Classical and Bayesian analyses of transmission experiments Jantien Backer and Thomas Hagenaars...
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Transcript of Classical and Bayesian analyses of transmission experiments Jantien Backer and Thomas Hagenaars...
Classical and Bayesian analyses
of transmission experiments
Jantien Backer and Thomas Hagenaars
Epidemiology, Crisis management & DiagnosticsCentral Veterinary Institute of Wageningen UR
The Netherlands
InFER2011, 30th of March 2011
2
Background
Transmission experiments typical in veterinary epidemiology controlled environment known inoculation moments infection process monitored by regular sampling
Analysis Maximum Likelihood Estimation:
• straightforward but discretizations and assumptions necessary Bayesian:
• more flexible (e.g. prior information, test characteristics) but more laborious
Transmission experiments ideally suited for comparison of analyses
3
Outline
Example transmission experiment MLE analysis Bayesian analysis
Comparison MLE and Bayesian analyses 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?
Summary
Next steps
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Transmission experiment
inoculated animal
infectious animal
contact (susceptible) animal
removed animal
day 0 day 1 day 2 - 20 day 21
vaccinated population of chickenschallenged with Highly Pathogenic Avian Influenza
H5N1(data J.A. van der Goot)
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0 1 2 3 4 5 6 7 8 9 10 14 17 21+ + + + †+ + + †+ + + + + + + + + + - + ++ + + + + + + + + - + + ++ + + + †
- + + + + + †- - + + + + + + + + + +- + - + + + + + †- + + + + + †- - + + + + + †
0 1 2 3 4 5 6 7 8 9 10 14 17 21+ + + + †+ + + + + + + †+ + + †+ + + + †+ + + †
+ - + + + + + + †- + + + + + †- - + - + + + + + †- - + + + + + + + †- - - - + + + + + + †
Transmission experiment
assumed: SIR model
infection interval
infectious interval
removal interval
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MLE analysis
determine loglikelihood function
maximize loglikelihood function MLE transmission rate parameter MLE infectious period distribution MLE reproduction number R0
construct 95% confidence interval from likelihood profile using likelihood ratio test
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MLE analysis
1 2
contact animals 1
logL
ln exp ln 1 expj j
j j
e e
j s e
I t dt I t dtN N
sj : start of contact
e1j : start of infection interval
e2j : end infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
probability of escaping infection
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation infectious period
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1 2
contact animals 1
logL
ln exp ln 1 expj j
j j
e e
j s e
I t dt I t dtN N
sj : start of contact
e1j : start of infection interval
e2j : end of infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation infectious period
probability of infection in interval (e1j, e2j)
MLE analysis
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MLE analysis
1 2
contact animals 1
logL
ln exp ln 1 expj j
j j
e e
j s e
I t dt I t dtN N
sj : start of contact
e1j : start of infection interval
e2j : end of infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation infectious period
10
MLE analysis
1 2
contact animals 1
infectious animals
logL , ,
ln exp ln 1 exp
1 ln ; , ln 1 ; ,
j j
j j
e e
j s e
j j j jj
I t dt I t dtN N
c g T c G T
sj : start contact
e1j : start infection interval
e2j : end infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation of infectious period
pdf infectious period distribution
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MLE analysis
1 2
contact animals 1
infectious animals
logL , ,
ln exp ln 1 exp
1 ln ; , ln 1 ; ,
j j
j j
e e
j s e
j j j jj
I t dt I t dtN N
c g T c G T
sj : start contact
e1j : start infection interval
e2j : end infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation of infectious period
cdf infectious period distribution
sj : start contact
e1j : start infection interval
e2j : end infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
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MLE analysis
1 2
contact animals 1
infectious animals
logL ,
ln exp ln 1 exp
1 ln ; , ln 1 ; ,
j j
j j
e e
j s e
j j j jj
I t dt I t dtN N
c g T c G T
sj : start contact
e1j : start infection interval
e2j : end infection interval
cj : censoring infectious period (boolean)
Tj : infectious period = ½ (r1j + r2j) - ½ (i1j + i2j)
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation of infectious period
13
MLE analysis
1 2
contact animals 1
infectious animals
logL , ,
ln exp ln 1 exp
1 ln ; , ln 1 ; ,
j j
j j
e e
j s e
j j j jj
I t dt I t dtN N
c g T c G T
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)
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Bayesian analysis
determine likelihood function
choose prior distributions uninformative Ga (0.01, 0.01)
adjust proposal distributions during convergence to achieve acceptance rate of 40% - 60%
MCMC chain (length 10000) 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
15
Bayesian analysis
contact animals
infectious animals
L , ,
exp
1 ; , 1 ; ,
j
j
ej
j s
j j j jj
I eI t dt
N N
c g T c G T
sj : start of contact
ej : infection moment
cj : censoring infectious period (boolean)
Tj : infectious period = (rj - ij)
β : transmission rate parameter
N : total number of animals
I(t) : number of infectious animals at time t
μ : average infectious period
σ : standard deviation of infectious period
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Bayesian analysis
medβ = 0.79 (0.39 - 1.40) medμ = 8.7 (6.5 – 12.5)
medσ = 5.9 (3.9 – 10.5)medR0 = 6.8 (3.2 – 13.4)
β = 0.82 (0.41 - 1.46) μ = 8.5 (6.4 – 12.2)
σ = 5.6 (3.7 – 9.9)R0 = 7.0 (3.3 – 13.7)
transmission parameter β average infectious period µ
standard deviation σ of infectious period distribution
reproduction number R0
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Comparison MLE and Bayesian analyses Simulated transmission experiments
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
# contact infections # contact infections # contact infections
R0 = 0.5 R0 = 2 R0 = 8
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Comparison MLE and Bayesian analyses
95% confidence intervalML estimate
95% credible intervalmedian parameter value
transmission parameter β
MLE coverage: 94/100MLE coverage: 94/100
Bayesian coverage: 91/100
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Comparison MLE and Bayesian analyses, R0 = 2
95% confidence intervalML estimate
95% credible intervalmedian parameter value
transmission parameter β
94/100
91/100
average infectious period
93/100
94/100
standard deviation infectious period distribution
95/100
97/100
reproduction number R0
92/100
92/100
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Comparison MLE and Bayesian analyses, R0 = 8
95% confidence intervalML estimate
95% credible intervalmedian parameter value
transmission parameter β
78/100
75/100
average infectious period
91/100
91/100
standard deviation infectious period distribution
91/100
91/100
reproduction number R0
80/100
77/100
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Comparison MLE and Bayesian analyses, R0 = 0.5
95% confidence intervalML estimate
95% credible intervalmedian parameter value
transmission parameter β
85/100
82/100
average infectious period
91/100
92/100
standard deviation infectious period distribution
88/100
89/100
reproduction number R0
83/100
83/100
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Summary
Results MLE and Bayesian analyses 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
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Next steps
Bayesian analysis include latent period estimation implement test characteristics extend to larger groups with unobserved infections
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Comparison MLE and Bayesian analyses: latent period
95% confidence intervalML estimate
95% credible intervalmedian parameter value
assumed 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
Thank you
This study was funded by the Dutch Ministry of Economic Affairs, Agriculture and Innovation