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General Overview

The famous case of John Snow in London is well known, as an investigation of a cholera outbreak and provided the first in proof of a linkage between the presence of a pathogen in water and a water borne current outbreak. Cholera has remained a major health risk to populations around the world. Vibrio cholerae (V. cholerae) is a gram negative bacterium, the causative agent of cholera, which causes very watery stool, typically referred to rice-watery stool.

Person to person transmission has been typically assumed to be minimal probably due the dose response parameters. However, the main risk posed from this pathogen has always been considered from primary exposure, contaminated water.

http://www.cdc.gov/cholera/index.html

Summary Data

Hornick et al. (1971) has a broad repository of both human and animal data. The most interesting data sets in this study are human exposure where the onset of symptoms as well as indication of asymptomatic infection were monitored. Additionally, both diarrhea and cholera diarrhea (very different in appearance) were also monitored as symptoms.

Cash et al. (1974) has only one dataset which showed a significant trend between dose and observed probability of response. Interestingly, this study determined that in human volunteer feeding trials, the reduction of pH in subject stomachs decreased the overall dose required for a similar response. This was accomplished using sodium bicarbonate, essentially ant-acid treatment, thus demonstrating the effectiveness of higher pH in some protection from V. cholerea.

Recommended Model

Experiment number 249 is recommended model as it has lowest ID50 among the models.

Exponential and betapoisson model.jpg

 

ID # of Doses Agent Strain Dose Units Host type Μodel Optimized parameters Response type Reference
126 6 Inaba 569B CFU human beta-Poisson a = 3.18E-01

LD50/ID50 = 6.82E+03
N50 = 6.82E+03
any diarrhea Kotak, B. G., et al. "Occurrence and toxicological evaluation of cyanobacterial toxins in Alberta lakes and farm dugouts." Water Research. 27 (1993): 3.
128 6 Inaba 569B CFU human beta-Poisson a = 1.10E-01

LD50/ID50 = 3.88E+07
N50 = 3.88E+07
cholera diarrhea McCullough, N. B., and C. W. Elsele. "Experimental Human Salmonellosis: I." Pathogenicity of Strains of Salmonella meleagridis and Salmonella anatum Obtained from Spray-Dried Whole Egg. 88 (1951): 3.
167 7 Inaba 569B (classical) CFU human beta-Poisson a = 1.31E-01

LD50/ID50 = 2.91E+09
N50 = 2.91E+09
diarrhea and culture positive Levine, MM, et al. "Reactogenicity, immunogenicity and efficacy studies of Escherichia coli type 1 somatic pili parenteral vaccine in man." Scandinavian journal of infectious diseases. Supplementum. 33 (1982): 83-95.
249 6 Inaba 569B CFU human beta-Poisson a = 2.50E-01

LD50/ID50 = 2.43E+02
N50 = 2.43E+02
infection Diringer, H., J. Roehmel, and M. Beekes. "Effect of repeated oral infection of hamsters with scrapie." Journal of General Virology. 79 (1998): 3.
35 7 Inaba 569B CFU human beta-Poisson a = 1.98E-01

LD50/ID50 = 6.36E+08
N50 = 6.36E+08
diarrhea or culture positive Hornick, R. B., et al. "The Broad Street pump revisited: response of volunteers to ingested cholera vibrios." Bulletin of the New York Academy of Medicine. 47 (1971): 10.
Experiment ID:
126
# of Doses:
6
Agent Strain:
Inaba 569B
Dose Units:
CFU
Host type:
human
Μodel:
beta-Poisson
Optimized parameters: a = 3.18E-01

LD50/ID50 = 6.82E+03
N50 = 6.82E+03
Reference:
Human Inaba Strain 569 B [1]
Dose Diarrhea No diarrhea Total
10 0 2 2
1000 0 4 4
1E+04 9 4 13
1E+05 6 2 8
1E+06 20 3 23
1E+08 2 0 2

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 55.2 51.9 5 3.84 
5.88e-13
11.1 
1.19e-10
Beta Poisson 3.3 4 9.49 
0.508
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 3.18E-01 1.51E-01 1.84E-01 2.00E-01 6.52E-01 7.34E-01 1.66E+00
N50 6.82E+03 1.68E+03 2.35E+03 2.75E+03 1.83E+04 2.25E+04 3.38E+04

 

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

Experiment ID:
128
# of Doses:
6
Agent Strain:
Inaba 569B
Dose Units:
CFU
Host type:
human
Μodel:
beta-Poisson
Optimized parameters: a = 1.10E-01

LD50/ID50 = 3.88E+07
N50 = 3.88E+07
Reference:
Human Inaba Strain 569 B [1]
Dose Cholera diarrhea No cholera diarrhea Total
10 0 2 2
1000 0 4 4
1E+04 0 13 13
1E+05 1 7 8
1E+06 6 17 23
1E+08 1 1 2

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 23.3 22.8 5 3.84 
1.81e-06
11.1 
0.000297
Beta Poisson 0.504 4 9.49 
0.973
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 1.1E-01 1.94E-02 3.52E-02 3.83E-02 6.23E+01 4.54E+02 1.10E+03
N50 3.88E+07 1.13E+06 1.47E+06 1.71E+06 7.30E+11 4.70E+12 6.70E+18

 

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

Experiment ID:
167
# of Doses:
7
Agent Strain:
Inaba 569B (classical)
Dose Units:
CFU
Host type:
human
Μodel:
beta-Poisson
Optimized parameters: a = 1.31E-01

LD50/ID50 = 2.91E+09
N50 = 2.91E+09
Reference:
Human Inaba 569B [2]
Dose Diarrhea and culture positive No diarrhea and culture positive Total
1E+04 0 2 2
1E+06 0 4 4
1E+07 0 4 4
1E+08 2 2 4
1E+09 0 2 2
1E+10 0 1 1
1E+11 2 0 2

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 14.9 7.58 6 3.84 
0.00589
12.6 
0.0207
Beta Poisson 7.36 5 11.1 
0.195
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 1.31E-01 1.04E-03 1.04E-03 1.04E-03 1.67E-01 1.67E-01 1.67E-01
N50 2.91E+09 3.23E+08 3.23E+08 3.23E+08 8.87E+254 8.87E+254 8.87E+254

 

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

Best Fit
Experiment ID:
249
# of Doses:
6
Agent Strain:
Inaba 569B
Dose Units:
CFU
Host type:
human
Μodel:
beta-Poisson
Optimized parameters: a = 2.50E-01

LD50/ID50 = 2.43E+02
N50 = 2.43E+02
Reference:
Human Inaba Strain 569B [1]
Dose Infected Non-infected Total
10 0 2 2
1000 3 1 4
1E+04 11 2 13
1E+05 7 1 8
1E+06 21 2 23
1E+08 2 0 2

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 92.4 91.2 5 3.84 
0
11.1 
0
Beta Poisson 1.16 4 9.49 
0.885
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.5E-01 1.14E-01 1.48E-01 1.66E-01 5.44E-01 6.55E-01 2.91E+00
N50 2.43E+02 1.88E+01 4.82E+01 6.35E+01 1.52E+03 2.08E+03 3.60E+03

 

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

Experiment ID:
35
# of Doses:
7
Agent Strain:
Inaba 569B
Dose Units:
CFU
Host type:
human
Μodel:
beta-Poisson
Optimized parameters: a = 1.98E-01

LD50/ID50 = 6.36E+08
N50 = 6.36E+08
Reference:

Please Note

The results from this experiment are not recommended for use, rather they are present for reference purposes. The confidence intervals are not complete for this experiment since the data is not allowing for a stable bootstrap outcome. It is recommended to use the recommended model as this data is likely just capable of passing the test of trend, but still likely suboptimal for dose response modeling purposes.

 

Human Inaba Strain 569B [1]
Dose Diarrhea or culture positive No diarrhea or culture positive Total
1E+04 0 2 2
1E+06 0 4 4
1E+07 0 4 4
1E+08 2 2 4
1E+09 1 1 2
1E+10 0 1 1
1E+11 2 0 2

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom
χ20.95,1 
p-value
χ20.95,m-k 
p-value
Exponential 15.1 10.4 6 3.84 
0.00126
12.6 
0.0193
Beta Poisson 4.72 5 11.1 
0.451
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 1.98E-01 9.92E-04 9.92E-04 1.04E-03 3.20E-01 3.65E-01 4.07E-01
N50 6.36E+08 8.26E+05 8.26E+05 1.18E+08 5.33E+207 5.33E+207 8.87E+254

 

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