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

4

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)

5

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

6

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

7

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



e2j : end of infection interval








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



e2j : end of infection interval








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








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










cdf infectious period distribution

sj : start contact





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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










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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

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


ej : infection moment


Tj : infectious period = (rj - ij)






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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




94/100

91/100

average infectious period

93/100

94/100

standard deviation infectious period distribution

95/100

97/100


92/100

92/100

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Comparison MLE and Bayesian analyses, R0 = 8




78/100

75/100


91/100

91/100


91/100

91/100


80/100

77/100

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Comparison MLE and Bayesian analyses, R0 = 0.5




85/100

82/100


91/100

92/100


88/100

89/100


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



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)


Thank you

[email protected]

This study was funded by the Dutch Ministry of Economic Affairs, Agriculture and Innovation

Classical and Bayesian analyses of transmission experiments Jantien Backer and Thomas Hagenaars...

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