Wesley Johnson, 1 Mark Thurmond, 2 Young Choi, 1 Andres Perez 2 1 Department of Statistics,

1. A Bayesian method to discriminate between FMDV-infected and uninfected animals, in vaccinated or unvaccinated populations Wesley Johnson,1 Mark Thurmond,2 Young Choi,1 Andres Perez2 1Department of Statistics, 2Department of Medicine and Epidemiology, University of California, Davis CA

2. Key questions • What is the probability that an individual animal is infected, given its serologic value? • What is the prevalence of infection in a herd, flock, or region, given the serologic values of a sample of the animals? • How are these questions addressed in the face of vaccination?

3. Problems in serologic detection of FMDV-infected and vaccinated animals • Discrimination between vaccinal and infection antibodies is not perfect • Loss of information in dichotomized serologic result (‘positive’ or ‘negative’) • Predictive values require an estimate of prevalence • Prevalence typically unknown

4. Problem in use of dichotomized serologic results (positive or negative) Dichotomization, based on a cutoff value • Ignores the scale of information in the serologic assay values • Gives same result to divergent values above (or below) threshold (eg. A and B, or C and D) • Gives different results to very similar values on each side of the cutoff value (eg. B and C) Negative Positive A B C D x 0 Cutoff value (O.D., S/P)

5. Example of Bayesian inference (eg. predictive value positive/negative with traditional dichotomized result) • Inputs: • Pr ( T+ | Inf) (eg. Se) • Pr ( T+ | Not Inf) (eg. 1-Sp) • Pr ( Inf) = True prevalence (unknown?) • Output: (using Bayes’ Theorem) • Pr ( Inf | T+) = prev*Se/(prev*Se +(1-prev)*(1-Sp)) • Pr (Inf|T-) = prev*(1-Se)/(prev*(1-Se)+ (1-prev)*Sp)

6. Bayesian PDA Functions for infected and uninfected animals Probability density

7. Probability of animal infection and estimating within herd prevalence • Sample 100 animals and obtain serologic values (S1,S2,….S100) • Model: f( Si | Inf) , f( Si | Not Inf) (previous picture) • Diagnosis for animal i: (Bayes Theorem) Pr(Inf|Si) =prev*f( Si | Inf)/ [prev*f( Si | Inf)+(1-prev)*f( Si | Not Inf)] • Prevalence unknown but can be estimated using the above data in the context of this modeling

8. Vaccination • Sample vaccinated animals. In the same way as above. Can obtain • Pr (Inf | Vac, S) • And sampling non-vaccinated animals, similarly obtain • Pr (Inf | NVac, S) • Can estimate prevalence among Inf and Non-inf populations sampled

9. Illustration • Assume appropriate (training) data have been collected to estimate f( S | Vac, Inf) (n=197) and f( S| Vac, No-Inf) (n=553) • Assume a sample of Vaccinated (Infection status unknown) Animals was taken (n = 115) • Add some “prior” scientific information about the distribution of serology values for infected and non-infected animals

10. And some prior information about the prevalence of infection in the sampled population • All elicited from clinician independently of the data being used • The following ELISA values were obtained for animals in the sample

12. The estimated prevalence of infection in the herd sampled was 0.19 (0.12, 0.27)

15. Problem can be generalized to incorporate additional covariate information, X • Model f( S | Inf, X) and f( S | Not Inf, X) • Calculate: Pr (Inf | S, X)

16. Data needed to assess Bayesian PDA for FMD • Serologic values (O.D.) for animal groups • Infected, vaccinated • Infected, unvaccinated • Uninfected, non-vaccinated • Uninfected, vaccinated • And a new sample of animals with unknown infection status • Covariate information for each animal • Time between vaccination and sampling • Number of previous vaccinations • Age • Species • Gender