Classical and Bayesian analyses
of transmission experiments
Jantien Backer and Thomas Hagenaars
Epidemiology, Crisis management & Diagnostics
Central Veterinary Institute of Wageningen UR
The Netherlands
InFER2011, 30th of March 2011
Background
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Transmission experiments
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typical in veterinary epidemiology
controlled environment
known inoculation moments
infection process monitored by regular sampling
Analysis
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Maximum Likelihood Estimation:
• straightforward but discretizations and assumptions necessary
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Bayesian:
• more flexible (e.g. prior information, test characteristics) but more laborious
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Transmission experiments ideally suited for comparison of analyses
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Outline
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Example transmission experiment
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MLE analysis
Bayesian analysis
Comparison MLE and Bayesian analyses
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simulated transmission experiments for low, medium and high R0
how does ML estimate and median of posterior distribution relate?
is the true value included in confidence and/or credible interval?
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Summary

Next steps
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Transmission experiment
day 0
day 1
day 2 - 20
day 21
inoculated animal
infectious animal
contact (susceptible) animal
removed animal
vaccinated population of chickens
challenged with Highly Pathogenic Avian Influenza
4H5N1
Transmission experiment
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assumed: SIR model
infection interval
infectious interval
removal interval
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5
MLE analysis

determine loglikelihood function
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maximize loglikelihood function
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MLE transmission rate parameter
MLE infectious period distribution
MLE reproduction number R0
construct 95% confidence interval
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from likelihood profile
using likelihood ratio test
6
MLE analysis
logL    

 
ln exp  

 N

j  contact animals


e1 j

sj


 
I  t  dt    ln 1  exp  

 N




e2 j

e1 j

I  t  dt  


probability of escaping infection
sj : start of contact
β : transmission rate parameter
e1j : start of infection interval
N : total number of animals
e2j : end infection interval
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
σ : standard deviation infectious period
7
MLE analysis
logL    

 
ln exp  

 N

j  contact animals


e1 j

sj


 
I  t  dt    ln 1  exp  

 N




e2 j

e1 j

I  t  dt  


probability of infection in interval (e1j, e2j)
sj : start of contact
β : transmission rate parameter
e1j : start of infection interval
N : total number of animals
e2j : end of infection interval
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
σ : standard deviation infectious period
8
MLE analysis
logL    

 
ln exp  

 N

j  contact animals


e1 j

sj


 
I  t  dt    ln 1  exp  

 N




e2 j

e1 j

I  t  dt  


sj : start of contact
β : transmission rate parameter
e1j : start of infection interval
N : total number of animals
e2j : end of infection interval
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
σ : standard deviation infectious period
9
MLE analysis
logL   ,  ,   

 
ln exp  

 N

j  contact animals




  e2 j

s I  t  dt   ln 1  exp   N e1 I  t  dt 
j
j





1  c j  ln  g Tj ;  ,    c j ln 1  G T j ;  ,  

j  infectious animals
e1 j
pdf infectious period distribution
sj : start contact
β : transmission rate parameter
e1j : start infection interval
N : total number of animals
e2j : end infection interval
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
σ : standard deviation of infectious period
10
MLE analysis
logL   ,  ,   

 
ln exp  

 N

j  contact animals




  e2 j

s I  t  dt   ln 1  exp   N e1 I  t  dt 
j
j





1  c j  ln  g Tj ;  ,    c j ln 1  G T j ;  ,  

j  infectious animals
e1 j
cdf infectious period distribution
sj : start contact
β : transmission rate parameter
e1j : start infection interval
N : total number of animals
e2j : end infection interval
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
σ : standard deviation of infectious period
11
MLE analysis
logL   ,   

 
ln exp  

 N

j  contact animals




  e2 j

s I  t  dt   ln 1  exp   N e1 I  t  dt 
j
j





1  c j  ln  g Tj ;  ,    c j ln 1  G T j ;  ,  

j  infectious animals
e1 j
sj : start contact
β : transmission rate parameter
e1j : start infection interval
N : total number of animals
e2j : end infection interval
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
σ : standard deviation of infectious period
12
MLE analysis
logL   ,  ,   

 
ln exp  

 N

j  contact animals




  e2 j

s I  t  dt   ln 1  exp   N e1 I  t  dt 
j
j





1  c j  ln  g Tj ;  ,    c j ln 1  G T j ;  ,  

j  infectious animals
e1 j
β = 0.82 (0.41 – 1.46) day-1
μ = 8.5 (6.4 – 12.2) days
σ = 5.6 (3.7 – 9.9) days
R0 = βμ = 7.0 (3.3 – 13.7)
13
Bayesian analysis

determine likelihood function

choose prior distributions
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adjust proposal distributions
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during convergence
to achieve acceptance rate of 40% - 60%
MCMC chain (length 10000)
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uninformative Ga (0.01, 0.01)
update infection, infectious and removal moments: Metropolis-Hastings sampling (normal
proposal distributions)
update β: Gibbs sampling
update μ and σ: Metropolis-Hastings sampling (gamma proposal distributions)
construct 95% credible interval

from posterior parameter distributions
14
Bayesian analysis
L   , ,  

 I ej 
j  contact animals

N
 
exp  
 N


s I  t  dt 
j

ej


1  c j  g T j ;  ,    c j 1  G T j ;  ,   


j  infectious animals

β : transmission rate parameter
sj : start of contact
N : total number of animals
ej : infection moment
I(t) : number of infectious animals at time t
cj : censoring infectious period (boolean)
μ : average infectious period
Tj : infectious period = (rj - ij)
σ : standard deviation of infectious period
15
Bayesian analysis
medβ = 0.79 (0.39 - 1.40)
medμ = 8.7 (6.5 – 12.5)
β = 0.82 (0.41 - 1.46)
μ = 8.5 (6.4 – 12.2)
transmission parameter β
average infectious period µ
medR0 = 6.8 (3.2 – 13.4)
medσ = 5.9 (3.9 – 10.5)
R0 = 7.0 (3.3 – 13.7)
σ = 5.6 (3.7 – 9.9)
reproduction number R0
standard deviation σ of infectious period distribution
16
Comparison MLE and Bayesian analyses

Simulated transmission experiments
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SIR model
5 inoculated animals with 5 contact animals, two replicates
transmission rate parameter β = (0.125, 0.5, 2) day-1
average infectious period μ = 4 days
standard deviation infectious period σ = 2√2 (shape parameter of 2)
reproduction number R0 = (0.5, 2, 8)
sampling intervals of one day
end of experiment at day 14
in total 100 simulated transmission experiments per scenario
R0 = 0.5
# contact infections
R0 = 2
# contact infections
R0 = 8
# contact infections
17
Comparison MLE and Bayesian analyses
MLE coverage: 94/100
Bayesian coverage: 91/100
transmission parameter β
95% confidence interval
ML estimate
95% credible interval
median parameter value
18
Comparison MLE and Bayesian analyses, R0 = 2
94/100
93/100
91/100
94/100
transmission parameter β
reproduction number R0
average infectious period
92/100
95/100
92/100
97/100
standard deviation infectious period distribution
95% confidence interval
ML estimate
95% credible interval
median parameter value
19
Comparison MLE and Bayesian analyses, R0 = 8
78/100
91/100
75/100
91/100
transmission parameter β
reproduction number R0
average infectious period
80/100
91/100
77/100
91/100
standard deviation infectious period distribution
95% confidence interval
ML estimate
95% credible interval
median parameter value
20
Comparison MLE and Bayesian analyses, R0 = 0.5
85/100
91/100
82/100
92/100
transmission parameter β
reproduction number R0
average infectious period
83/100
88/100
83/100
89/100
standard deviation infectious period distribution
95% confidence interval
ML estimate
95% credible interval
median parameter value
21
Summary
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Results MLE and Bayesian analyses
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maximum likelihood estimate similar to median value of posterior
confidence interval comparable to credible interval
inclusion of true value in confidence and credible intervals comparable
22
Next steps
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Bayesian analysis

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include latent period estimation
implement test characteristics
extend to larger groups with unobserved infections
23
Comparison MLE and Bayesian analyses: latent period
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assumed
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SEIR model
average latent period of 2 days (and shape parameter of 4)
reproduction number R0 = 2
average latent period of all infected animals
(with informative gamma prior)
reproduction number R0
95% confidence interval
ML estimate
95% credible interval
median parameter value
24
Thank you
This study was funded by the Dutch Ministry of
Economic Affairs, Agriculture and Innovation
jantien.backer@wur.nl