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

http://www.iaeme.com/IJCIET/index.asp 324 editor@iaeme.com
International Journal of Civil Engineering and Technology (IJCIET)
Volume 7, Issue 1, Jan-Feb 2016, pp. 324-336, Article ID: IJCIET_07_01_027
Available online at
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=7&IType=1
Journal Impact Factor (2016): 9.7820 (Calculated by GISI) www.jifactor.com
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication
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 44o
c,
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 44o
c 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)
V
V
 )(
1
x 


t
t
K
(5)







t
t
KxCV
V o
x
ln)(ln
)(
(6)
V
V
K
t
t
t
t
V
V
K
C
C x
oo
x
x
x






 lnlnln
)(
)(
)(
(7)
V
KV
t
t
C
C x
ox
x
o







)(
)(
(8)
V
V
t
tK
x
x
x
o
o
C
C 






ln
)(
)(

(9)
V
V
tK
x
x x
o
C
C 1ln
)(
)( 
 
(10)
otK
xC
1ln
)(

 
(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
tv
VxxCtv
VxxC   )()(
(13)
Taking Laplace transform of the equation










 S
V
Vo
C
V
SV
Co o
o
)(
)(
(14)
C (o) V (o) + S C (o) V = V
C (o) V (o) + S C (o) V - 0V (15)
Applying quadratic expression we have
Where a = VO, b = Sv and –Bv
=> v
VSSv
C x
2
422
)(


2
42
)(


SSv
C x
(16)
But
V
VC x)(
for S in (16) give
2
42
)(
2
)(
)(

 V
V
V
VC
C
xx
x
(17)
=> 2
41 2
2
)(

 V
V
C
x
x
(18)
=>
tt
AC x







 








 


2
41
1
2
41
1
)(


(19)
o
o
tK
xC
1ln
)(

 

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
t
t
xC 






 







 

 2
41
112
41
1
)(



(20)
But V
L
t 
(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
41
112
41
1 



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
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
ConcentrationMg/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
ConcentrationMg/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
ConcentrationMg/l
Distance (mm)
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
ConcentrationMg/l
Time Per Day
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
ConcentrationMg/l
Distance (m)
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
ConcentrationMg/l
Time Per Day
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.
REFERENCES
[1] Biggs J, Williams P, Whitfield M, Nicolet P, Weatherby A (2005). 15years of
pond assessment in Britain: results and lessons learned from the work of pond
conservation. Aquatic conservation; Marine Freshwater Ecosystems. 15: 693-714.
[2] Huct M (1986). Textbook of Fish Culture 2nd Edn, Fish News Book Ltd,
England.vide Study on the physiochemical properties of Water of Mouri River,
Khulna Bangladesh, Pak. J. Biol. Sci. 10(5): 710-717
[3] GESAMP (1985). Review of a potentially harmful substances-cadium, lead and
tin IMO/FAO/UNESCO/WMO/IAEA/UN/UNEP Rep. Stud.22: p. 116.
[4] Mishra KD (1991). Impact of sewage and industrial pollution on physicochemical
characteristics of water in River Betwa at Vidisha, Madhya Pradesh, India. J.
Environ. Health, 25: 288-299
[5] Ehiagbonare .J E*and Ogunrinde, Y.O; (2010) Physico-chemical analysis of fish
pond water in Okada and its environs, Nigeria African Journal of Biotechnology
Vol. 9(36), pp. 5922-5928, 6
[6] Dwivedi, B.K.and G.C.Pandey (2002). Physico-chemical factors and algal
diversity of two ponds inFaizabad, India Poll.Res.21(3):361-370
[7] Forenshell, G. 2001. Setting basin design. Western Regional Aquaculture Center,
WRAC-106. USA: 6pp.
[8] Gray, A.V. 1989. Case study on water quality modelling of Dianchi lake, Yunnan
province, southwest China.Water Sci.Technol. 40:35–43.
[9] Miller and siemmens 2003 Anal. Chem., 31, 426-428
[10] Pandey, (2003). Trends in eutrophication research and control. Hydrol. Proc., 10
(2), 131-295 (165 pages).
[11] Patil Dilip B, Tijare Rajendra V (Dept Chem, Govt Sci Coll, Gadchiroli 442605).
Investigation of pollution mystery of suspected carcinogen Cr (VI) and its
control. J Indl Polln Contl, 17(1) (2001), 43-47 [4 Ref].
[12] Okoye, P. A. C., Enemuoh, R. E. and Ogunjiofor, J. C., 2002. Traces of heavy
metals in Marine crabs. J. Chem. Soc. Nigeria, vol. 27 (1) 76-77
[13] Schulz, K. and B. Howe, 2003. Uncertainity and sensistivity analysis of water
transport modeling in a layered siol profile using fuzzy set theory. J.
Hydroinform., 1: 127-138
[14] Katsuro, A., Yashiko, K., Yoshinnori, S., Takashi, T., and Hayao, S., 2004.
Heavy – metal Distribution in River Waters and sediment a “firefly village”,
Shikou, Japan: Application of multivariate Analysis. analytical science, 20, 79-
84.

[15] Sandoyin, 1991: Eutrophication trends in the water quality of the Rhode River. J.
Mar. Biol. Assoc., 54, 825-855 (31 pages).
[16] Lowel and Thompson F., 1992. Biodiversity of vibrios. Microbiol. Mol. Biol.
Rev., 68: 403-431
[17] Ghose F.; Basu, P. (1968). Eutrophication trends in the water quality of the
Rhode River. J. M Biol. Assoc., 54, 825-855 (31 pages).
[18] Gupta, G. K., Shukle, R., (2006). Physiochemical and Bacteriological Quality in
Various Sources of Drinking Water from Auriya District (UP) Industrial Area.
Pollution Research, 23 (4): 205-209.
[19] Eluozo. S. N and Ode T, Mathematical Model to Monitor Stiff Clay Compression
Index in Wet Land Area of Degema. International Journal of Advanced Research
in Engineering and Technology, 6(12), 2015, pp. 59-72.
[20] Eluozo. S. N and Ode T, Modeling and Simulation of Compression Strength for
Firm Clay in Swampy Area of Ahoada East. International Journal of Advanced
Research in Engineering and Technology, 6(12), 2015, pp. 73-85.
[21] Eluozo. S. N and Ode T, Mathematical Model to Predict Compression Index of
Uniform Loose Sand in Coastal Area of Degema, Rivers State of Nigeria.
International Journal of Advanced Research in Engineering and Technology,
6(12), 2015, pp. 86-103.

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

Recommended

Recommended

More Related Content

What's hot

What's hot (19)

Viewers also liked

Viewers also liked (19)

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

Similar to DEVELOPMENT OF MATHEMATICAL MODEL TO PREDICT THE TRANSPORT OF E.COLI IN A NATURAL POND (20)

More from IAEME Publication

More from IAEME Publication (20)

Recently uploaded

Recently uploaded (20)

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