Pathogen Specific Persistence Modeling Data


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October 26, 2017

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Mitchell, J. and Akram, S. 2017. Pathogen Specific Persistence Modeling Data. In: J.B. Rose and B. Jiménez-Cisneros, (eds) Global Water Pathogen Project.  http://www.waterpathogens.org (M. Yates (eds) Part 4 Management of Risk from Excreta and Wastewater)  http://www.waterpathogens.org/book/pathogen-specific-persistence-modeling-data Michigan State University, E. Lansing, MI, UNESCO.  
https://doi.org/10.14321/waterpathogens.53

Acknowledgements: K.R.L. Young, Project Design editor; Website Design (http://www.agroknow.com); K. Dean, E. Willis and A. Wissler,  literature review and reviewed and organized data.  

Last published: October 26, 2017
Authors: 
Jade Mitchell (Michigan State University)Sina Akram (Michigan State University)

Summary

Persistence modeling facilitates the accurate simulation of different stages of growth, survival, and death of microorganisms in environmental matrices by describing the changes in population size of microorganisms over time. The most commonly used model for simulating persistence patterns of pathogens is the first order exponential one-parameter model. However, persistence curves for many microorganisms do not follow this classic linear trend, as evidenced by decades of studies across the growing field of predictive microbiology and for microbes in many different matrices. Therefore, it is essential for an evaluation of linear and non-linear curves (models) in order to provide an accurate description of both pathogen and matrix specific persistence (or inactivation).

Seventeen linear and nonlinear persistence models were used to find the best models for describing the persistence of water microbes - bacteria, viruses, bacteriophages, bacteroidales, and protozoa in human urine, wastewater, freshwater, marine water, groundwater matrices, biosolids and manure. A total number of 30 datasets were used in this study containing 180 different pathogen (or indicator)/matrix combinations to find the best fitting models through linear regression techniques to describe persistence subject to various conditions. Like the exponential decay model, these models contain general parameter(s) to mathematically describe the relationship between reductions in microbial populations with time. The models do not contain explanatory variables to isolate the effects of environmental conditions (i.e. temperature, UV exposure) on inactivation. Overall, three models (JM2, JM1, and Gamma) were found to be the best fitting models across the entire data set and represented 59%, 34% and 25% of the data sets respectively. JM2 fit the persistence data the best across environmental matrices except in human urine, and groundwater in which JM1 performed the best at describing the persistence patterns. Across pathogens, JM2 was the best model for bacteria, bacteriophages, and bacteroidales. However, viruses were best fit by JM1. The models which best describe the persistence pattern of each pathogen or indicator in a matrix under different treatments and their corresponding parameters are presented in this chapter. In addition, T90 and T99 values, which are commonly used to specify the time required for a pathogen concentration to decrease by one and two log units, respectively, is reported for all the datasets and compared between matrices and microorganisms types. While this metric is often used to describe pathogen persistence, it is only relevant in the linear region of the persistence curve and will be misleading if an incorrect model is assumed or a model other than the best fitting model is used to estimate these values. Therefore, the results in this chapter that contains the best fitting models and parameters along with the associated calculation of T90 and T99 for various pathogen/matrix combinations can reduce uncertainty in estimations of pathogen population size in water environments over time.

1.0 Introduction

Mathematical models are commonly used to describe the microbial inactivation of pathogens persisting in environmental matrices and to predict population sizes for subsequent human health risk calculations, which may lead to treatment decisions. Persistence modeling facilitates the accurate simulation of the stages of growth, survival, and death of microorganisms in different matrices by describing the mathematical relationship between the population size of microorganisms and time through regression techniques to estimate kinetic parameters (constants). While T90 and T99 values are commonly used to indicate the time required for a pathogen concentration to decrease by one and two log units, respectively, these metrics are only relevant in the linear region of a persistence curve and may be misleading if more complex persistence patterns accurately describe a specific pathogen in a specific matrix (i.e. curves with shoulders or tails). The significance of utilizing mathematical models to describe kinetics is in their ability to describe the persistence pattern of specific pathogens in different environments enabling engineers and policy makers to predict the absolute microorganism populations at any given time through interpolation or extrapolation, not just the population size relative to the initial conditions like the T90 or T99.

The most commonly used model for simulating persistence patterns of pathogens is the first order exponential one-parameter model, which was originally developed to describe the inactivation of chemical disinfectants (Chick 1908). However, as predictive microbial modeling has developed as a filed over many years, persistence curves for microorganisms in a number of environments were observed that do not follow this classic linear pattern. .Therefore, accurate description of non-linear persistence curves is essential. A common explanation for this non-linearity in persistence is that the population of microorganisms may consist of several sub-populations, each with different inactivation kinetics. Along with linear curves, curves with “shoulder” (a delay before attenuation begins), curves with “tailing” (attenuation slows with time), and sigmoidal curves (both a shoulder and a tailing) are the four most typically observed models for bacterial decay (Xiong et al. 1999). Shoulders in curves represent the smooth initial inactivation, while tailing can represent an intrinsic resistance of some microorganisms or that they are protected by various factors. Figure 1 shows a schematic representation of different patterns of persistence of microorganisms.

Figure 1. Schematic representation of four different persistence patterns

As the first order model cannot describe more complex persistence patterns such as shoulders, tailing, and sigmoidal curves noted above, several other models were developed and tested over time.

Table 1 shows the list of the microbial persistence models, which were utilized for the studies described in this chapter as well as their corresponding equations. These models are mostly empirical and have three or fewer model parameters.

Table 1. Models utilized in this study

Model

Abbreviation

Equationa

Reference

Exponential

Ep

$$\frac{Nt}{N0}=exp(-k_{1}t)$$

Chick, 1908

Logistic

lg1

$$\frac{Nt}{N0}=\frac{2}{1+exp(-k_{1}t)}$$

Kamau et al., 1990

Fermi

lg2

$$\frac{Nt}{N0}=\frac{1}{1+exp(-k_{1}(t-k_{2}))}$$

Peleg, 1995

Exponential damped

Epd

$$\frac{Nt}{N0}=10^{exp(-k_{1}t\times exp(-k_{2}t))}$$

Cavalli-Sforza et al., 1983

Juneja and Marks 1

JM1

$$\frac{Nt}{N0}=1-(1-exp(-k_{1}t))^{k_{2}}$$

Juneja et al., 2001

Juneja and Marks 2

JM2

$$\frac{Nt}{N0}=\frac{1}{1+exp(k_{1}+k_{2}log(t))}$$

Juneja et al., 2006

Gompertz 2

Gz

$$\frac{Nt}{N0}=exp\left [ \frac{-k_{1}}{k_{2}}exp((k_{2}t)-1) \right ]$$

Wu et al., 2004

Weibull

Wb

$$\frac{Nt}{N0}=10^{-((t/k_{1})^k_{2})}$$

Peleg, 2003

Lognormal

ln

$$\frac{Nt}{N0}=1-\left \{ (ln(t)-k_{1}/k_{2}) \right \}$$

Aragao et al., 2007

Gamma

Gam

$$\frac{Nt}{N0}=exp\left \{(t^{k_{1}-1})exp^{(\frac{-t}{k_{2}})} \right \}$$

van Gerwen and Zwietering, 1998

Broken-line

Bi

$$\frac{Nt}{N0}=exp(-k_{1}t), t<k_{3}$$

Muggeo, 2003

Broken-line 2

Bi2

$$\frac{Nt}{N0}=exp(-k_{1}t+k_{2}(t-k_{3})), t \geq k_{3}$$

Muggeo, 2003

Double exponential

Dep

$$\frac{Nt}{N0}=k_{3}exp(-k_{1}t)+(1-k_{3})exp(-k_{2}t)$$

Gerard Abraham et al., 1990

Gompertz 3

Gz3

$$\frac{Nt}{N0}=10^{k_{1}exp\left [ -exp(\frac{-k_{2}exp(1)(k_{3}-t)}{k_{1}}+1)\right ] }$$

Gil et al., 2011

Gompertz-Makeham

Gzm

$$\frac{Nt}{N0}=10^{(-k_{3}t-k_{1}/k_{2}(exp(k_{2}t)-1))}$$

Jodrá, 2009

Sigmoid type A

sA

$$\frac{Nt}{N0}=10^{( (k_{1}t) / \left \{ (1+k_{2}t)(k_{3}-t) \right \} )}$$

Peleg, 2006

Sigmoid type B

sB

$$\frac{Nt}{N0}=10^{- (k_{1} \times t^{k_{3}}) / (k_{2}+t^{k_{3}}) }$$

Peleg, 2006

aNt denotes the number of organisms remaining at time t; N0 denotes the initial number of organisms; k1, k2, and k3 are model parameters. Model parameters, ki, are constants in each model that have different units depending on the model.
 

2.0 Best-Fitting Models

A literature review was conducted as described in the previous chapter, “Persistence of Pathogens in Sewage and Other Water Types”. However, it was also expanded to include papers with die-off data presented additional water matrix terms: sludge, urine and manure. For this analysis, raw data of microorganism concentration vs. time was obtained from the original author of the peer-reviewed journal article or digitized from the figures in the publications. A total of 95 studies contained acceptable data for modeling, which contained 304 individual data sets describing a pathogen and matrix combination under various environmental conditions (urine, n=9; freshwater, n=131; wastewater, n=16; biosolids, n=42; marine, n=60; groundwater, n=46). While the environmental conditions - temperature, the presence of UV light or indigenous microbiota, for example - are known to influence decay rates, the focus of this analysis is not to describe empirical data sets with explanatory variables through regression modeling as this is reported in the original papers. The purpose of this chapter is to summarize pathogen specific decay rates using generalized persistence models which best-fit decay data sets for specific water environments under specific conditions.

Seventeen previously established linear and nonlinear persistence models were evaluated to determine the best models for describing the persistence of different bacteria, viruses, bacteriophages, bacteroidales, and protozoa in human urine, wastewater, biosolids and manure, freshwater, marine water, groundwater matrices. The chapter includes a selected representation of pathogens, marker or indicator data sets across the 6 environmental matrices noted above. A total number of 30 studies are summarized in this chapter which includes 180 different pathogen or indicator/matrix combinations. Indicator bacteria are typically used to detect the level of fecal contamination in environments and are generally not pathogenic to human health while pathogens are microorganisms which can produce diseases. Table 2 shows the best fitting models for all collected data. JM2, JM1, and gamma models were the best fit for 58.7%, 33.7% and 25.0% of the experimental data better than the other tested models respectively. As JM2 fit the persistence data the best in all matrices except in human urine, and groundwater where JM1 performed the best at describing the persistence patterns.

Table 2. Best fitted models to different matrices

Matrices

Modela

Percent of the combinations for which the model was the best fit

All Dataset

JM2

58.7

JM1

33.7

Gam

25.0

Epd

20.7

Dep

20.2

Ep

19.0

ln

18.5

Human Urine

JM1

66.7

Ep

44.4

Epd

33.3

Wastewater

JM2

50.0

Gam

33.3

ln

33.3

Freshwater

JM2

76.0

JM1

34.7

Epd

21.3

Marine Water

JM2

50.0

Dep

34.1

Gam

25.0

Groundwater

JM1

50.0

Dep

50.0

ln

50.0

Manure & Biosolids

JM2

47.5

JM1

35.0

lg1

30.0

aModels described in Table 1

Table 3 shows the best fitting models for different pathogen and indicator types. JM2 was the best model for bacteria, Bacteriophages, and bacteroidales. For viruses, JM1 best described the persistence curves.

Table 3. Best fitted models to different types of microorganisms

Microbe Group

Modela

Percent of the Combinations for Which the Model was the Best Fit

Bacteria

JM2

54.0

JM1

31.0

Gamma

21.8

Bacteriophages

JM2

61.4

JM1

38.6

Gamma

29.5

Viruses

JM1

57.1

Gamma

52.4

JM2

52.4

Bacteroidales

JM2

74.2

ln

29.0

JM1

29.0

aModels described in Table 1

Table 4 shows the best fitting models for specific pathogens and indicators. The JM2 model best described the persistence curves for E.Coli, Enterococci, Salmonella, and HF183. The persistence was best described for MS2 by the JM1 model, and exponential model was the best fitting model for Adenovirus.

Table 4. Best fitted models to specific pathogens and indicators

Indicator or Pathogen Type

Number of Pathogen and Indicator/Matrix Combinations

Modela

Percent of the Pathogen and Indicator/Matrix Combinations for Which the Model was the Best Fit

E. coli

26

JM2

65.4

Ln

30.8

JM1

26.9

Enterococci

12

JM2

50.0

Ep

33.3

JM1

25.0

Salmonella

18

JM2

31.8

Gzm

27.3

Epd

18.2

HF183b

9

JM2

100.0

JM1

33.3

Gam

11.1

Adenovirus

7

Ep

57.1

JM2

42.9

JM1

42.9

MS2 bacteriophage

15

JM1

80.0

Epd

53.3

Gam

53.3

aFor model description see Table 1;  bBacteroides human source tracking gene target

3.0 Summary of T90 and T99 data

The time that the concentration of a type of microorganism in a specific environment decreases by one and two log units is called T90 and T99 respectively. These values are calculated and summarized using the best fitting models and their corresponding parameters for every pathogen (or indicator)/matrix combination. The predicted number of days needed to achieve 90% and 99% decay rates (T90 and T99) of pathogens and indicators in different matrices are summarized in Tables 5 and 6. The tables also show the range of variations and the standard deviations in T90 and T99 values calculated for every pathogen (or indicator)/matrix combination.

Table 5. Range, average and standard deviations in T90 and T99 values in different matrices

Matrix

Number of Pathogen
and Indicator/Matrix
Combinations

 

T90 Days

T99 Days

Human Urine

9

Range

0.1 to 2.6

0.3 to 5.1

Average (SDa)

0.7 (0.8)

1.4 (1.5)

Wastewater

11

Range

0.1 to 62.5

0.2 to 85.9

Average (SD)

21.1 (25.6)

32.6 (33.6)

Freshwater

72

Range

0.1 to 125.8

0.4 to 278.5

Average (SD)

15.6 (29.4)

31.8 (60.8)

Marine Water

34

Range

0.1 to 127.8

0.2 to 160.8

Average (SD)

12.1 (24.3)

23.5 (36.1)

Groundwater

5

Range

0.1 to 20.8

0.7 to 99.8

Average (SD)

11.6 (8.4)

54.0 (40.3)

Manure & Biosolids

38

Range

0.2 to 120.7

0.5 to 290.6

Average (SD)

8.6 (19.6)

19.3 (46.7)

aSD: Standard Deviation

Table 6. Range, average and standard deviations in T90 and T99 values for different pathogen or indicator types

Pathogen or
Indicator

Number of Pathogen
and Indicator/Matrix
Combinations

 

T90 Days

T99 Days

Bacteria

71

Range

0.1 to127.8

0.2 to 290.6

Average (SDa)

11.9 (23.5)

23.9 (51.3)

Bacteroidalesb

33

Range

0.4 to 17.2

0.9 to 77.0

Average (SD)

2.9 (3.0)

7.5 (13.9)

Bacteriophage

43

Range

0.1 to 125.8

0.3 to 251.6

Average (SD)

17.9 (33.2)

33.4 (62.0)

Protozoa

2

Range

11.4 to 55.2

69.8 to 85.9

Average (SD)

33.3 (31.0)

77.9 (11.3)

Viruses

18

Range

0.1 to 56.4

0.4 to 121.7

Average (SD)

13.4 (17.4)

36.6 (35.5)

aSD: Standard Deviation; bGene persistence

4.0 Persistence Modeling in Pathogen/Matrix Combinations

The results from the different studies on the persistence of various types of pathogenic human bacteria, viruses, bacteroidales, protozoa, and bacteriophages in different environments of human urine, wastewater, freshwater, marine water, groundwater, and biosolids were collected to find the best-fit mathematical models for different pathogen (or indicator)/matrix combinations.

4.1 Persistence Modeling in Human Urine

The model parameters for different pathogen combinations in a human urine matrix along with the corresponding fitted parameters are presented in Table 7. Pathogens studied consisted of adenovirus and MS2 bacteriophage. Data were obtained from a yet unpublished study conducted by Dr. Tamar Kohn from Swiss Federal Institute of Technology in Lausanne.

Table 7. Best fitted models and fitting parameters for MS2 bacteriophage (virus indicator) and adenovirus in human urine matrix

Virus Type

T90 (T99) Days

Temp

̊C

Best Fit Modela

k1

k2

Data

Pointsb

Other Factors

Other

Modelsc

Adenovirus

0.2 (0.4)

35

JM2

9.77

5.36

10

NR

None

MS2

2.6 (5.1)

20

Depd

-0.42

0.90

10

pH=8.47; NH3=15.8

None

MS2

1.1 (1.8)

35

JM1

3.42

4.74

8

pH=8.15; NH3=19.1

None

MS2

0.8 (1.6)

35

Ep

2.92

NA

8

pH=8.19; NH3=24.6

JM1

MS2

0.9 (1.7)

35

JM1

3.10

1.73

4

pH=8.19; NH3=24.4

Epd, Gam, Bi3, lg1

MS2

0.2 (0.4)

35

JM1

11.62

0.81

6

pH=8.72; EC=33.6; NH3=81

Ep, Epd

MS2

0.1 (0.3)

35

JM1

10.16

0.22

6

pH=8.79; EC=33.0; NH3=106

Gam

MS2

0.5 (1.0)

35

JM2

4.45

3.71

8

pH=8.49; EC=16.0; NH3=28.2

Bi3, Ep, Epd

MS2

0.3 (0.7)

35

Ep

6.95

NA

10

pH=8.48; EC=33.6; NH3=71.1

JM1, Gam

aFor model description see Table 1; bNumber of data points modeled in the experiment; cOther models that provided an equally statistical best-fit; dBest fit a model with three parameters, where k3= 0.00000001

4.2 Persistence Modeling in Wastewater

The model parameters for different pathogen combinations in wastewater matrices along with the corresponding fitting parameters are presented in Table 8. Studied pathogens consisted of bacteria, viruses, bacteroidales, and protozoa and matrices consisted of treated and untreated wastewaters.

Table 8. Best fitted models and fitting parameters for pathogens in wastewater matrices

Agent

T90 (T99) Days

Temp

̊C

Best Fit Modela

(Other Modelsb)

k1

k2

Data Pointsc

Other Factors

Reference

INDICATOR or PATHOGEN

Enterococci

13.0 (60.1)

10

JM2

(Ep, Epd, ln)

-1.82

1.57

11

Light

Walters et al., 2009

Enterococci

1.5

(2.8)

13

lg1

(Ep)

1.90

NA

9

Light

Walters and Field, 2009

E. coli

9.4 (12.6)

6

Gz3e

-9.50

0.40

7

NRd

Czajkowska et al., 2008

E. coli

4.5

(5.8)

24

JM2

-12.55

9.76

7

NR

Czajkowska et al., 2008

Salmonella Thompson

9.7 (19.3)

25

Ep

0.24

NA

14

NR

Ravva and Sarreal, 2014

Salmonella enterica

62.5 (63.1)

15

sAf

-703.9

-541.2

14

Darkness

Boehm et al., 2012

Salmonella enterica

0.12 (0.22)

15

JM2

(ln, JM1,Gam)

 

 

35

Light

Boehm et al., 2012

Humbacg

1.1

(2.0)

10

JM2

1.76

4.20

11

Light

Walters et al., 2009

PROTOZOA AND VIRUSES

Cryptosporidium parvum

55.2 (85.9)

5.2

Gam

9.69

3.27

7

Sunlight

Jenkins et al., 2013

Norovirus

19.1 (26.6)

20

JM2

(Gz3, Gam, ln, JM1, Sb)

-18.98

7.19

4

NR

Skraber et al., 2009

Norovirus

56.4 (80.1)

4

JM1

(JM2, Gam, ln)

0.10

28.18

4

NR

Skraber et al., 2009

aFor model description see Table 1; bOther models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dNR: Not Reported; eBest fit a model with three parameters, where k3=7.74; fBest fit a model with three parameters, where k3=63.8; gA gene target

4.3 Persistence Modeling in Manure and Biosolids

The model parameters for different pathogen combinations in different manure and biosolids matrices along with the corresponding fitting parameters are presented in Table 9. Studied pathogens consisted of bacteria and viruses, bacteroidales, and bacteriophages and matrices consisted of different biosolid types of composted manure, sludge, manure, and freshwaters contaminated with feces.

Table 9. Best fitted models and fitting parameters for indicators and pathogens in manure and biosolids matrices

Agent

T90

(T99)

Days

Temp

̊C

Best Fit Modela

(Other Modelsb)

k1

k2

Data Pointsc

Other Factors

Reference

BACTERIA

E.coli

30.3 (46.5)

20

JM1

(ln, Gam, JM2)

0.14

8.43

6

NRd

Klein et al., 2011

E.coli

4.5 (14.0)

37

JM1

(Dep)

0.23

0.23

5

NR

Klein et al., 2011

E.coli

0.4 (1.8)

50

JM2

(JM1)

3.52

1.67

6

NR

Klein et al., 2011

E.coli

5.0 (7.4)

24

Ln

1.14

0.37

6

NR

Czajkowska et al., 2008

E.coli

8.3 (13.4)

6

Ln

1.54

0.45

6

NR

Czajkowska et al., 2008

Clostridium sporogenes

120.7 (290.6)

20

Gam

(Ep, lg1, Epd, JM1, JM2, ln)

85.39

0.53

4

Composted manure

Klein et al., 2011

Clostridium sporogenes

25.0 (50.0)

37

Ep

(lg1, Epd, JM1, Gam)

0.09

N/Ae

4

Composted manure

Klein et al., 2011

Clostridium sporogenes

11.5 (23.0)

50

Ep

(Epd, JM1, lg1)

0.20

N/A

5

Composted manure

Klein et al., 2011

Clostridium sporogenes

3.1 (6.1)

60

Depg

(Gzm)

-0.12

0.75

5

Composted manure

Klein et al., 2011

Listeria monocytogenes

30.3 (37.0)

20

JM1

(lg2, JM2, Gam)

0.35

3981.15

6

NR

Klein et al., 2011

Listeria monocytogenes

7.3 (14.5)

37

Ep

(JM1, lg1, JM2, Gam)

0.31

NA

6

NR

Klein et al., 2011

Listeria monocytogenes

4.3 (6.6)

50

Epd

0.33

-0.12

4

NR

Klein et al., 2011

Enterococci

4.5 (5.6)

13

jm2

-14.93

11.32

11

In Dark Treatment

Walters and Field, 2009

Enterococci

1.6 (2.8)

13

lg1

(Ep)

1.90

N/A

12

In Light Treatment

Walters and Field, 2009

Enterococci

8.9 (17.4)

Ambient temperature

Gz3h

(Dep)

-2.56

0.15

9

Shading

Oladeinde et al., 2014

Enterococci

7.9 (13.4)

Ambient temperature

Gz3i

-2.40

0.25

9

No Shading

Oladeinde et al., 2014

BACTERIA SOURCE TRACKING GENE MARKERSf

CF128 DNA

1.6 (3.2)

13

Dep

(Epd)

-0.57

1.45

29

In Dark Treatment

Walters and Field, 2009

CF128 DNA

2.6 (3.3)

13

JM2

(ln, Ep, lg1)

-6.63

9.38

24

In Light Treatment

Walters and Field, 2009

CF128 RNA

1.4 (2.9)

13

JM2

(Wb, ln)

1.15

3.21

18

In Dark Treatment

Walters and Field, 2009

CF128 RNA

0.6 9 (1.1)

13

JM2

(ln, JM1)

4.20

3.51

19

In Light Treatment

Walters and Field, 2009

CF193 DNA

1.4 (2.8)

13

Ep

(lg1, Bi3)

1.66

N/A

40

In Dark Treatment

Walters and Field, 2009

CF193 DNA

0.6 (1.3)

13

Ln

-1.23

0.63

24

In Light Treatment

Walters and Field, 2009

CF193 DNA

3.1 (4.1)

Ambient temperature

Gam

(ln)

0.21

10.31

6

NR

Liang et al., 2012

CF193 RNA

0.4 (0.9)

13

JM2

(ln, Wb)

5.06

3.43

25

In Dark Treatment

Walters and Field, 2009

CF193 RNA

0.6 (0.9)

13

JM2

(ln)

5.16

5.06

25

In Light Treatment

Walters and Field, 2009

CF193 RNA

4.7 (77.0)

Ambient temperature

Ln

(JM1, JM2, Gam, Bi3, lg1)

0.22

0.28

5

NR

Liang et al., 2012

Genbac

5.8 (13.2)

Ambient temperature

Depj

0.05

0.41

10

NR

Oladeinde et al., 2014

Cow M3

3.6 (14.5)

Ambient temperature

JM2

-0.01

1.72

10

No shading

Oladeinde et al., 2014

Cow M3

4.5 (14.2)

Ambient temperature

Depk

0.05

0.54

10

Shading

Oladeinde et al., 2014

Rum-2-Bac

2.4 (8.2)

Ambient temperature

JM2

0.48

1.96

10

No shading

Oladeinde et al., 2014

Rum-2-Bac

4.6 (11.6)

Ambient temperature

Depl

(JM2)

0.05

0.52

10

Shading

Oladeinde et al., 2014

BACTERIOPHAGE AND VIRUSES

MS2

0.6 (1.3)

35

Ep

(Epd, Gam, JM1, JM2, Wb)

3.63

N/A

7

Manure, pH=8.08;

ECl=2.0 (mS/cm);

NH4=99 (mM)

Decrey and Kohn, 2017

MS2

0.2 (0.5)

35

Epd

(JM1, JM2, Gam)

11.94

0.33

9

Manure, pH=8.05;

EC=4.6 (mS/cm);

NH4=505 (mM)

Decrey and Kohn, 2017

MS2

3.2 (3.6)

35

Bi3m

(Epd)

1.75

-3.28

7

Manure, pH=8.23;

EC=1.7 (mS/cm);

NH4=121(mM)

Decrey and Kohn, 2017

MS2

3.1 (7.3)

35

Ln

(Epd, JM1, JM2, Gam)

0.10

0.81

7

Sludge, pH=7.76;

EC=6.7 (mS/cm);

NH4=28 (mM)

Decrey and Kohn, 2017

MS2

5.7 (10.0)

35

lg1

(JM1, Gam)

0.54

2.28

11

Sludge, pH=7.76;

EC=6.7 (mS/cm);

NH4=28 (mM)

Decrey and Kohn, 2017

Adenovirus

13.7 (27.4)

35

Ep

(lg1)

0.17

N/A

13

Manure, pH=8.23;

EC=1.7 (mS/cm);

NH4=121(mM)

Decrey and Kohn, 2017

Adenovirus

2.4 (4.8)

35

Ep

(lg1)

0.97

N/A

10

Sludge, pH=7.76;

EC=6.7 (mS/cm);

NH4=28 (mM)

Decrey and Kohn, 2017

aFor model description see Table 1; bOther models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dNot reported; eN/A not applicable as model only has one parameter; fBelonging to the order of the Bacteroidales; gBest fit a model with three parameters, where k3=0.000001; hBest fit a model with three parameters, where k3=2.14; iBest fit a model with three parameters, where k3=3.84; jBest fit a model with three parameters, where k3=0.01; kBest fit a model with three parameters, where k3=0.02; lBest fit a model with three parameters, where k3=4.17; mEC Electrical conductivity

4.4 Persistence Modeling in Freshwater

The model parameters for different pathogen combinations in different freshwater matrices along with the corresponding fitting parameters are presented in Tables 10a,10b, and 10c. Studied pathogens consisted of bacteria, viruses, bacteroidales, bacteriophages, and protozoa and matrices consisted of different freshwater types of lakes and rivers.

Table 10a. Best fitted models and fitting parameters for bacteria indicators and pathogens in freshwater matrices

Indicator or

Pathogen

T90

(T99)

Days

Temp

ºC

Best Fit Modela

(Other Models)b

k1

k2

Data Pointsc

Other Factors

Reference

E. coli

0.01

(0.4)

14.1

JM1

(Gam, JM2)

0.16

0.00

4

Indigenous Microbiota and Ambient Sunlight

Korajkic et al., 2014

E. coli

6.6

(9.6)

14.1

JM2

(JM1, ln, Gam, Epd)

0.79

20.42

4

Indigenous Microbiota

Korajkic et al., 2014

E. coli

2.7

(6.6)

14.1

Dep

(Ep, Epd, JM2)

-3.97

0.85

4

Ambient Sunlight

Korajkic et al., 2014

E. coli

0.01

(0.4)

14.1

lg1

(Ep, JM1, ln)

0.25

N/Ad

4

Without a Treatment

Korajkic et al., 2014

E. coli

11.8

(21.2)

25

Ln

(Epd, JM2)

3.80

0.14

5

Control

Dick et al., 2010

E. coli

1.1

(2.0)

25

JM2

1.63

4.32

5

Light

Dick et al., 2010

E. coli

1.2

(2.4)

25

JM2

1.65

3.33

5

Sediment

Dick et al., 2010

E. coli

1.1

(2.7)

15

Epd

(JM2)

2.24

0.10

5

N/Ad

Dick et al., 2010

E. coli

3.2

(6.1)

25

JM2

-2.11

3.72

5

Reduced Predation

Dick et al., 2010

Entero1ae

3.8

(5.4)

14.1

JM2

(Gam, JM1)

-7.19

6.99

4

Indigenous Microbiota and Ambient Sunlight

Korajkic et al., 2014

Entero1ae

4.1

(5.6)

14.1

JM2

(JM1, Gamma, lg1)

-8.76

7.74

4

Indigenous Microbiota

Korajkic et al., 2014

Entero1ae

3.9

(5.5)

14.1

JM2

-6.77

6.65

4

Ambient Sunlight

Korajkic et al., 2014

Entero1ae

4.2

(5.6)

14.1

JM2

(JM1)

-8.97

7.83

4

Without a Treatment

Korajkic et al., 2014

Enterococci

0.9

(2.0)

14.1

Epd

(JM1, JM2, Gam)

2.63

0.07

4

Indigenous Microbiota and Ambient Sunlight

Korajkic et al., 2014

Enterococci

2.5

(3.7)

14.1

JM2

(Ep, Lg1)

-3.53

6.14

4

Indigenous Microbiota

Korajkic et al., 2014

Enterococci

0.6

(3.0)

14.1

JM1

(Gam)

0.80

0.11

4

Ambient Sunlight

Korajkic et al., 2014

Enterococci

4.0

(6.1)

14.1

JM2

(JM1)

-6.03

5.88

4

Without a Treatment

Korajkic et al., 2014

Fecal streptococci

2.6

(2.8)

24

Epd

0.0002

-3.15

6

Phosphate buffered freshwater with sunlight

Fujioka et al., 1981

Salmonella enterica

80.1

(82.1)

15

sAf

(Dep, Gzm, sB)

-820.2

-201.7

16

Dark

Boehm et al., 2012

Bacteroides fragilis

17.2

(31.0)

10

 

0.17

NA

5

NRg

Balleste and Blanch, 2010

aFor model description see Table 1; bOther models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dN/A not applicable as model only has one parameter; eMolecularly based fecal indicator bacteria; fBest fit a model with three parameters, where  k3=84.2; gNR: Not Reported

Table 10b. Best fitted models and fitting parameters for bacterial indicator and microbial source tracking genes in freshwater matrices

Agent

T90 (T99) Days

Temp

̊C

Best Fit Modela (Other Models)b

k1

k2

Data Pointsc

Other Factors

Reference

BACTERIA

ENT 23s

23.5 (278.5)

41

JM2 (ln, Ep, Epd)

-0.87

0.97

11

In Dark Treatment

Walters et al., 2009

ENT 23s

13.0 (60.1)

41

JM2 (Ep, ln, Epd)

-1.82

1.57

11

In Light Treatment

Walters et al., 2009

BACTERIA SOURCE TRACKING GENETIC MARKERSd

GenBac3

3.5 (4.4)

14.1

JM2 (JM1)

-10.65

10.30

4

Indigenous Microbiota and Ambient Sunlight

Korajkic et al., 2014

GenBac3

3.5 (4.5)

14.1

JM2 (JM1)

-9.84

9.57

4

Indigenous Microbiota

Korajkic et al., 2014

GenBac3

3.6 (4.8)

14.1

JM2

-8.30

8.23

4

Ambient Sunlight

Korajkic et al., 2014

GenBac3

3.8 (5.1)

14.1

JM2

-9.30

8.57

4

Neither

Korajkic et al., 2014

HF183

3.2 (3.9)

14.1

JM2

-12.26

12.29

4

Indigenous Microbiota and Ambient Sunlight

Korajkic et al., 2014

HF183

3.3 (4.1)

14.1

JM2 (JM1)

-9.86

10.22

4

Indigenous Microbiota

Korajkic et al., 2014

HF183

3.2 (4.0)

14.1

JM2

-10.01

10.58

4

Ambient Sunlight

Korajkic et al., 2014

HF183

3.4 (4.4)

14.1

JM2 (JM1)

-10.01

9.92

4

Neither

Korajkic et al., 2014

HF183

0.5 (1.3)

25

JM2

3.92

2.37

5

Control

Dick et al., 2010

HF183

0.6 (1.4)

25

JM2

3.58

3.09

5

Light

Dick et al., 2010

HF183

0.6 (1.3)

25

JM2

3.88

2.95

5

Sediment

Dick et al., 2010

HF183

0.6 (2.0)

15

JM2 (Gam, JM1)

3.12

2.09

5

Reduced Temperature

Dick et al., 2010

HF183

1.5 (2.6)

25

JM2

0.60

4.24

5

Reduced Predation

Dick et al., 2010

HumM2

3.2 (4.3)

14.1

JM2

2.26

153.58

4

Indigenous Microbiota and Ambient Sunlight

Korajkic et al., 2014

HumM2

3.5 (4.4)

14.1

JM2 (JM1)

-10.27

10.04

4

Indigenous Microbiota

Korajkic et al., 2014

HumM2

3.2 (4.0)

14.1

JM2

-10.15

10.55

4

Ambient Sunlight

Korajkic et al., 2014

HumM2

3.5 (4.5)

14.1

JM2 (JM1)

-10.37

9.96

4

Neither

Korajkic et al., 2014

BifAd

4.07(8.26)

Ambient temperature

JM2

-2.56

3.39

7

NRe

Jeanneau et al., 2012

Humbac

0.66(21.06)

Ambient temperature

Epd

3.80

0.14

4

NR

Dick et al., 2010

B. thetaiotamicron

6.7 (13.3)

10

Ep (JM1, Gam, ln, JM2)

0.34

NA

8

NR

Balleste and Blanch, 2010

aFor model description see Table 1; bOther models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dBelonging to the order of the Bacteroidales; eNR: Not Reported

 

Table 10c. Best fitted models and fitting parameters for bacteriophage and viruses in freshwater

Agent

T90

(T99)

Days

Temp

̊C

Best Fit Modela

(Other Models)b

k1

k2

Data Pointsc

Reference

VIRUSES

Enterovirus

12.9

(25.7)

37

Ep

(JM1, lg1, Epd)

0.18

N/Ad

11e

Walters et al., 2009

Enterovirus

4.3

(21.2)

37

JM1

(Gam)

0.12

0.11

10f

Walters et al., 2009

Rotavirus

31.1

(60.3)

15

JM2

(Epd, Bi)

-10.27

3.63

8

Espinosa et al., 2008

BACTERIOPHAGE

F+DNA, f1

32.2

(60.6)

4

JM2

-11.00

3.80

12

Long and Sobsey, 2004

F+DNA, f1

2.7

(5.4)

20

Depg

0.18

0.85

9

Long and Sobsey, 2004

F+DNA, fd

31.1

(50.5)

4

JM2

-14.86

4.96

10

Long and Sobsey, 2004

F+DNA, fd

3.5

(5.5)

20

JM2

-4.72

5.47

7

Long and Sobsey, 2004

F+DNA, M13

104.8

(188.4)

4

lg1

(Ep, ln, JM2, JM1)

0.03

N/A

10

Long and Sobsey, 2004

F+DNA, M13

13.4

(28.9)

20

Epd

(JM2, Gam)

0.18

0.00

11

Long and Sobsey, 2004

F+DNA, OW

104.8

(188.4)

4

lg1

(Ep, ln, JM2, JM1, Gam)

0.03

N/A

12

Long and Sobsey, 2004

F+DNA, SD

9.8

(16.1)

4

JM2

-8.92

4.87

9

Long and Sobsey, 2004

F+DNA, SD

2.0

(3.0)

20

JM2

-1.82

5.89

5

Long and Sobsey, 2004

F+DNA, ZJ2

37.0

(69.3)

4

JM2

-11.63

3.83

11

Long and Sobsey, 2004

F+DNA, ZJ2

3.3

(6.7)

20

Depi

(JM2)

0.00

0.69

7

Long and Sobsey, 2004

F+RNA, Dm3

1.7

(6.6)

4

JM1

(Gam, Dep, Epd)

0.43

0.16

8

Long and Sobsey, 2004

F+RNA, Dm3

32.8

(42.3)

20

Epd

3.62

0.10

7

Long and Sobsey, 2004

F+RNA, Go1

125.8

(251.6)

4

Ep

(Bi, JM2)

0.02

N/A

11

Long and Sobsey, 2004

F+RNA, Go1

8.0

(13.3)

20

JM2

(Gz3)

-7.63

4.73

11

Long and Sobsey, 2004

F+RNA, MS2

109.8

(197.4)

4

lg1

0.03

N/A

12

Long and Sobsey, 2004

F+RNA, MS2

9.1

(18.9)

20

Dep

0.09

0.26

11

Long and Sobsey, 2004

F+RNA, SG1

69.0

(138.1)

4

Bi

(Epd, JM2, Wbl, ln,Gam, JM1)

0.03

0.03

12

Long and Sobsey, 2004

F+RNA, SG1

7.2

(13.1)

20

JM2

-5.80

4.04

10

Long and Sobsey, 2004

F+RNA, SG4

2.5

(5.0)

4

Depj

(Gz3)

0.00

0.92

8

Long and Sobsey, 2004

F+RNA, SG4

0.4

(0.9)

20

JM2

4.77

2.77

5

Long and Sobsey, 2004

F+RNA, SG42

1.4

(3.4)

4

JM2

1.32

2.69

9

Long and Sobsey, 2004

F+RNA, SG42

0.5

(1.0)

20

JM2

4.75

4.08

3

Long and Sobsey, 2004

F+RNA, sp2

19.3

(37.3)

4

JM2

(Dep)

-8.56

3.63

12

Long and Sobsey, 2004

F+RNA, sp2

4.2

(8.5)

20

Depk

0.05

0.54

9

Long and Sobsey, 2004

FRNAPH

3.76

(16.62)

18.5

JM2

0.06

1.61

3

Jeanneau et al., 2012

aFor model description see Table 1; bOther models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dN/A not applicable as model only has one parameter; eExperiment conducted in the dark; fExperiment conducted in the light; gBest fit a model with three parameters, where k3=0.0008; iBest fit a model with three parameters, where k3=0.000005; jBest fit a model with three parameters, where k3=0.00017; kBest fit a model with three parameters, where k3=0.00005

4.5 Persistence Modeling in Marine Water

The model parameters for different pathogen combinations in different marine water matrices along with the corresponding fitting parameters are presented in Table 11. Studied pathogens consisted of bacteria and viruses, and matrices consisted of different marine water types of seawater, laboratory prepared saltwater and isolated estuarine waters.

Table 11. Best fitted models and fitting parameters for indicators and pathogens in marine water matrices

Agent

T90

(T99) Days

Temp

̊C

Best Fit Modela

(Other Models)b

k1

k2

Data Pointsc

Other Factors

Reference

BACTERIA

Clostridium perfringens

2.1 (3.6)

22-24

Gz3e

-2.41

0.91

11

Beach Sand
Microcosms

Zhang et al., 2015

Clostridium perfringens

2.0 (3.9)

22-24

JM2

(Dep)

-0.16

3.51

11

Seawater
Microcosms

Zhang et al., 2015

E. coli

1.4 (2.6)

20

JM2

0.70

4.00

5

NRd

Chandran and Hatha, 2005

E. coli

0.4 (5.1)

30

JM2

2.99

0.98

5

NR

Chandran and Hatha, 2005

E. coli

1.1 (2.1)

Ambient temperature

JM2

1.70

3.79

5

Sunlight

Chandran and Hatha, 2005

E. coli

21.7 (22.9)

27

Depf

-0.02

-0.03

12

NR

Miyagi et al., 2001

E. coli

20.0 (22.6)

27

Depg

0.02

0.00

12

NR

Miyagi et al., 2001

E. coli

38.0 (68.3)

21

lg1

(Bi2)

0.08

N/A

20

No UV radiation

Beckinghausen et al., 2014

E. coli

19.5 (33.1)

21

JM2

(JM1, ln, Gam, Wb, sB, lg1)

-11.26

4.53

7

UV radiation and
algae association

Beckinghausen et al., 2014

E. coli

9.8 (13.6)

21

Epd

(Dep)

0.48

0.07

7

UV radiation and no algae association

Beckinghausen et al., 2014

E. coli

3.8 (8.9)

Ambient temperature

JM2

(Ep, ln, lg1)

-1.65

2.86

6

Water column

Korajkic et al., 2013

E. coli

13.8 (39.3)

Ambient temperature

JM2

(ln, Ep, sB, JM1, Gam, lg1)

-3.83

2.30

6

Sediments

Korajkic et al., 2013

Enterococci

5.3 (10.7)

22-24

Deph

-0.29

0.43

11

Beach Sand
Microcosms

Zhang et al., 2015

Enterococci

2.0 (2.9)

22-24

JM2

-2.45

6.60

11

Seawater
Microcosms

Zhang et al., 2015

Fecal streptococci

5.9 (10.2)

24 ± 2

JM1

(Gam, lg1, Dep, JM2)

0.54

2.44

5

Without Sunlight

Fujioka et al., 1981

Fecal streptococci

2.3 (2.7)

24 ± 2

sBi

(Wb, Gam)

53402.75

1026397.77

5

With Sunlight

Fujioka et al., 1981

Salmonella Heidelberg

0.1 (0.2)

15

Ep

27.79

N/A

21

In Light Treatment

Boehm et al., 2012

Salmonella Mbandaka

7.9 (13.4)

15

Depj

(Bi, Gzm)

0.01

0.01

14

In Dark Treatment

Boehm et al., 2012

Salmonella Mbandaka

0.1 (0.2)

15

lg1

(JM2)

22.66

N/A

22

In Light Treatment

Boehm et al., 2012

Salmonella Typhimurium

127.8 (160.8)

15

Epd

0.00

-0.01

13

In Dark Treatment

Boehm et al., 2012

Salmonella Typhimurium

1.8 (2.6)

20

sBk

2.77

17.34

5

NR

Chandran and Hatha, 2005

Salmonella Typhimurium

0.4 (1.0)

30

JM2

(JM2)

4.56

2.88

5

NR

Chandran and Hatha, 2005

Salmonella Typhimurium

0.8 (1.7)

Ambient temperature

Ep

(JM2, Epd)

2.76

N/A

5

Sunlight

Chandran and Hatha, 2005

Salmonella Typhimurium

103.9 (113.3)

43

jm1

0.25

18201386478

50

Seawater w/o
Alginic Acid

Davidson et al., 2015

Salmonella Typhimurium

61.9 (70.4)

43

jm2

-75.18

18.75

45

Seawater with
Alginic Acid

Davidson et al., 2015

Salmonella Typhimurium

108.0 (121.0)

43

Epd

0.00

-0.04

60

Filtered Seawater
w/o Alginic Acid

Davidson et al., 2015

Salmonella Typhimurium

132.5 (145.8)

43

Ln

(Wb, Gam, Bi3, Gz3, Gzm)

4.82

0.09

86

Filtered Seawater
with Alginic Acid

Davidson et al., 2015

Salmonella Typhimurium

100.9 (201.9)

12

Bi

(Bi3)

0.02

-0.36

44

Unfiltered seawater with Alginic acid

Davidson et al., 2015

Salmonella Typhimurium

61.9 (70.4)

12

JM2

(ln, JM1, Gam)

-75.18

18.75

44

Unfiltered seawater without Alginic acid

Davidson et al., 2015

Salmonella Typhimurium

41.3 (82.6)

21

Epd

0.22

0.10

20

No UV radiation

Beckinghausen et al., 2014

Salmonella Typhimurium

416.2 (832.5)

21

Gzml

(Dep)

0.01

-4*10-7

7

UV radiation and
algae association

Beckinghausen et al., 2014

Salmonella Typhimurium

15.7 (21.3)

21

lg2

0.43

10.55

7

No UV radiation and algae association

Beckinghausen et al., 2014

Vibrio fluvialis

3.9 (18.2)

Ambient temperature

JM2

(Dep, Bi3)

0.11

1.55

12

Sediment

Amel et al., 2008

Vibrio fluvialis

1.0 (2.0)

Ambient temperature

Dep

0.03

2.29

12

No Sediment

Amel et al., 2008

Helicobacter pylori

 

9.4 (16.8)

37

lg1

(Dep, Gam, JM1, JM2, ln)

0.06

-0.24

7

NR

Konishi et al., 2007

VIRUSES

Adenovirus

55.7 (121.7)

37

JM1

(Gam, JM2, Dep)

0.03

0.67

6

No UV irradiation

de Abreu Correa et al., 2012

Adenovirus

0.1 (4.6)

37

JM1

(Gam, Dep)

0.06

0.01

6

UV irradiation

de Abreu Correa et al., 2012
 

Hepatitis A

0.2 (55.6)

37

JM1

(Gam, JM2, Dep)

0.02

0.02

6

No UV irradiation

de Abreu Correa et al., 2012
 

Hepatitis A

0.1 (9.6)

37

JM1

(Gam, JM2, Bi3, Dep)

0.02

0.01

6

UV irradiation

de Abreu Correa et al., 2012
 

Murine Norovirus-1

0.1 (31.2)

37

JM1

(Gam, ln, JM2, Epd)

0.02

0.01

6

No UV irradiation

de Abreu Correa et al., 2012
 

Murine Norovirus-2

1.7 (4.7)

37

JM2

0.87

2.39

4

UV irradiation

de Abreu Correa et al., 2012

Poliovirus

8.9 (17.9)

15

Ep

(Epd, JM1, Gam)

0.26

NR

5

NR

Enriquez et al., 1995

a For model description see Table 1; b Other models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dNot Reported; eBest fit a model with three parameters, where k3=0.97; fBest fit a model with three parameters, where  k3=6.94; gBest fit a model with three parameters, where k3=2.70; hBest fit a model with three parameters, where  k3=0.000002; iBest fit a model with three parameters, where  k3=3.61; jBest fit a model with three parameters, where k3=2.24; kBest fit a model with three parameters, where k3=4.0; lBest fit a model with three parameters, where  k3=0.000003; The T90 and T99 values were eliminated in decay rate analysis (section 2.0) due to considerably different experimental conditions.

4.6 Persistence Modeling in Groundwater

The model parameters for different pathogen combinations in different groundwater matrices along with the corresponding fitting parameters are presented in Table 12. Studied pathogens consisted of viruses, a protozoa, and a bacteriophage.

Table 12. Best fitted models and fitting parameters for pathogens in groundwater

Indicator or

Pathogen Agent

Group

T90

(T99)

Days

Temp

̊C

Best Fit Modela

(Other Models)b

K1

K2

Data Pointsc

Reference

Cryptosporidium

Protozoa

11.4

(69.8)

NRd

Depe

0.04

1.04

8

Sidhu and Toze, 2012

Adenovirus

Virus

18.1

(99.8)

NR

JM2

(ln, Epd, JM1, Gam, Dep, Ep)

-1.87

1.40

5

Sidhu and Toze, 2012

Echovirus

Virus

20.9

(75.2)

12

JM2

(Ep, ln)

-3.49

1.87

10

Yates et al., 1990

Poliovirus

Virus

7.5

(24.6)

12

Gam

(JM1, ln, Dep)

10.24

0.24

13

Yates et al., 1990

MS2

Bacteriophage

0.1

(0.7)

NR

JM1

(Wb)

0.95

0.01

6

Yates et al., 1990

aFor model description see Table 1; bOther models that provided an equally statistical best-fit; cNumber of data points modeled in the experiment; dNR: Not reported; eFit the model with three parameters K3=0.16

5.0 Models and Methodological Approach

As the first order model could not describe the persistence pattern of various pathogen/matrix combinations, several other models were developed and tested over time. The logistic model was developed originally to describe the sigmoid shaped decay curves in microbiology (Kamau et al. 1990). The Fermi model was applied originally to describe the influence of electric field intensity on the persistence of a microbial population (Peleg 1995). In addition, an exponentially damped polynomial model can be used to describe tailing survival curves (Cavalli-Sforza et al. 1983). Juneja and Marks developed two models, denoted here as JM1 and JM2, to describe the fate of foodborne pathogens in food processing operations (Juneja et al. 2003; Juneja et al. 2006). JM1 was mostly utilized to simulate the decay which had a convex shape and describe the tail in decay curves over long periods. JM2 was frequently used to simulate the non-linear persistence curves of thermal inactivation rates. Different variations of Gompertz models, denoted Gz2 and Gz3, were developed to predict the number of microorganisms persisting under stressed environmental conditions (Wu et al. 2004; Gil et al. 2011). Along with the logistic model, Gompertz functions are frequently used to fit sigmoidal kinetics (Membre et al. 1997). Gz2 and Gz3 models can describe log-linear kinetics, shoulders, and tailing effects (Zwietering et al. 1990; Chhabra et al. 1999). Weibull and lognormal models are commonly used for thermal and non-thermal disinfection in different matrices. Weibull model can predict linear, concave, and convex and sigmoidal curves (Coroller et al. 2006). The other commonly used persistence model is Gamma. This is a simple model with few parameters and is commonly being utilized to simulate the microbial persistence in water matrices under varying environmental conditions (van Gerwen & Zwietering 1998). The broken line models of Bi and Bi2 were originally developed to simulate multiple break-points in decay curves (Muggeo 2003). The double exponential model that was originally developed to simulate thermal inactivation is able to represent linear and biphasic persistence curves (Abraham et al. 1990). On the other hand, sigmoidal models of sA and sB are typically being used to describe concave inactivation curves (Peleg 2006).

This chapter specifies and discusses the most appropriate mathematical models to describe the persistence patterns of various types of pathogens in different matrices, and compares the best fit models and decay rates in the specific pathogen (or indicator)/matrix combinations.

Maximum likelihood estimation and Bayesian Information Criterion (BIC) values were used in order to assess the goodness of fit among the 17 persistence models which were fit to each pathogen (or indicator)/matrix combination. The models with the lowest absolute BIC values were selected as the best fitting models. Differences less than 2 in BIC values are not considered strong evidence for model selection. If the difference between the lowest BIC values of some models in a pathogen/matrix combination was less than 2, all these models were considered as the best fit models in this chapter.

BIC is defined as:

BIC = k ln (n) - 2 ln (Lm)

where k is the total number of parameters, n is the number of data points in the observed data (x), and Lm is the maximum likelihood of the model. Lm is defined as:

Lm = p(x|θ, Μ)

where θ are the parameter values which maximize the likelihood function, and M is the model used.

Raw data from different studies were obtained/extracted, analyzed, and fit to various persistence models using R statistical language (R Development Core Team, 2013). Data-capturing software, GetData Graph Digitizer (http://www.getdata-graph-digitizer.com), was used to digitize figures in the published papers where data values were not specifically stated.

References

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