Contents
 1 Pathogenic Escherichia coli
 1.1 Overview
 1.2 Summary of data and models
 1.3 Optimization Output for experiment 98
 1.4 Optimization Output for experiment 39
 1.5 Optimization Output for experiment 40
 1.6 Optimization Output for experiment 42
 1.7 Optimization Output for experiment 43
 1.8 Optimization Output for experiment 165
 1.9 Optimization Output for experiment 38, 39, 40, 42, 99, 144
 1.10 Optimization Output for experiment 214, 216, 217
 1.11 Optimization Output for experiment 142, 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, 172
 1.12 Optimization Output for experiment 38, 42, 99, 165
 1.13 Optimization Output for experiment 153, 157, 159, 214, 216, 217
 1.14 Optimization Output for experiment 154, 156, 158, 160, 219, 220, 221
 1.15 Optimization Output for experiment 39, 40
 1.16 Optimization Output for experiment 96, 100, 166
 1.17 References
Pathogenic Escherichia coli
Author: Kyle S. Enger
Overview
Escherichia coli typically resides as a symbiotic bacterium in the mammalian large intestine, benefiting itself as well as the host. However, there are several wellestablished pathotypes of diseasecausing E. coli^{[1]}^{[2]}:
 Enteropathogenic (EPEC)
 Attaches to small intestinal wall and produces ‘attaching and effacing lesions’, in which microvilli are destroyed and the bacteria become perched on pedestals on the surface of the epithelial cell. This ability is encoded on the locus of enterocyte effacement (LEE) pathogenicity island.
 Causes an inflammatory response and diarrhea, but seldom in persons older than 2y; it can also be isolated from healthy older persons.
 Primarily found in developing countries.
 Enterohemorrhagic (EHEC)
 This is discussed in more detail in its own chapter.
 Enterotoxigenic (ETEC)
 Attaches to small intestinal wall.
 Produces a heatlabile (LT) and/or heatstable (ST) toxins, both of which cause secretion from the small intestinal wall, leading to mild to severe watery diarrhea. LT is an immunogenic multisubunit protein similar to cholera enterotoxin. ST is a nonimmunogenic polypeptide containing 1819 amino acids.
 Primarily found in developing countries and is a major cause of diarrhea in weaned infants, as well as traveler’s diarrhea.
 Can be shed even by immune asymptomatic individuals.
 Enteroaggregative (EAEC)
 Loosely classified group, some of which may be nonpathogenic.
 Produces a thick biofilm (‘stacked brick’ configuration) in the small or large intestines.
 Thought to cause persistent diarrhea (lasting >14 days).
 Can produce many different secretory toxins and cytotoxins, but not ST or LT.
 Enteroinvasive (EIEC)
 Actually invades the epithelial cells of the large intestine, where it multiplies.
 Usually produces watery diarrhea similar to that of EPEC and ETEC, sometimes inflammatory colitis or dysentery.
 Particularly closely related to Shigella sp. (which are now thought to be subgroups of E. coli); much pathogenesis from EIEC and Shigella sp. is mediated by the pWR100 virulence plasmid.
 Diffusely adherent (DAEC)
 Attaches to the small intestinal wall and induces formation of projections which wrap around the bacterium.
Enterotoxigenic Escherichia coli (ETEC) is the most common type of diarrheagenic E. coli.^{[3]} It may also be the most common cause of childhood diarrhea in the developing world, responsible for approximately 1/7 of diarrheal episodes in children aged less than 1 year and almost 1/4 of diarrheal episodes in 14 year olds.^{[4]} It can also cause severe dehydrating choleralike disease in adults.^{[3]} Diagnosis is complicated since many other Gramnegative bacteria produce similar toxins, so toxins as well as the E. coli bacterium must be tested for in order to yield accurate results.^{[4]}
ETEC can often be detected in apparently healthy people. In developing countries among healthy 011 month olds, and 14 year olds, 11.7% and 7.1%, respectively, are estimated to be colonized with ETEC.^{[4]}
Feeding studies of ETEC or EPEC in healthy volunteers first gave 23g of NaHCO_{3}, which neutralizes stomach acid and reduces the infectious dose.^{[5]} However, it has been suggested that food as a vehicle has a similar acidneutralizing effect, so feeding studies given with NaHCO_{3} may better represent natural foodborne infection.^{[5]} ETEC and EPEC generally have high ID_{50}, and partly as a consequence of this, they do not appear to be transmitted persontoperson; a study of ETECinfected volunteers cohoused with uninfected volunteers did not result in any transmission of infection.^{[6]} Food was all served individually to the volunteers over the course of the experiment, so there was no opportunity for ETEC to spread via that route.^{[6]}
http://www.cdc.gov/ecoli/index.html
Summary of data and models
There are many human feeding studies of various E. coli types and strains, which can be pooled in various ways to yield different dose response models. Many of these have small sample sizes and cannot be used on their own to reliably fit a dose response model. Most datasets for E. coli infections describe high levels of infection resulting from high doses. Lower doses remain to be investigated, and dose response models for infection are therefore uncertain. Another important factor is whether the dose was given with bicarbonate, which would neutralize some stomach acid and possibly increase infectivity.
Haas, Rose, and Gerba (1999)^{[7]} fitted betaPoisson models to several pooled datasets describing the disease response from ETEC, EPEC and EIEC. Among these datasets were the EPEC strains O111^{[8]} and O55, as well as EIEC strains 4608 and 1624^{[9]} with diarrhea as the end point. However, it mixed data from experiments in which bacteria were given with and without bicarbonate.
The best available dataset using infection as a response comes from an experiment (98) with 3 dose levels, feeding EIEC to adult humans.^{[9]}
Powell et al. (2000)^{[10]} pooled 3 human trial datasets^{[11]}^{[12]} for EPEC to produce a betaPoisson model and a Weibullgamma model.
Additional pooling analyses for this chapter were conducted on the basis of pathotype (ETEC or EPEC), whether the dose was given with bicarbonate, and the nature of the response (disease or infection), incorporating more data from the literature than the previous two published models. Since some combinations of these factors lacked data, analyses could only be done for ETEC disease (buffered or unbuffered), EPEC disease (buffered), ETEC infection (unbuffered), and EPEC infection (buffered). The pooled datasets for infection contained mostly positive responses, and therefore their behavior at low doses is very uncertain. For the pooled analyses describing disease, datasets were excluded if they contributed significantly (P < 0.05) to the 2 log likelihood of the model given the data.^{[7]} Two experiments^{[9]} examining diarrhea from EIEC were also pooled.
Experiment serial number 
Reference 
Host type 
Agent strain 
Route 
# of doses 
Dose units 
Response 
Best fit model 
Optimized parameter(s) 
LD_{50}/ID_{50}

98* 
^{[9]} 
human 
EIEC 1624 
oral (in milk) 
3 
CFU 
positive stool isolation 
betaPoisson 
α = 1.55E01 , N_{50} = 2.11E+06 
2.11E+06

39 
^{[9]} 
human 
EIEC 4608 
oral (in milk) 
3 
CFU 
mild to severe diarrhea 
exponential 
k = 9.7E09 
7.14E+07

40 
^{[9]} 
human 
EIEC 1624 
oral (in milk) 
3 
CFU 
mild to severe diarrhea 
exponential 
k = 1.22E08 
5.7E+07

42 
^{[13]} 
human 
ETEC O55 (in paper as “type 55, B5”) 
oral 
4 
CFU 
slight to severe illness 
betaPoisson 
α = 8.7E02 , N_{50} = 2.05E+05 
2.05E+05

43 
^{[8]} 
human 
ETEC O111 (in paper as "E. coli 111, B4") 
oral 
4 
CFU 
slight to severe illness 
betaPoisson 
α = 2.63E01 , N_{50} = 3.56E+06 
3.56E+06

165 
^{[5]} 
human 
ETEC 2144 (ST) 
oral (in milk) 
3 
CFU 
diarrhea or vomiting 
betaPoisson 
α = 2.5E01 , N_{50} = 9.1E+07 
9.1E+07

38, 39, 40, 42, 99, 144 
^{[9]}^{[13]}^{[14]}^{[7]} 
human 
ETEC B7A 
oral (in milk) 
15 
CFU 
mild to severe diarrhea 
betaPoisson 
α = 1.78E01 , N_{50} = 8.6E+07 
8.6E+07

214, 216, 217 
^{[12]}^{[11]}^{[10]} 
human 
EPEC B1718 (serotype O11:NM) 
oral (with NaHCO3) 
8 
CFU 
diarrhea 
betaPoisson 
α = 2.21E01 , N_{50} = 6.85E+07 
6.85E+07

142, 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, 172 
^{[15]}^{[14]}^{[16]}^{[17]}^{[18]}^{[6]} 
human 
ETEC B7A 
oral (with NaHCO3) 
19 
CFU 
diarrhea 
betaPoisson 
α = 7.54E02 , N_{50} = 1.7E+06 
1.7E+06

38, 42, 99, 165 
^{[9]}^{[13]}^{[5]} 
human 
ETEC B7A 
oral (in milk) 
11 
CFU 
mild to severe diarrhea 
betaPoisson 
α = 2.06E01 , N_{50} = 1.28E+08 
1.28E+08

153, 157, 159, 214, 216, 217 
^{[19]}^{[12]}^{[11]} 
human 
EPEC E2348/69 (O127:H6) 
oral (w. 2g NaHCO3) 
11 
CFU 
diarrhea 
betaPoisson 
α = 1.62E01 , N_{50} = 9.98E+07 
9.98E+07

154, 156, 158, 160, 219, 220, 221 
^{[19]}^{[20]}^{[11]} 
human 
EPEC E2348/69 (O127:H6) 
oral (w. 2g NaHCO3) 
13 
CFU 
shedding in feces 
exponential 
k = 1.95E06 
3.56E+05

39, 40 
^{[9]} 
human 
EIEC 4608 
oral (in milk) 
6 
CFU 
mild to severe diarrhea 
exponential 
k = 1.07E08 
6.5E+07

96, 100, 166 
^{[21]} 
human 
ETEC B7A 
oral (in milk) 
7 
CFU 
positive stool isolation 
betaPoisson 
α = 3.75E01 , N_{50} = 1.78E+05 
1.78E+05

*This model is preferred in most circumstances. However, consider all available models to decide which one is most appropriate for your analysis.


Optimization Output for experiment 98
Escherichia coli (EIEC 1624) in the human model data ^{[9]}
Dose 
Positive stool isolation 
No positive stool isolation 
Total

1E+04 
0 
5 
5

1E+06 
5 
4 
9

1E+08 
3 
2 
5


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

28.6

27.2

2

3.84 1.82e07

5.99 6.18e07

Beta Poisson

1.38

1

3.84 0.24

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

1.55E01

1.26E03 
1.26E03 
2.84E02 
1.84E+01 
1.29E+02 
1.90E+02

N_{50}

2.11E+06

1.73E+05 
2.95E+05 
2.95E+05 
7.85E+08 
9.22E+140 
9.22E+140


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 39
Escherichia coli (EIEC 4608) in the human model data ^{[9]}
Dose 
Mild to severe diarrhea 
No mild to severe diarrhea 
Total

1E+04 
0 
5 
5

1E+06 
0 
5 
5

1E+08 
5 
3 
8


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

0.0986

0.00188

2

3.84 1

5.99 0.952

Beta Poisson

0.1

1

3.84 0.751

Exponential is preferred to betaPoisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

k 
9.7E09 
2.85E09 
2.86E09 
4.66E09 
2.04E08 
2.04E08 
5.07E08

ID50/LD50/ETC* 
7.14E+07 
1.37E+07 
3.40E+07 
3.40E+07 
1.49E+08 
2.43E+08 
2.44E+08

*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter) Exponential model plot, with confidence bounds around optimized model
Optimization Output for experiment 40
Escherichia coli (EIEC 1624) in the human model data ^{[9]}
Dose 
Mild to severe diarrhea 
No mild to severe diarrhea 
Total

1E+04 
0 
5 
5

1E+06 
1 
8 
9

1E+08 
3 
2 
5


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

2.99

2.98

2

3.84 0.0845

5.99 0.224

Beta Poisson

0.0156

1

3.84 0.901

Exponential is preferred to betaPoisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

k 
1.22E08 
1.97E09 
2.19E09 
4.40E09 
4.03E08 
1.17E07 
2.50E07

ID50/LD50/ETC* 
5.7E+07 
2.77E+06 
5.92E+06 
1.72E+07 
1.58E+08 
3.17E+08 
3.52E+08

*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter) Exponential model plot, with confidence bounds around optimized model
Optimization Output for experiment 42
Escherichia coli (ETEC O55 (in paper as “type 55, B5”)) in the human model data ^{[13]}
Dose 
Slight to severe illness 
No slight to severe illness 
Total

1.43E+08 
6 
2 
8

1.73E+09 
5 
2 
7

5.33E+09 
6 
2 
8

1.6E+10 
7 
1 
8


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

35

34.5

3

3.84 4.2e09

7.81 1.21e07

Beta Poisson

0.486

2

5.99 0.784

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

8.7E02

9.87E04 
1.02E03 
1.67E02 
3.57E01 
4.67E01 
7.83E01

N_{50}

2.05E+05

3.40E09 
4.31E06 
2.31E04 
1.30E+08 
2.09E+08 
5.13E+08


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 43
Escherichia coli (ETEC O111 (in paper as “E. coli 111, B4”)) in the human model data ^{[8]}
Dose 
Slight to severe illness 
No slight to severe illness 
Total

7E+06 
7 
4 
11

5.31E+08 
8 
4 
12

6.5E+09 
11 
0 
11

9E+09 
12 
0 
12


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

39.8

33.4

3

3.84 7.5e09

7.81 1.19e08

Beta Poisson

6.38

2

5.99 0.0412

Neither the exponential nor betaPoisson fits well; betaPoisson is less bad.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

2.63E01

9.92E04 
7.95E02 
1.00E01 
4.71E01 
5.53E01 
1.48E+01

N_{50}

3.56E+06

6.89E01 
1.15E+03 
2.08E+03 
1.85E+07 
2.49E+07 
4.18E+07


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 165
Escherichia coli (ETEC 2144 (ST)) in the human model data ^{[5]}
Dose 
Diarrhea or vomiting 
No diarrhea or vomiting 
Total

1E+06 
0 
4 
4

1E+08 
3 
2 
5

1E+10 
4 
1 
5


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

15.9

15.3

2

3.84 9e05

5.99 0.000359

Beta Poisson

0.531

1

3.84 0.466

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

2.5E01

9.94E04 
9.94E04 
9.94E04 
1.76E+02 
7.32E+02 
7.32E+02

N_{50}

9.1E+07

3.05E+04 
4.09E+04 
5.00E+04 
3.20E+08 
6.91E+08 
6.91E+08


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 38, 39, 40, 42, 99, 144
E. coli disease (ETEC, EPEC, EIEC) in the human model data
Dose 
Mild to severe diarrhea 
No mild to severe diarrhea 
Total

1E+04 
0 
5 
5

1E+04 
0 
5 
5

1E+06 
0 
5 
5

1E+06 
1 
8 
9

1E+08 
1 
4 
5

1E+08 
5 
3 
8

1E+08 
3 
2 
5

1E+08 
2 
3 
5

1.43E+08 
6 
2 
8

2.7E+08 
9 
7 
16

1.73E+09 
5 
2 
7

5.33E+09 
6 
2 
8

1E+10 
4 
1 
5

1E+10 
3 
2 
5

1.6E+10 
7 
1 
8


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

119

113

14

3.84 0

23.7 0

Beta Poisson

6.56

13

22.4 0.924

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

1.78E01

9.39E02 
1.09E01 
1.19E01 
3.21E01 
3.63E01 
4.77E01

N_{50}

8.6E+07

1.75E+07 
2.62E+07 
3.25E+07 
2.63E+08 
3.23E+08 
5.21E+08


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 214, 216, 217
EPEC disease in the human model data
Dose 
Diarrhea 
No diarrhea 
Total

1E+06 
0 
4 
4

1E+06 
1 
4 
5

1E+08 
1 
4 
5

5E+08 
3 
2 
5

2.5E+09 
6 
0 
6

1E+10 
3 
2 
5

1E+10 
5 
0 
5

2E+10 
2 
0 
2


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

31.8

20.6

7

3.84 5.69e06

14.1 4.37e05

Beta Poisson

11.2

6

12.6 0.0813

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

2.21E01

9.31E02 
1.18E01 
1.27E01 
1.25E+00 
7.41E+02 
1.29E+04

N_{50}

6.85E+07

6.14E+06 
1.11E+07 
1.52E+07 
6.32E+08 
7.69E+08 
1.09E+09


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 142, 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, 172
EPEC disease in the human model data
Dose 
Diarrhea 
No diarrhea 
Total

1E+06 
3 
3 
6

1E+07 
6 
9 
15

1E+08 
6 
1 
7

1E+08 
7 
4 
11

1E+08 
7 
5 
12

1E+08 
9 
3 
12

1E+08 
6 
3 
9

1E+08 
4 
2 
6

1E+08 
3 
1 
4

1E+08 
4 
0 
4

2.7E+08 
9 
7 
16

5E+08 
19 
8 
27

1E+09 
5 
3 
8

1E+09 
7 
1 
8

1E+09 
5 
5 
10

1E+10 
8 
0 
8

1E+10 
5 
4 
9

1E+10 
9 
3 
12

1E+10 
9 
5 
14


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

412

393

18

3.84 0

28.9 0

Beta Poisson

19.1

17

27.6 0.322

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

7.54E02

8.28E03 
1.11E02 
1.51E02 
1.46E01 
1.59E01 
1.84E01

N_{50}

1.7E+06

2.80E11 
1.91E06 
1.44E03 
2.34E+07 
3.09E+07 
4.67E+07


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 38, 42, 99, 165
ETEC disease, unbuffered, in the human model data
Dose 
Mild to severe diarrhea 
No mild to severe diarrhea 
Total

1E+06 
0 
4 
4

1E+08 
1 
4 
5

1E+08 
2 
3 
5

1E+08 
3 
2 
5

1.43E+08 
6 
2 
8

1.73E+09 
5 
2 
7

5.33E+09 
6 
2 
8

1E+10 
4 
1 
5

1E+10 
3 
2 
5

1E+10 
4 
1 
5

1.6E+10 
7 
1 
8


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

68.4

62.9

10

3.84 2.22e15

18.3 8.97e11

Beta Poisson

5.53

9

16.9 0.786

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

2.06E01

1.75E02 
1.17E01 
1.32E01 
3.79E01 
4.29E01 
5.57E01

N_{50}

1.28E+08

3.53E+05 
3.17E+07 
4.09E+07 
3.83E+08 
4.65E+08 
6.86E+08


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 153, 157, 159, 214, 216, 217
ETEC infection, unbuffered, in the human model data
Dose 
Diarrhea 
No diarrhea 
Total

1E+06 
0 
4 
4

1E+06 
1 
4 
5

1E+08 
1 
4 
5

5E+08 
3 
2 
5

2.5E+09 
6 
0 
6

1E+10 
9 
1 
10

1E+10 
9 
5 
14

1E+10 
3 
2 
5

1E+10 
5 
0 
5

2E+10 
2 
0 
2

2.3E+10 
14 
5 
19


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

57.8

43.4

10

3.84 4.51e11

18.3 9.36e09

Beta Poisson

14.4

9

16.9 0.108

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

1.62E01

8.23E02 
9.98E02 
1.08E01 
3.69E01 
4.21E01 
5.36E01

N_{50}

9.98E+07

7.29E+06 
1.55E+07 
2.20E+07 
8.50E+08 
1.06E+09 
1.57E+09


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
Optimization Output for experiment 154, 156, 158, 160, 219, 220, 221
EPEC, infection, buffered, in the human model data
Dose 
Shedding in feces 
No shedding in feces 
Total

1E+06 
3 
1 
4

1E+06 
5 
0 
5

1E+06 
4 
1 
5

1E+08 
5 
0 
5

1E+08 
5 
0 
5

9E+08 
8 
0 
8

1E+10 
10 
0 
10

1E+10 
9 
0 
9

1E+10 
14 
0 
14

1E+10 
5 
0 
5

1E+10 
5 
0 
5

1E+10 
5 
0 
5

2.3E+10 
19 
0 
19


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

1.98

2.71e06

12

3.84 0.999

21 0.999

Beta Poisson

1.98

11

19.7 0.999

Exponential is preferred to betaPoisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

k 
1.95E06 
8.47E07 
1.03E06 
1.25E06 
2.64E06 
2.64E06 
2.64E06

ID50/LD50/ETC* 
3.56E+05 
2.63E+05 
2.63E+05 
2.63E+05 
5.53E+05 
6.73E+05 
8.18E+05

*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter) Exponential model plot, with confidence bounds around optimized model
Optimization Output for experiment 39, 40
EIEC disease, unbuffered, in the human model data
Dose 
Mild to severe diarrhea 
No mild to severe diarrhea 
Total

1E+04 
0 
5 
5

1E+04 
0 
5 
5

1E+06 
0 
5 
5

1E+06 
1 
8 
9

1E+08 
5 
3 
8

1E+08 
3 
2 
5


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

3.19

2.24

5

3.84 0.134

11.1 0.67

Beta Poisson

0.951

4

9.49 0.917

Exponential is preferred to betaPoisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

k 
1.07E08 
3.63E09 
4.79E09 
5.77E09 
2.04E08 
2.28E08 
3.21E08

ID50/LD50/ETC* 
6.5E+07 
2.16E+07 
3.04E+07 
3.40E+07 
1.20E+08 
1.45E+08 
1.91E+08

*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter) Exponential model plot, with confidence bounds around optimized model
Optimization Output for experiment 96, 100, 166
Dose response data
Dose 
Positive stool isolation 
No positive stool isolation 
Total

1E+06 
3 
1 
4

1E+08 
4 
1 
5

1E+08 
5 
0 
5

1E+08 
5 
0 
5

1E+10 
5 
0 
5

1E+10 
5 
0 
5

1E+10 
5 
0 
5


Goodness of fit and model selection
Model 
Deviance 
Δ 
Degrees of freedom 
χ^{2}_{0.95,1} pvalue 
χ^{2}_{0.95,mk} pvalue

Exponential

64.1

61.4

6

3.84 4.77e15

12.6 6.67e12

Beta Poisson

2.71

5

11.1 0.745

BetaPoisson fits better than exponential; cannot reject good fit for betaPoisson.


Optimized parameters for the betaPoisson model, from 10000 bootstrap iterations
Parameter

MLE estimate

Percentiles

0.5% 
2.5% 
5% 
95% 
97.5% 
99.5%

α

3.75E01

1.29E01 
1.34E01 
1.34E01 
9.97E+00 
9.97E+00 
1.07E+01

N_{50}

1.78E+05

3.63E01 
3.63E01 
3.63E01 
2.46E+06 
3.48E+06 
6.09E+06


Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters. beta Poisson model plot, with confidence bounds around optimized model
References
 ↑ Kaper JB, Nataro JP & Mobley HL (2004) Pathogenic Escherichia coli. Nature Reviews. Microbiology. 2(2), pp.123140. Full text
 ↑ Nataro JP & Kaper JB (1998) Diarrheagenic Escherichia coli. Clinical Microbiology Reviews. 11(1), pp.142201. Full text
 ↑ ^{3.0} ^{3.1} Qadri F, et al. (2005) Enterotoxigenic Escherichia coli in developing countries: epidemiology, microbiology, clinical features, treatment, and prevention. Clinical Microbiology Reviews. 18(3), pp.465483. Full text
 ↑ ^{4.0} ^{4.1} ^{4.2} Wennerås C & Erling V (2004) Prevalence of enterotoxigenic Escherichia coliassociated diarrhoea and carrier state in the developing world. Journal of Health, Population, and Nutrition. 22(4), pp.370382. Abstract
 ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} ^{5.4} Levine MM, et al. (1977) Diarrhea caused by Escherichia coli that produce only heatstable enterotoxin. Infection and Immunity. 17(1), pp.7882. Full text
 ↑ ^{6.0} ^{6.1} ^{6.2} Levine MM, et al. (1980) Lack of persontoperson transmission of enterotoxigenic Escherichia coli despite close contact. American Journal of Epidemiology. 111(3), pp.347355. Full text
 ↑ ^{7.0} ^{7.1} ^{7.2} Haas CN, Rose JB & Gerba CP (1999) Quantitative Microbial Risk Assessment. John Wiley & Sons, Inc. Cite error: Invalid
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tag; name "Haas_1999" defined multiple times with different content
 ↑ ^{8.0} ^{8.1} ^{8.2} Ferguson WW & June RC (1952) Experiments on feeding adult volunteers with Escherichia coli 111, B4, a coliform organism associated with infant diarrhea. American Journal of Hygiene. 55(2), pp.155169. Full text
 ↑ ^{9.00} ^{9.01} ^{9.02} ^{9.03} ^{9.04} ^{9.05} ^{9.06} ^{9.07} ^{9.08} ^{9.09} ^{9.10} ^{9.11} DuPont HL, et al. (1971) Pathogenesis of Escherichia coli diarrhea. The New England Journal of Medicine. 285(1), pp.19. Full text Cite error: Invalid
<ref>
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 ↑ ^{10.0} ^{10.1} Powell MR (2000) Doseresponse envelope for Escherichia coli O157:H7. Quantitative Microbiology. 2, pp.141163. Full text
 ↑ ^{11.0} ^{11.1} ^{11.2} ^{11.3} Levine MM, et al. (1978) Escherichia coli strains that cause diarrhoea but do not produce heatlabile or heatstable enterotoxins and are noninvasive. Lancet. 1(8074), pp.11191122. Abstract Cite error: Invalid
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tag; name "Levine_1978" defined multiple times with different content
 ↑ ^{12.0} ^{12.1} ^{12.2} Bieber D, et al. (1998) Type IV pili, transient bacterial aggregates, and virulence of enteropathogenic Escherichia coli. Science (New York, N.Y.). 280(5372), pp.21142118. Full text Cite error: Invalid
<ref>
tag; name "Bieber_1998" defined multiple times with different content
 ↑ ^{13.0} ^{13.1} ^{13.2} ^{13.3} June RC, Ferguson WW & Worfel MT (1953) Experiments in feeding adult volunteers with Escherichia coli 55, B5, a coliform organism associated with infant diarrhea. American Journal of Hygiene. 57(2), pp.222236. Full text Cite error: Invalid
<ref>
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 ↑ ^{14.0} ^{14.1} Graham DY, Estes MK & Gentry LO (1983) Doubleblind comparison of bismuth subsalicylate and placebo in the prevention and treatment of enterotoxigenic Escherichia coliinduced diarrhea in volunteers. Gastroenterology 85(5), pp.10171022. Abstract
 ↑ Coster TS et al. (2007) Immune response, ciprofloxacin activity, and gender differences after human experimental challenge by two strains of enterotoxigenic Escherichia coli. Infection and Immunity. 75(1), pp.252259. Full text
 ↑ Levine MM, et al. (1979) Immunity to enterotoxigenic Escherichia coli. Infection and Immunity. 23(3), pp.729736. Full text
 ↑ Clements ML, et al. (1981) Lactobacillus prophylaxis for diarrhea due to enterotoxigenic Escherichia coli. Antimicrobial Agents and Chemotherapy 20(1), pp.104108. Full text
 ↑ Levine MM, et al. (1982) Reactogenicity, immunogenicity and efficacy studies of Escherichia coli type1 somatic pili parenteral vaccine in man. Scandinavian Journal of Infectious Diseases. Suppl. 33, pp.8395. Abstract
 ↑ ^{19.0} ^{19.1} Tacket CO, et al. (2000) Role of EspB in experimental human enteropathogenic Escherichia coli infection. Infection and Immunity. 68(6), pp.36893695. Full text Cite error: Invalid
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 ↑ Donnenberg MS, et al. (1993) Role of the eaeA gene in experimental enteropathogenic Escherichia coli infection. The Journal of Clinical Investigation. 92(3), pp.14121417. Full text
 ↑ COMPLETE REF HERE