Variola major: Dose Response Models Dose response experiments

General Overview

Variola major (V. major) is the causative agent of smallpox. Smallpox is a disease within the poxvirus group, which causes diseases with similar manifestation and symptoms with very different severities. Chickenpox, a disease with very low mortality associated with childhood is within the poxvirus group as well, however, does not have the severity or lethality of smallpox.

Summary Data

Marshall and Verone (1961)  challenged suckling mice with varying doses of V. major intraperitoneally. Four different ages of suckling mice were exposed (10-hour, 35-hour, 85-hour, and 145-hour old mice) and death was the recorded response. The four different experiments are shown in Table 1. 

Table 1:

Experiment	Age (Hours)	Dose (PFU)	Number of Mice	Number of Positive Responses	Number of Negative Responses
1	10	100000	9	9	0
	10	10000	9	9	0
	10	1000	9	4	5
	10	100	9	3	6
	10	10	8	1	7
2	35	1000000	11	11	0
	35	100000	8	8	0
	35	10000	8	3	5
	35	1000	8	5	3
	35	100	7	1	6
3	85	2000000	9	9	0
	85	200000	9	7	2
	85	20000	8	4	4
	85	2000	7	1	6
4	145	5000000	9	8	1
	145	500000	9	6	3
	145	50000	8	0	8
	145	5000	8	0	8

Weir and Haas (2009) fit the conventional exponential and beta-Poisson models to each individual dataset and a pooled dataset of all ages combined. Neither model provided a good fit to the pooled dataset, and it was determined that age was significantly influencing the dose response relationship. A modification of the exponential and beta-Poisson models was explored to evaluate the influence of incorporating age-dependent parameters. Equation 1 is the age-modification made to the exponential k parameter, such that k* is calculated with k₀, k₁ and the age of the mouse in hours. Similarly, Equations 2 and 3 and the age-modifications to the approximate beta-Poisson parameters of α^* and N50*. 

 Eq. 1

 Eq. 2

 Eq. 3

The age-dependent models were fit to the pooled datasets (n=18). Incorporating an age dependency improved the accuracy of the beta-Poisson model by 72%, and thus the age dependent beta-Poisson with parameters α0, α1, N50₀ and N50₁ is the recommended model for use as shown in Table 2. The modified beta-Poisson models for each age group are shown in Figure 1 and the bootstrapped parameter estimates are shown in Figure 2. More information about methods can be found in Weir & Haas (2009) . 

 

Table 2:

ID	# of Doses	Agent Strain	Dose Units	Host type	Μodel	Optimized parameters	Response type	Reference
1	5	V. major	PFU	Suckling Mice (0-24 hours)	Age-Dependent Beta-Poisson	α0 = 0.377; α1 = 0.00264; N500 = 184.683; N501 = 0.0501	Death
2	5	V. major	PFU	Suckling Mice (24-48 hours)
3	4	V. major	PFU	Suckling Mice (48-120 hours)
4	4	V. major	PFU	Suckling Mice (120-168 hours)

 

 

Figure 1: Modified beta Poisson dose-response models for each age group, encapsulated in overall confidence bounds for the modified model.

 

Figure 2: Comparison of bootstrapped age dependent parameters after 10,000 iterations.