character of distri-bution of N in these domains.

Each branch in Fig. 2a can be described by a distri-bution of (5) type but with different parameter values(Table 1).

Interestingly, the value of the dimensionless com-plex kt ≈ 6.6 is the same for both branches of the distri-bution, that is, it does not change during a year. If weassume that the resulting rate of bacterial populationgrowth k varies insignificantly within a year, the migra-tion time t must remain the same. We can concludefrom here that the sources of bacterial pollution that actduring the year are the same and related with the dis-charges of certain industrial, agricultural, or municipalfacilities.

Φ x( ) 1

2π---------- e t

2/2– td

∞–

x

∫=

The fluctuation intensity σ of the rate of increase inbacterial population in summer is 2.5 times greater thanin winter. This is difficult to derive from Fig. 1a but isclear when logarithmic scale is used, as can be seenfrom Fig. 3. The variation of logarithm of coliform pop-ulation below the horizontal line (i.e., in summer) isequal to two orders, while that above the line (in winter)is less than one order of magnitude.

Sulfite-reducing clostridia. Figure 2b gives anapproximation by lognormal distribution with twobranches. The first branch lies in the domain of low val-ues of the index (<10 units) and corresponds toautumn–winter season. Because of the lack of data inthis field, we can only say that this branch exists but itsquantitative characteristics are unreliable. The secondbranch lies within the interval of 10–100 units andembraces all seasons. As can be seen from Table 1, thevalue of kt complex for clostridia (the second branch ofthe distribution) is about 1.9 times less than forcoliforms. If we assume that the source of supply ofthese microorganisms to water is the same (whichimplies the same migration time t), the obtained differ-ence between kt values for them can be explained onlyby the lesser resultant rate of clostridia populationincrease as compared with coliforms. This conclusionregarding k parameter is confirmed by the fact that theclostridia population is an order of magnitude less thanthat of coliforms as can be seen from comparison ofFigs. 1a and 1b (clearly, after both microorganism pop-ulations are converted to the same dimensions).

The fluctuation intensity σ of clostridia growth ratein the autumn–winter period is 3.8 times as large as thatin other periods.

Total microbial count. As it was with coliforms,the distribution of bacterial population has two distinctbranches (Fig. 2c). One of these accounts for the periodof rains and spring flood, and the other describes low-flow periods (winter and summer). The intersection of

Table 1. Parameters of distributions (R2 is the determination coefficient)

Index Period α β kt σ R2

Coliforms Summer (N = 20–800) 0.573 3.786 6.61 0.480 0.998

Winter (N = 800–6000) 1.442 9.592 6.65 0.190 0.998

Clostridia Autumn–winter (N < 10) 0.261 2.521 9.66 0.872 0.967

All seasons (N = 10–100) 1.655 5.754 3.48 0.229 0.985

Total microbial count

Low-flow period (N = 20–160) 2.231 10.285 4.61 0.148 0.990

Rains, spring floods (N >160) 0.896 3.528 3.94 0.398 0.996

Coliphages Summer (N < 10) 0.289 1.607 5.56 1.038 0.929

Winter, spring floods (N = 10–250) 0.919 3.185 3.47 0.413 0.994

Streptococci Summer (N = 1–300) 0.443 1.946 4.39 0.761 0.991

Winter, spring floods (N = 300–2300) 1.186 6.099 5.14 0.263 0.988

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WATER RESOURCES Vol. 33 No. 6 2006

DOLGONOSOV et al.

4

2

0

–2

–41 10 100 1000 10000

Total coliforms, CFU/100 m

1

2

3

2

1

0

–1

–2

–31 10 100 1000

Sulfite-reducing clostridia, CFU/20 m

1

2

4

2

0

–2

–410 100 1000 10000

TMC, CFU/m

1

2

(‡)

(b)

(c)

Fig. 2. Probability distribution of microorganism population. The abscissa is the inverse normal distribution. Distribution in thedomain of (1) low and (2) high values of indices.

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MODEL OF FLUCTUATIONS IN BACTERIOLOGICAL INDICES 645

branches corresponds to the bacterial population of 160units. The values of distribution parameters are given inTable 1. Unlike coliforms, the values of kt complex varywith changing environmental conditions within 3.9–4.6(±8% with respect to the mean). This is due to varia-tions in the values of parameter k because of the heter-ogeneity of bacteria included in the total microbialcount and the different response of these bacteria tochanges in the environmental conditions.

The fluctuation intensity σ of the growth rate in theperiod of rains and spring flood is 2.7 times as large asthat in low-flow periods.

Coliphages. The probability distribution of coli-phage population is given in Fig. 2d. The lower branchof their distribution is unreliable because of the lack ofdata. The main branch lies in the population interval of10–250 units where most of data are concentrated.Because of the trophic relationship between coliphages

and coliforms, both species should originate from thesame pollution source. Since the value of kt complexfor coliphages is less than that for coliforms (Table 1),the resultant growth rate of coliphage population alsoshould be less than that for the host cells. This is con-firmed by the comparison of populations of thesemicroorganisms, which, as can be seen from Figs. 1a,1d, differ by an order of magnitude (the number ofcoliforms is greater).

The fluctuation intensity σ of the coliphage popula-tion (on the main branch of distribution) is close to thatfor coliforms in summer.

Fecal streptococci. As in previous cases, the distri-bution of their population can be closely approximatedby a lognormal law with two branches, the first ofwhich corresponds to the summer season and the sec-ond, to winter season and spring floods (Fig. 2e). Thetransfer from one branch to another takes place at

3

2

1

0

–1

–21 10 100 1000

Fecal streptococci, CFU/100 ml

1

2

Coliphages, PFU/100 ml

2

1

0

–1

–21 10 100 1000 10000

1

2

(d)

(e)

Fig. 2. (Contd.)

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WATER RESOURCES Vol. 33 No. 6 2006

DOLGONOSOV et al.

N = 300 units. The values of distribution parameters aregiven in Table 1.

Variations in kt complex over seasons around themean annual value are narrow (±8%). This means thatsources of bacterial pollution within the year are thesame (domestic discharges and discharges from stockfarms).

The fluctuation intensity σ of the growth rate of bac-terial population in summer is 2.9 times as large as thatin winter and during spring floods, which is very simi-lar to that observed in the case of coliform. Indeed, vari-ations in the streptococci population is less than anorder of magnitude in summer and more than twoorders of magnitude in winter (as becomes evident inlogarithmic scale as it was made for coliforms).

PREDICTION OF THE EXCEEDANCE PROBABILITY OF ADMISSIBLE LEVELS

Prediction Procedure

The established regularities allow us to pose theproblem of predicting the level of bacterial pollution.Formally, this means that the probability p = P{X > N}that the bacterial index X exceeds the specified limit Nis to be found. This can be readily obtained from theknown distribution function

(6)p 1 F N( ).–=

In the case of lognormal distribution (5), we findfrom here

(7)

where erfc is the complementary error function. Thevariable p is commonly referred to as exceedance prob-ability.

The exceedance probability can be interpreted as theproportion of time p within which the random variableexceeds the specified limit N. If a year is considered, thetotal duration of unfavorable (above the limit) periodswithin this period is equal to T = 365p days. The knowl-edge of the periods of above-limit pollution of the watersource is of critical importance for planning water pro-tection measures. Once it is determined within whattime the above-limit values of bacterial indices willexist, it becomes possible to plan the necessary engi-neering measures and create the reserves of reagents atthe water station.

When designing treatment facilities for water sup-ply systems, it is important to know for what load (interms of source water quality) these facilities are to berated. The principal condition is to ensure the requiredpurification degree under extremely unfavorable sourcewater quality. However, if a facility is rated for veryhigh but extremely unlikely load, the investments to itsconstruction will be unreasonably high. Therefore, it is

p12---erfc α N β–( )ln( ),=

Total coliforms, CFU/100 ml

1000

100

10

Oct

. 1, 2

001

Apr

.19,

200

2

Nov

. 5, 2

002

May

24,

200

3

Dec

. 10,

200

3

June

27,

200

4

Jan.

13,

200

5

Aug

. 1, 2

005

Fig. 3. Population of coliform bacteria in the logarithmic scale. The line at the level of 800 units divides the two branches of theprobability distribution that are shown in Fig. 2a.

WATER RESOURCES Vol. 33 No. 6 2006

MODEL OF FLUCTUATIONS IN BACTERIOLOGICAL INDICES 647

expedient to establish the least admissible exceedanceprobability p and next find the threshold values Np ofquality indices for the source water for which the treat-ment facilities have to be rated. Np is referred to asp-quantile of the distribution and can be found, in thegeneral case, by solving equation (6) with respect to N,and, in particular case of lognormal distribution, fromequation (7).

Example of the Application of the Procedure

The exceedance probability of different bacteriolog-ical indices for the Moskva River Water Source isshown in Fig. 4. Note the satisfactory agreementbetween the calculation results and empirical data as anindirect point in favor of the model principles.

Specified thresholds (limits) for some indices werealready mentioned above. Let us consider the exceed-ance probability of these thresholds and evaluate themean number of days in the year when these standardsare violated.

In the case of total coliforms, calculations show(Fig. 4a) that the exceedance probability of the speci-fied threshold N = 1000 units is p = 0.36, and the totalduration of such exceedance T = 128 days per year.

A more strict threshold was specified for thermotol-erant coliforms N = 100 units. The probability of viola-tion of this standard can be evaluated from the relation-ship of p vs N for total coliforms (Fig. 4a), since theoverwhelming majority of coliforms in the MoskvaRiver Water Source are thermotolerant species. Calcu-lations show that this limit can be exceeded with a prob-ability of p = 0.87. Therefore, the period of above-limitwater pollution in terms of this index will averageT = 318 days per year.

In the case of coliphages, the probability that thespecified limit N = 10 units will be exceeded isp = 0.86; therefore, the mean duration of the above-limit pollution (Fig. 4d) will amount to T = 314 days peryear.

The results presented here should be regarded asmean annual. Deviations in either side can take place inreality.

As mentioned above, the high levels of pollution inthe design process are convenient for prediction withthe use of quantiles. The quantiles calculated for severalspecified thresholds p based on the lognormal distribu-tion found in this study are given in Table 2. Dependingon what value of p is taken as the least admissibleexceedance probability, the calculations for structuresand processes should be carried out based on the valuesof indices that are given in the appropriate line ofTable 2. For example, suppose that p = 0.001. Con-verted for the year, this value yields the duration ofabove-limit pollution of 0.365 day or about 9 h. Withinthis period, water supply can be effected at the expenseof water stored in reservoirs. At this exceedance thresh-old, the choice of the process scheme and the calcula-

tion of treatment facilities should be carried out for theload given in the bottom line of Table 2.

CONCLUSIONS

Bacteriological indices of water quality are consid-ered. Specific features of the behavior of bacterial pol-lution in the process of its migration in the aquatic envi-ronment are identified and used for the construction ofmodel.

Based on the equation of population dynamics, amodel of fluctuations of the population of microorgan-isms in the aquatic environment in the process ofmigration is proposed. The obtained stochastic differ-ential equation describes a stochastic process with mul-tiplicative noise and leads to a lognormal probabilitydistribution for microorganism population.

Processing of time series of bacteriological charac-teristics shows the theoretical distributions to be inagreement with empirical data in a wide range of indexvalues, which embraces one to three orders of magni-tude.

A procedure is developed for probabilistic predic-tion of the bacterial pollution level in a water source.Quantiles of bacteriological index distributions ofwater quality are calculated for specified exceedancethresholds.

The application of the model to the analysis of waterquality in the Moskva River shows that the bacterialpollution of the water source is inadmissibly high,except for short-time periods, usually in summer. Thespecified limit is exceeded on the average within 128days per year for total coliforms, within 318 days forthermotolerant coliforms, and within 314 days forcoliphages. The maximum exceedance factor of thelimit in individual periods (mostly during floods)reaches 6 for total coliforms, 60 for thermotolerantcoliforms, and 28 for coliphages. To improve the envi-ronmental situation, it is necessary to develop an effec-tive protection system of the water source, which willensure a substantial decrease in the anthropogenic loadonto the catchment area, the reclamation of anthropo-genically disturbed landscapes, the removal of hazard-

Table 2. Quantiles Np for distributions of bacterial indices(in measurement units accepted for each index)

p Coliforms Clostridia TMC Coliph-ages

Strepto-cocci

0.1 1885 75 214 128 495

0.05 2430 98 322 190 670

0.01 3840 152 687 390 1280

0.005 4630 181 910 515 1460

0.001 6600 255 1630 890 3770

648

WATER RESOURCES Vol. 33 No. 6 2006

DOLGONOSOV et al.

1.0

0.8

0.6

0.4

0.2

010 100 1000 10000

Total coliforms, CFU/100 ml

1.0

0.8

0.6

0.4

0.2

01 10 100 1000

Sulfite-reducing clostridia, CFU/20 ml

1.0

0.8

0.6

0.4

0.2

010 100 1000 10000

TMC, CFU/ml

(‡)

(b)

(c)

Fig. 4. The exceedance probability of bacteriological indices of water quality. The ordinate is the exceedance probability, theabscissa is microorganism concentration.

WATER RESOURCES Vol. 33 No. 6 2006

MODEL OF FLUCTUATIONS IN BACTERIOLOGICAL INDICES 649

ous plants, thorough treatment of wastewater, the pre-vention of construction in water protection zones, andthe rehabilitation of disturbed aquatic ecosystem.

ACKNOWLEDGMENTS

This study was supported by the Russian Founda-tion for Basic Research, project no. 06–05–64464.

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