DEVELOPMENT OF MATHEMATICAL MODEL TO PREDICT THE TRANSPORT OF E.COLI IN A NATURAL POND

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 7, Issue 1, Jan-Feb 2016, pp. 324-336, Article ID: IJCIET_07_01_027

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ISSN Print: 0976-6308 and ISSN Online: 0976-6316

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DEVELOPMENT OF MATHEMATICAL

MODEL TO PREDICT THE TRANSPORT OF

E.COLI IN A NATURAL POND

Eluozo, S. N

Subaka Nigeria Limited, Port Harcourt, Rivers State of Nigeria

F.E.Ezeilo

Department of Civil Engineering,

Rivers State University of Science and Technology,

Port Harcourt, Rivers State, Nigeria

ABSTRACT

Development of mathematical model to predict the rate of microbial

depositions (E.coli) in a natural pond has been carried out. The models were

developed to monitor the rate of concentration at different periods, with

respect to the length of the pond at various sample station. Results of the

theoretical values were compared with the experimental analysis. The analysis

was thoroughly done to determine the physiochemical parameters of the pond.

Microbial traces were found from the experimental analysis at different

periods up to hundred days. The developed model compared favourably well

with the experimental values. The values explain the rate of microbial growth

and level of lag phase condition. The growth rate of the microbes were found

to be higher because there is high deposition of substrate for growth and

energy, while at some periods it degrades showing that the substrates have

reduced in concentration including the inhibition from the pH. In some cases

when the microbes developed lag phase condition it may be as a result of

other environmental factors. Finally, the growth rates are between fifty and

hundred days, showing that there is constant regeneration of the microbes

including other environmental factors. This condition calls for regulation of

waste dump at the study location. This will reduce the concentration of the

microbes. The pollution that deposited at the pond should be remediated to

prevent the death rate of aquatic habitat that may not favour some marine

habitats. The pollution will reduce the surface water pH and cause more harm

to the marine habitats.

Key words: Mathematical Model E.Coli and Natural Pond

Development of Mathematical Model To Predict The Transport of E.Coli In A Natural Pond


Cite this Article: Eluozo, S. N and F.E.Ezeilo, Development of Mathematical

Model To Predict The Transport of E.Coli In A Natural Pond, International

Journal of Civil Engineering and Technology, 7(1), 2016, pp. 324-336.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=7&IType=1

1. INTRODUCTION

Pond is known to be man-made or natural water body which is between 1 m2 and 2 ha

(~5 acres or 20,000 m2) in area, this types holds water for four months of the year or

more (Biggs et al., 2005). The superiority of water usually means the constituent

which must be present for finest growth of aquatic organisms. The determinant of

high-quality growth in water body includes dissolved oxygen, hardness, turbidity,

alkalinity, nutrients, temperature, etc. Conversely, other parameters like biological

oxygen demand, and chemical oxygen demand indicate contamination stage of a

given water body. In most water bodies, a variety of chemical parameters, these occur

in low concentrations. This concentration level increases due to human behavior, and

lack of environmental instruction (Mishra, 1991). The fish ponds in several areas

experience lots of biodegradations, pollutions are generated from different activities

of man, consequently, it begins to reduce the constituent of the water bodies that

affect the marine habitats, and some published data reflect adverse effects at

concentrations higher than acceptable limit (GESAMP, 1985). The productivity

depends on physiochemical characteristics of the water body (Huct, 1986). There is

dearth of information on the fish ponds. The purpose of this present investigation was

to determine the values of the major physiochemical parameters of fish ponds and it’s

environ. Furthermore, to determine if there is any build up of toxic substances which

could lead to bio-accumulation and magnification leading to health implications.

(Ehiagbonare and Ogunrinde 2010) Majority of water obtainable on the earth is saline

in nature; only small amount is fresh water. Freshwater has become a panic product

due to over exploitation and pollution (Ghose and Basu 1968; Gupta and Shukle;

2006; Patil and Tijare, 2001; Singh and Mathur, 2005). contamination is caused when

a vary in the physical, substance or biological situation in the environment

destructively affect quality of human life as well as other animals’ life and plant

(Lowel and Thompson, 1992; Okoye et al., 2002). Industrial, sewage, metropolitan

wastes are been continuously added to water bodies hence influence the

physiochemical quality of water making them unfit for use of domestic animals and

other organisms (Dwivedi and Pandey, 2002). Uncontrolled domestic waste water

release into pond as resulted in eutrophication of ponds as confirmation by

considerable algal bloom, dissolve oxygen reduction in the subsurface water leads to

large fish kill and other oxygen requiring organism (Pandey, 2003) Effluent is release

into environment with improved concentration of nutrient, sediment and toxic

substances may have a severe negative collision on the value and life forms of the

getting water body when discharge untreated or partially treated (Forenshell, 2001;

Miller and Siemmens 2003; Schulz and Howe, 2003). Water pollution by overflow

has become a question of substantial public and scientific worry in the light of

evidence of their extreme toxicity to human health and to biological ecosystems

(Katsuro et al., 2004). The incidence of heavy of metals in industrial and municipal

sewage effluents constitute a major source of the heavy metals entering aquatic

media. Hence there should be regular evaluation of these sewage effluents to ensure

that adequate measures are taken to reduce pollution level to the minimum.

Worldwide water bodies are primary means for disposal of waste, especially the

effluents from industrial, municipal sewage and agricultural practices that are near

Eluozo, S. N and F.E.Ezeilo


them. This effluent can alter the physical, chemical, and biological nature of receiving

water body (Sandoyin, 1991). The initial effect of waste is to degrade physical quality

of the water. Later biological degradation becomes evident in terms of number,

variety and organization of the living organism in the water (Gray, 1989).

2. MATERIALS AND METHOD

Sample at different station were collected at a natural pond, the same were in batch

reactor to culture the growth of the microbes at different period physiochemical

analysis were carried out determine the concentration of the microbes at various time,

procedures for the analysis are stated bellow.

Bacteriological Methodology: Membrane filtration. Testing of Water (WHO, 1993,

1996, 1998)

Principle of Method: A 100ml water sample was filtered through membrane filters.

The membranes, with the coliform organism (E. coli) on it, are then cultured on a pad

of sterile selective broth containing lactose and an indicator. After incubation, the

number of colonies of coliform (E. coli) were counted. This gives the presumptive

number of E. coli in the 100ml water sample.

Choice of Technique: The method is recommended for its accuracy, speed of result,

and because it can be performed in the field.

3. REQUIRED

1. Sterile filtration unit for holding 47mm diameter membrane filters with suction

device (wagteck international)

2. Sterile grid membrane filters of 47mm diameter with a pore size of 0.45um (oxide).

3. Sterile 47mm diameter cellulose pads (both culture medium to be added just before

use).

4. Sterile Petri dishes 50-60mm diameter

5. Sterile membrane lauryl sulphate broth (lactose sodium lauryl sulphate broth)

6. Autoclaving unit, blunt ended forceps, sterile bottles, grease pencil, incubator at 44oc,

Bunsen burner, Petri-dish holders and oblique light source.

4. PROCEDURE

1. Assembling the Filtration Unit: The sterile broth is aseptically added to the

cellulose pad in a Petri-dish. The membrane filter is aseptically removed from the

sterile pack using a flame sterilized blunt forceps and placed on the filter base with

the grid-side uppermost and centrally. Next, the filter lid was screwed into place.

2. Suction Filtration of water sample: 100ml of the different water samples were

thoroughly mixed by inverting the bottles several times and gently poured into the

assembled filtration unit.

The water was drawn into the filter membrane by suction using the hand held

pressure pump.

A blunt-ended forceps was sterilized by naked Bunsen flame, cooled and the

membranes were aseptically removed from the filtration unit after unscrewing the lid

of the filtration unit.

The membranes were placed, grid-side uppermost, on the culture medium pads in the

Petri-dishes, ensuring there were no air bubbles trapped under the membranes.



The Petri-dishes were closed and the top of the lids were labeled with the code

numbers of the water samples and volumes of water used using a grease pencil.

3. Incubation of Samples: The Petri-dishes were packed in a Petri dish holder with lids

uppermost and placed inside the incubator at 44oc for 12 – 16 hours. Examination,

count and calculation of E.coli colonies:

Following incubation and using oblique lighting, the membranes were examined one

after the other for yellow lactose fermenting colonies, 1-3mm in diameter. The

number of colonies if any was counted. Any plink and small colonies less than 1mm

in diameter were ignored. Number of colonies too numerous to count were reported

as “too numerous to count” (indicative of gross contamination).

To calculate the presumptive E. coli count/100ml water sample, the number of

colonies counted per membrane was multiplied by 1

Developed Mathematical Model

(1)

(2)

(3)

(4)

VV )(

1x

t

tK

(5)

t

tKxC

VV o

xln)(ln

)(

(6)

V

VK

t

t

t

t

V

VK

C

Cx

oo

x

x

x

lnlnln

)(

)(

)(

(7)

V

KV

t

t

C

Cx

ox

x

o

)(

)(

(8)

VV

ttK

x

xx

o

oC

C

ln

)(

)(

(9)

VV

tK

x

x x

oC

C 1ln

)(

)(

(10)

otK

xC1ln

)(

(11)



Where

(12)

This implies that the contaminant is more proportional to time.

The model can be applied in waste dump site. But considering the equation in a

condition of Batch System of a pond

tvVxxC

tvVxxC )()(

(13)

Taking Laplace transform of the equation

S

V

VoC

V

SVCo o

o)(

)(

(14)

C (o) V (o) + S C (o) V = V

C (o) V (o) + S C (o) V - 0V (15)

Applying quadratic expression we have

Where a = VO, b = Sv and –Bv

=> v

VSSvC x

2

422

)(

2

42

)(

SSvC x

(16)

But V

VC x)(for S in (16) give

2

42

)(2

)(

)(

V

V

V

VC

C

xx

x

(17)

=> 2

412

2

)(

V

V

C

x

x

(18)

=>

tt

AC x

2

411

2

411

)(

(19)

o

o

tK

xC1ln

)(



Integrating the boundary condition in this condition at t = 0, C(o) = (0), t = (0)

Therefore equation (2) becomes

0 = A + B => A = - B

If A = 1 => B = 1

Therefore the model equation can be expressed as

tt

xC

2

41112

411

)(

(20)

But V

Lt

(21)

5. RESULTS AND DISCUSSION

Table 1 Comparison of theoretical and experimental values at various distances

Distance Theoretical values Experimental values

1.5 1.40E-03 1.80E-03

3 3.00E-03 3.26E-03

4.5 4.76E-03 4.42E-03

6 6.78E-03 6.75E-03

7.5 8.93E-03 8.36E-03

9 1.10E-02 1.19E-02

10.5 1.30E-02 1.37E-02

12 1.60E-02 1.58E-02

13.5 1.90E-02 1.85E-02

15 2.00E-02 2.02E-02

Table 2 comparison of Theoretical and experimental values at various Time

Time Theoretical values Experimental

values

10 1.40E-03 1.80E-03

20 3.00E-03 3.26E-03

30 4.76E-03 4.42E-03

40 6.78E-03 6.75E-03

50 8.93E-03 8.36E-03

60 1.10E-02 1.19E-02

70 1.30E-02 1.37E-02

80 1.60E-02 1.58E-02

90 1.90E-02 1.85E-02

100 2.00E-02 2.02E-02

V

LV

L

xC

2

41112

411



Table 3 comparison of theoretical and experimental values at various distances

Distance Theoretical values

(different velocity Experimental values

1.5 2.13E-04 2.80E-04

3 4.43E-04 4.26E-04

4.5 3.40E-05 3.42E-04

6 4.89E-05 4.75E-04

7.5 3.00E-05 3.36E-04

9 4.31E-05 1.19E-04

10.5 6.10E-03 6.37E-03

12 1.60E-02 1.58E-02

13.5 2.30E-02 2.85E-02

15 2.10E-02 2.02E-02

Table 4 comparison of theoretical and experimental values at various Time

Time Theoretical values (different

velocity Experimental values

10 2.13E-04 2.80E-04

20 4.43E-04 4.26E-04

30 3.40E-05 3.42E-04

40 4.89E-05 4.75E-04

50 3.00E-05 3.36E-04

60 4.31E-05 1.19E-04

70 6.10E-03 6.37E-03

80 1.60E-02 1.58E-02

90 2.30E-02 2.85E-02

100 2.10E-02 2.02E-02

Table 5 comparison of theoretical and experimental values at various Time

Time Theoretical values at

different velocity Experimental values

10 5.68E-04 4.88E-05

20 9.17E-04 3.23E-05

30 1.24E-03 2.66E-03

40 1.37E-03 3.45E-03

50 1.24E-03 2.66E-03

60 9.41E-04 9.51E-04

70 9.26E-04 9.51E-05

80 9.23E-04 1.31E-04

90 8.02E-04 8.12E-03

100 1.20E-01 1.17E-01



Table 6 comparison of theoretical and experimental values at various distances

Distance Theoretical values at

different velocity Experimental values

1.5 5.68E-04 4.88E-05

3 9.17E-04 3.23E-05

4.5 1.24E-03 2.66E-03

6 1.37E-03 3.45E-03

7.5 1.24E-03 2.66E-03

9 9.41E-04 9.51E-04

10.5 9.26E-04 9.51E-05

12 9.23E-04 1.31E-04

13.5 8.02E-04 8.12E-03

15 1.20E-01 1.17E-01

Figure I Comparison of theoretical and experimental values at various distances

y = -6E-06x3 + 0.0002x2 + 0.0001x + 0.0014 R² = 0.9976

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

0 5 10 15 20

Co

nce

ntr

atio

n M

g/l

Distance (MM)

Theoretical values

Experimental values

Poly. (Experimental values)



Figure 2 Comparison of theoretical and experimental values at various distances


y = -2E-08x3 + 4E-06x2 + 2E-05x + 0.0014 R² = 0.9976

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

0 20 40 60 80 100 120

Co

nce

ntr

atio

n M

g/l

Time Per Day

Theoretical values

Experimental values


y = -1E-05x4 + 0.0005x3 - 0.0047x2 + 0.0172x - 0.0176

R² = 0.9437

-5.00E-03

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

3.50E-02

0 5 10 15 20

Co

nce

ntr

atio

n M

g/l

Distance (mm)

Theoretical values (different velocity

Experimental values






y = -7E-09x4 + 2E-06x3 - 0.0001x2 + 0.0026x - 0.0176

R² = 0.9437

-5.00E-03

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

3.50E-02

0 20 40 60 80 100 120

Co

nce

ntr

atio

n M

g/l

Time Per Day

Theoretical values (different velocity

Experimental values


y = 0.0003x3 - 0.006x2 + 0.0346x - 0.0486 R² = 0.8454

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

0 5 10 15 20

Co

nce

ntr

atio

n M

g/l

Distance (m)

Theoretical values at different velocity

Experimental values





6. RESULTS AND DISCUSSION

Figure 1, shows that the natural pond regeneration of microbes observed a gradual

process, where the optimum values were recorded at fifteen metres from the station at

1.5 metres. The level of growth shows that there is high concentration of substrate that

may have caused the rapid growth of the microbes at that distance at the period of 50

days. Both parameters were found to compare favourably well with the experimental

values. While in figure 2 lag phase were experienced between 1.5 metres to 9 metres

from 10 days to 30 days and suddenly as the substrate regenerate, rapid growth were

recorded from 10.5 metres and 13.5 metres. The experimental result maintained the

same condition showing the level of comparison with the theoretical values. Finally

the concentration result of both parameters recorded 0.023mg/l, it also implies that the

PH values of the water were between the acceptable levels that may have allowed the

growth not inhibiting the microbes. Figure 3 developed some slight variation in the

microbes’ growth in the pond. This condition explains the variation in the constituent

of some physiochemical parameters which sometimes faviour the rate of

concentration. And also inhibit the growth rate of the microbes in the pond; both

values also compared favourably well. While figures 4 and 5 developed its lag phase

between 1.5 to 10.5 metres at the period of 10 to 90 days, it also suddenly increased in

growth rate at the distance between 13.5 metres at the period of 90 to 100 days. This

can be attributed to the rate of regeneration of the microbes and other constituent that

may give them energy for rapid increase and growth. The rate of PH level may also

y = 0.0003x3 - 0.006x2 + 0.0346x - 0.0486 R² = 0.8454

-2.00E-02

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

0 5 10 15 20

Co

nce

ntr

atio

n M

g/l

Time Per Day

Theoretical values at different velocity

Experimental values




contribute to the rate of growth if it cannot inhibit the microbes at that time. The

analytical model compared favourably well with the experimental values this proof

the authenticity of the predictive model simulated.

7. CONCLUSION

Development of mathematical model to predict the rate of microbial depositions

(E.coli) in a natural pond has been carried out. The behaviour of the microbes in the

pond was thoroughly explained from the results. High concentrations were found

between 9 to 13.5 metres at the period of 20 - 90 days, while the lag phase occurred

between 1.5 metres to 9 metres. The condition can be attributed to the rate of substrate

deposition in the pond while the figure with respect to time the lag was found between

10 to 50 days. The theoretical and experimental values compared favourably well.

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Documents

DEVELOPMENT OF MATHEMATICAL MODEL TO PREDICT THE TRANSPORT OF E.COLI IN A NATURAL POND