Microbial Risk Assessment Guideline

• The Exponential Model
• The beta-Poisson Model
• The Gamma distribution
• The Beta distribution
• Probability of exposure and infection
• Generalized Framework for organisms

• single-hit familty of dose-response model

\[ p(d) = 1 - e^{-r d} \] where

• \( p(d) \) is the cumulative probability of infection in the exposed population
• \( d \) is the pathogen dose in infectious units (organisms)

• \( r \) is the probability of infection given ingestion of one organism.

  • Assume the same probability of infection for each individual.
  • Good fit for a number of hte human pathogen-challenge datasets.

• assigns a distrbution to \( r \) to represent the variability in the pathogen-host interaction

• The beta-Poisson cumulative probability is

\[ p(d) = 1 - M(\alpha, \alpha + \beta, -d) \] where

• \( p(d) \) is the cumulative probability of infection in the exposed population
• \( d \) is the pathogen dose in infectious units (organisms)
• \( \alpha \), \( \beta \) are the beta distribution parameters.
• \( M \) is the confluent hypergeometric function

\[ p(d) = 1 - (1 + d/\beta)^{-\alpha} \] where

• \( p(d) \) is the cumulative probability of infection in the exposed population
• \( d \) is the pathogen dose in infectious units (organisms)
• \( \alpha \), \( \beta \) are the beta distribution parameters.
• \( M \) is the confluent hypergeometric function

\[ f(d | \alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} d^{\alpha - 1} (1 - d)^{\beta - 1}, 0 \leq d \leq 1 \] where

• \( f \) is the density function
• \( d \) is the pathogen dose in infectious units (organisms)
• \( \alpha \), \( \beta \) are the beta distribution parameters.
• \( \Gamma \) is the gamma function \[ \Gamma(t) = \int_{0}^{\infty} x^{t-1} e^{-x} dx \]

• The risk of infection applies only to the susceptible population

\[ (1 - fraction) \times F(d, \theta) \] where \( F \) is the dose-response function and \( \theta \) is the parameter vector.

• $P_{exp}(\( j \) | Dose) = Probability of having \( j \) pathogenic microbes in an ingested dose.

• $P_{Inf}(\( j \) | Inf) = Conditional probability of infection from \( j \) pathogens ingested.

• Probability of exposure

\[ P_{exp}(j) = \frac{\mu^j}{j!} e^{-\mu} \]

\[ P_{Inf}(j) = 1 - e^{-j \mu} \]

• Estimate changes in annual illness counts based on changes in the frequency of occurrence among food commodities. \[ P(ill) = P(ill | exp) P(exp) \]
  
  • \( P(ill) \) = prob. illness from a product-pathogen pairing across a population
  • \( P(ill|exp) \) = prob. exposure to a random contaminated serving will produce illness
  • \( P(exp) \) = frequency of exposure to the pathogen on a per serving basis.

\[ I_{Avoided} ~ Poisson\left[\left( 1 - \frac{P_{new}(exp)}{P_{initial}(exp)} \right) \lambda_{ill} \right] \] where

• \( I_{Avoided} \) annual illness avoided.
• \( \lambda_{ill} \) annual rate of illnesses.
• \( P_{new}(exp) \) = frequency of new exposure to the pathogen on a per serving basis.
• \( P_{initial}(exp) \) = frequency of initially exposure to the pathogen on a per serving basis.

• Organisms ingested –> organisms survive to colonize –> sufficient colonies to cause effect

\[ P_d(d) = \sum_{k_{min} = 1}^{\infty} \sum_{k = k_{min}}^{\infty} \sum_{j = k}^{\infty} P_1(j | d) P_2(k|j) P(k_{min}) \] where

• \( P(k_{min}) \) = fraction of subjects that require \( k_{min} \) original organisms to survive in order to become infected
• \( P_1(j|d) \) = fraction of subjects ingesting from an average dose \( d \) who actually ingest \( j \) organisms (Poisson)
• \( P_2(k|j) \) = fraction of subjects ingesting \( j \) organisms in which \( k \) organisms survive (binomial; beta-binomial)

\[ f(P_i) = \frac{T_i!}{P_i!(T_i - P_i)!} \pi_{i}^{P_i} (1 - \pi_i)^{T_i - P_i} \] where

• \( T_i \) = total subjects
• \( P_i \) = all positive subjects
• All subjects exposed to (poisson average) dose \( d_i \)
• \( \pi_i \) = positive probability

• Use likelihood criteria \[ \ln(L) = \sum_{i = 1}^{N} \ln(f_i(P_i)) \]
• \( \pi_i^0 \) = \( \frac{P_i}{T_i} \)