Measles Elimination in Italy

Del Fava, E., Shkedy, Z., Bechini, A., Bonanni, P. and Manfredi, P., 2012. Towards measles elimination in Italy: Monitoring herd immunity by Bayesian mixture modelling of serological data. Epidemics, 4(3), pp.124-131.

• Measles outbreaks continue to take place in Tuscany (Italy)
• School immunization campaign (2004-2005)
• Antibody counts (serum samples) taken prior (2003) and after (2005-2006) campaign

• How effective was the school immunization campaign?
• Which cohorts still exhibited susceptibility, or were weakly immune, after the campaign?

Contreras Carrasco, Oscar. “Gaussian Mixture Models Explained.” Towards Data Science, 2 June 2019, towardsdatascience.com/gaussian-mixture-models-explained-6986aaf5a95.

\( Y_{i} \sim N(\mu_{j}(T_{ij}), \sigma_{j}^{2}) \)

\( T_{ij} \sim Categorical(\pi_{j}(a_{i})) \)

\( \mu_{j} \stackrel{iid}{\sim} U(Y_{min}, Y_{max}) \)

\( \sigma_{j} \sim Inv-Gamma(0.01, 100) \)

\( (\pi_{1}(a), ..., \pi_{J}(a)) \sim Dirichlet(\alpha_{1}=1, ..., \alpha_{J}=1) \)

for \( j=1, ..., J \) components, \( i = 1, ..., n \) subjects and \( a=1, ..., a_{max} \) ages.

• \( Y_{i} \) is antibody concentration, \( log_{10}mUI/mL \)
• \( T_{i} \) is a latent classification random variable that is equal to 1 if subject \( i \) belongs to group \( j \), and zero otherwise.
• \( \mu_{j} \) and \( \sigma_{j}^{2} \) are the component-specific locations and scales, respectively
• \( \pi_{j}(a_{i}) \) are the mixing weights representing the probability that an individual aged \( a_{i} \) belongs to the \( j \)-th mixture component.
• \( Inv-Gamma() \) and \( Dirichlet \) selected to create flat priors
• must have \( \sum_{j=1}^{J} \pi_{j}(a_{i})=1 \)
• Number of components, \( J \), equals 3 for 2003 samples and 4 for 2004-2005 samples

(Remark: Dirichlet distribution is conjugate for the categorical distribution.)

• JAGS used for Gibbs Sampling
• No details for number of chains, burn-in, length or thinning
• Smoothing for every \( m \) saved replicate of the prevalence (\( \pi \)) using a GAM with logit link function: \( log(\pi_{j}(a)^{m}) / (1 - log(\pi_{j}(a)^{m}) = \beta_{0} + \beta_{1}f(a) \)

where P-splines are applied as a smoother for the nonparametric function \( f(a) \)

• Penalized expected deviance (PED) used as selection criterion for the best goodness-of-fit among possible models

• group of susceptible individuals about 13-14 years old
• larger group of weakly immune (“low responders”) individuals about 20 years of age
  

\( \mu_{j} \sim U(Y_{min}, Y_{max}) \) for \( \mu_{1}, ..., \mu_{J} \)

\( \tau_{j} = 1/\sigma_{j}^2 \sim \Gamma(\epsilon, \epsilon) \)

Can be rewritten as: \( g(Y_{i}) = \sum_{j=1}^{J} \pi_{j}(a_{i}) N(Y_{i} | \mu_{j}, \sigma_{j}^{2}) \), for \( i=1, ..., n \)