1. 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, 30 th of March 2011

2. 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. 3 Outline Example transmission experiment MLE analysis Bayesian analysis Comparison MLE and Bayesian analyses simulated transmission experiments for low, medium and high R 0 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. 4 Transmission experiment inoculated animal infectious animal contact (susceptible) animal removed animal day 0 day 1day 2 - 20 day 21 vaccinated population of chickens challenged with Highly Pathogenic Avian Influenza H5N1 (data J.A. van der Goot)

5. 5 012345678910141721 ++++† +++† ++++++++++-++ +++++++++-+++ ++++† -+++++† --++++++++++ -+-+++++† -+++++† --+++++† 012345678910141721 ++++† +++++++† +++† ++++† +++† +-++++++† -+++++† --+-+++++† --+++++++† ----++++++† Transmission experiment assumed: SIR model infection interval infectious interval removal interval

6. 6 MLE analysis determine loglikelihood function maximize loglikelihood function MLE transmission rate parameter MLE infectious period distribution MLE reproduction number R 0 construct 95% confidence interval from likelihood profile using likelihood ratio test

7. 7 MLE analysis s j : start of contact e1 j : start of infection interval e2 j : end infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j ) 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

8. 8 s j : start of contact e1 j : start of infection interval e2 j : end of infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j ) 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 (e1 j, e2 j ) MLE analysis

9. 9 s j : start of contact e1 j : start of infection interval e2 j : end of infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j ) 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. 10 MLE analysis s j : start contact e1 j : start infection interval e2 j : end infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j ) 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

11. 11 MLE analysis s j : start contact e1 j : start infection interval e2 j : end infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j ) 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 s j : start contact e1 j : start infection interval e2 j : end infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j )

12. 12 MLE analysis s j : start contact e1 j : start infection interval e2 j : end infection interval c j : censoring infectious period (boolean) T j : infectious period = ½ (r1 j + r2 j ) - ½ (i1 j + i2 j ) 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. 13 MLE analysis 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 R 0 = βμ = 7.0 (3.3 – 13.7)

14. 14 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. 15 Bayesian analysis s j : start of contact e j : infection moment c j : censoring infectious period (boolean) T j : infectious period = (r j - i j ) 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

16. 16 Bayesian analysis medβ = 0.79 (0.39 - 1.40) medμ = 8.7 (6.5 – 12.5) medσ = 5.9 (3.9 – 10.5)medR 0 = 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) R 0 = 7.0 (3.3 – 13.7) transmission parameter βaverage infectious period µ standard deviation σ of infectious period distributionreproduction number R 0

17. 17 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 R 0 = (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 R 0 = 0.5R 0 = 2R 0 = 8

18. 18 Comparison MLE and Bayesian analyses 95% confidence interval ML estimate 95% credible interval median parameter value transmission parameter β MLE coverage: 94/100 Bayesian coverage: 91/100

19. 19 Comparison MLE and Bayesian analyses, R 0 = 2 95% confidence interval ML estimate 95% credible interval median 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 R 0 92/100

20. 20 Comparison MLE and Bayesian analyses, R 0 = 8 95% confidence interval ML estimate 95% credible interval median parameter value transmission parameter β 78/100 75/100 average infectious period 91/100 standard deviation infectious period distribution 91/100 reproduction number R 0 80/100 77/100

21. 21 Comparison MLE and Bayesian analyses, R 0 = 0.5 95% confidence interval ML estimate 95% credible interval median 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 R 0 83/100

22. 22 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

23. 23 Next steps Bayesian analysis include latent period estimation implement test characteristics extend to larger groups with unobserved infections

24. 24 Comparison MLE and Bayesian analyses: latent period 95% confidence interval ML estimate 95% credible interval median parameter value assumed SEIR model average latent period of 2 days (and shape parameter of 4) reproduction number R 0 = 2 average latent period of all infected animals (with informative gamma prior) reproduction number R 0

25. Thank you jantien.backer@wur.nl This study was funded by the Dutch Ministry of Economic Affairs, Agriculture and Innovation