The model uses the following notation:

**M** = Herd size = 25

**Lambda** = pats/cow/day = 12

**Tested** = number of pats tested = 10

**Positive** = number of tested pats that were positive = 2

**Se** = P(test positive is pat infected) = Beta(18+1,20-18+1) from binomial theory

**Sp** = P(test negative is pat not infected) = Beta(19+1,20-19+1) from binomial theory

**BadCows** = Prior distribution for infected cows = IntUniform(0,M)

A discrete uniform distribution representing an uninformed prior

**InfectedPats** = Number of ’fresh’ infected cow pats

=IF(BadCows=0,0,RiskPoisson(BadCows*Lambda))

The IF statement is there to prevent @RISK’s Poisson distribution from producing an error when its mean = 0. The Poisson distribution is modeling the number of pats that are laid by infected animals (BadCows), assuming that pats are laid randomly in time and that each cow lays on average ’Lambda’ pats per day

**NotInfectedPats** = Number of ’fresh’ pats from cows not infected

=IF(M=BadCows,0,RiskPoisson((M-BadCows)*Lambda))

The IF statement is there to prevent @RISK’s Poisson distribution from producing an error when its mean = 0. The Poisson distribution is modeling the number of pats that are laid by non-infected animals (M-BadCows), assuming that pats are laid randomly in time and that each cow lays on average ’Lambda’ pats per day. It assumes that cows lay the same average number of pats whether infected or not (maybe a big assumption).

**TotalPats** = Total number of pats produced that day

= InfectedPats + InfectedPats

**InfectedInSample** = Number of infected pats in the sample

=IF(InfectedPats=0,0,RiskHypergeo(Tested,InfectedPats,TotalPats))

Recognises that this is a Hypergeometric sample from a total of TotalPats, of which InfectedPats’ are infected, and Tested’ are sampled from this population.

**TruePos** = Number of infected pats that tested positive

IF(InfectedInSample=0,0,RiskBinomial(InfectedInSample,Se))

Each infected pat has probability Se of testing positive. The IF statement is there to prevent @RISK’s Binomial distribution from producing an error when the number of trials = 0.

**FalsePos** = Number of non-infected pats that tested positive

=IF(Tested-InfectedInSample=0,0,RiskBinomial(Tested-InfectedInSample,1-Sp))

Each non-infected pat has probability (1-Sp) of testing positive.

**Posterior** = estimated number of infected cows in herd

=RiskOutput(”Posterior”)+IF(TruePos+FalsePos=Positive,BadCows,NA())

Accepts values from the prior ’BadCows’ if the total number of positive pats (TruePos+FalsePos) equals the number observed ’positive’. The NA function generates an error that is not then processed by @RISK (i.e. plays no part in calculation of output statistics) beyond noting that the error was produced.

The simulation model with an IntUniform(0,25) prior produces the following posterior: