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General Overview

Coronaviruses cause acute and chronic respiratory, enteric, and central nervous system (CNS) diseases in humans and many species of animals. Coronaviruses are divided into three groups based on the genome sequences, including SARS-CoV (a member of group II) as well as murine hepatitis virus (MHV), bovine coronavirus, porcine hemagglutinating encephalomyelitis virus (HEV), equine coronavirus, and human coronavirues OC43 and NL63, which also cause respiratory infections. SARS-CoV, the causal pathogen of severe acute respiratory syndrome (SARS), caused a large outbreak of this severe pneumonia occurred in Hong Kong in 2003 and rapidly spread throughout the world. SARS-CoV can infect and replicate in mice, ferrets, hamsters, cats, and several species of nonhuman primates (cynomolgus and rhesus macaques, African green monkeys, and marmosets). MHV that infects both mice and rats often has been studied as a suitable model of human coronavirus diseases, according to Watanabe et al.[1].

Summary Data

DeDiego et al.[2] challenged four groups of the transgenic mice intranasally with graded doses of rSARS-CoV and the survival was monitored for 13 days.

De Albuquerque et al.[3] inoculated A/J mice with MHV-1 intranasally via intranasal route and monitored the survival for 21 days.

 

Exponential and betapoisson model.jpg

Summary

By increasing the number of data points, the pooling narrows the range of the confidence region of the parameter estimates and enhances the statistical precision.

ID # of Doses Agent Strain Dose Units Host type Μodel Optimized parameters Response type Reference
260 4 rSARS-CoV PFU mice hACE-2 exponential k = 2.97E-03
LD50/ID50 = 2.33E+02
 		 death

DeDiego et al.[2]  
260, 261 8 rSARS-CoV PFU mice hACE-2 and A/J exponential k = 2.46E-03
LD50/ID50 = 2.82E+02
 		 death  
261 4 MHV-1 PFU A/J mice exponential k = 2.14E-03
LD50/ID50 = 3.24E+02
 		 death De Albuquerque et al.[3]

 

Optimization Output for Exp. 260:

Experiment ID: 260

# of Doses: 4 

Agent Strain: rSARS-CoV

Dose Units: PFU

Host type: mice hACE-2

Μodel: exponential

Optimized parameters:
k = 2.97E-03
LD50/ID50 = 2.33E+02
 

mice/rSARS-CoV strain model data (DeDiego et al.[2])
Dose Dead Survived Total
240 1 2 3
800 3 0 3
2400 2 0 2
12000 6 0 6

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value		 χ20.95,m-k 
p-value
Exponential 0.968 -0.000923 3 3.84 
1		 7.81 
0.809
Beta Poisson 0.969 2 5.99 
0.616
Exponential is preferred to beta-Poisson; 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 2.97E-03 1.90E-03 1.90E-03 1.90E-03 2.97E-03 2.97E-03 2.97E-03
ID50/LD50/ETC* 2.33E+02 2.33E+02 2.33E+02 2.33E+02 3.64E+02 3.64E+02 3.64E+02
*Not a parameter of the exponential model; however, it facilitates comparison with other models.

 

Optimization Output for Exp. 260, 261:

Best Fit

Experiment ID: 260, 261

# of Doses: 8

Agent Strain: rSARS-CoV

Dose Units: PFU

Host type: mice hACE-2 and A/J 

Μodel: exponential

Optimized parameters:
k = 2.46E-03
LD50/ID50 = 2.82E+02

Pooled dose response model data 
Dose Dead Survived Total
5 0 5 5
50 1 4 5
240 1 2 3
500 3 2 5
800 3 0 3
2400 2 0 2
5000 5 0 5
12000 6 0 6

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value		 χ20.95,m-k 
p-value
Exponential 1.75 -0.00181 7 3.84 
1		 14.1 
0.972
Beta Poisson 1.75 6 12.6 
0.941
Exponential is preferred to beta-Poisson; 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 2.46E-03 1.07E-03 1.28E-03 1.35E-03 4.59E-03 5.27E-03 6.80E-03
ID50/LD50/ETC* 2.82E+02 1.02E+02 1.32E+02 1.51E+02 5.13E+02 5.43E+02 6.47E+02
*Not a parameter of the exponential model; however, it facilitates comparison with other models.

 

Optimization Output for Exp. 261:

Experiment ID: 261

# of Doses: 4

Agent Strain: MHV-1

Dose Units: PFU 

Host type: A/J mice

Μodel: exponential

Optimized parameters:
k = 2.14E-03
LD50/ID50 = 3.24E+02
 

Mice/MHV-1 strains model data (De Albuquerque et al.[3])
Dose Dead Survived Total
5 0 5 5
50 1 4 5
500 3 2 5
5000 5 0 5

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value		 χ20.95,m-k 
p-value
Exponential 0.606 0.0689 3 3.84 
0.793		 7.81 
0.895
Beta Poisson 0.537 2 5.99 
0.765
Exponential is preferred to beta-Poisson; 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 2.14E-03 6.25E-04 6.55E-04 9.06E-04 6.58E-03 6.58E-03 9.86E-03
ID50/LD50/ETC* 3.24E+02 7.03E+01 1.05E+02 1.05E+02 7.65E+02 1.06E+03 1.11E+03
*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

 

Recommended Model

It is recommended that the pooled experiments 260 and 261 should be used as the best dose-response model. Both strains are common in outbreaks. The pooling narrows the range of the confidence region of the parameter estimates and enhances the statistical precision.

References

  1. Watanabe, T., Bartrand T. A., Weir M. H., Omura T., & Haas C. N. (2010).  Development of a Dose-Response Model for SARS Coronavirus. Risk Analysis: An International Journal. 30(7), 1129–1138.
  2. DeDiego, M. L., Pewe L., Alvarez E., Rejas M. T., Perlman S., & Enjuanes L. (2008).  Pathogenicity of severe acute respiratory coronavirus deletion mutants in hACE-2 transgenic mice. Virology.. 376, 2.
  3. Albuquerque, N. D., Baig E., Ma X., Zhang J., He W., Rowe A., et al. (2006).  Murine Hepatitis Virus Strain 1 Produces a Clinically Relevant Model of Severe Acute Respiratory Syndrome in A/J Mice. Journal of Virology. 80, 21.
ID # of Doses Agent Strain Dose Units Host type Μodel Optimized parameters Response type Reference
260 4 rSARS-CoV PFU mice hACE-2 exponential
k = 2.97E-03
LD50/ID50 = 2.33E+02

death Albuquerque, N D., et al. "Murine Hepatitis Virus Strain 1 Produces a Clinically Relevant Model of Severe Acute Respiratory Syndrome in A/J Mice." Journal of Virology. 80 (2006): 21.
260, 261 0 rSARS-CoV PFU mice hACE-2 and A/J exponential
k = 2.46E-03
LD50/ID50 = 2.82E+02

death
261 4 MHV-1 PFU A/J mice exponential
k = 2.14E-03
LD50/ID50 = 3.24E+02

death Albuquerque, N D., et al. "Murine Hepatitis Virus Strain 1 Produces a Clinically Relevant Model of Severe Acute Respiratory Syndrome in A/J Mice." Journal of Virology. 80 (2006): 21.
Experiment ID:
260
# of Doses:
4
Agent Strain:
rSARS-CoV
Dose Units:
PFU
Host type:
mice hACE-2
Μodel:
exponential
Optimized parameters:
k = 2.97E-03
LD50/ID50 = 2.33E+02

Reference:
mice/rSARS-CoV strain model data [2]
Dose Dead Survived Total
240 1 2 3
800 3 0 3
2400 2 0 2
12000 6 0 6

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value		 χ20.95,m-k 
p-value
Exponential 0.968 -0.000923 3 3.84 
1		 7.81 
0.809
Beta Poisson 0.969 2 5.99 
0.616
Exponential is preferred to beta-Poisson; 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 2.97E-03 1.90E-03 1.90E-03 1.90E-03 2.97E-03 2.97E-03 2.97E-03
ID50/LD50/ETC* 2.33E+02 2.33E+02 2.33E+02 2.33E+02 3.64E+02 3.64E+02 3.64E+02
*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

Best Fit
Experiment ID:
260, 261
# of Doses:
0
Agent Strain:
rSARS-CoV
Dose Units:
PFU
Host type:
mice hACE-2 and A/J
Μodel:
exponential
Optimized parameters:
k = 2.46E-03
LD50/ID50 = 2.82E+02

Reference:
Pooled dose response model data [1]
Dose Dead Survived Total
5 0 5 5
50 1 4 5
240 1 2 3
500 3 2 5
800 3 0 3
2400 2 0 2
5000 5 0 5
12000 6 0 6

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value		 χ20.95,m-k 
p-value
Exponential 1.75 -0.00181 7 3.84 
1		 14.1 
0.972
Beta Poisson 1.75 6 12.6 
0.941
Exponential is preferred to beta-Poisson; 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 2.46E-03 1.07E-03 1.28E-03 1.35E-03 4.59E-03 5.27E-03 6.80E-03
ID50/LD50/ETC* 2.82E+02 1.02E+02 1.32E+02 1.51E+02 5.13E+02 5.43E+02 6.47E+02
*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

Experiment ID:
261
# of Doses:
4
Agent Strain:
MHV-1
Dose Units:
PFU
Host type:
A/J mice
Μodel:
exponential
Optimized parameters:
k = 2.14E-03
LD50/ID50 = 3.24E+02

Reference:
Mice/MHV-1 strains model data [3]
Dose Dead Survived Total
5 0 5 5
50 1 4 5
500 3 2 5
5000 5 0 5

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value		 χ20.95,m-k 
p-value
Exponential 0.606 0.0689 3 3.84 
0.793		 7.81 
0.895
Beta Poisson 0.537 2 5.99 
0.765
Exponential is preferred to beta-Poisson; 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 2.14E-03 6.25E-04 6.55E-04 9.06E-04 6.58E-03 6.58E-03 9.86E-03
ID50/LD50/ETC* 3.24E+02 7.03E+01 1.05E+02 1.05E+02 7.65E+02 1.06E+03 1.11E+03
*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