Measles Elimination in Italy

Eric Thompson
April 26, 2021

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, pp.124-131.

Scientific Context

  • Measles outbreaks continue to take place in Italy and worldwide
  • School immunization campaign in Italy (2004-2005)
  • Antibody counts (serum samples) taken in:
    • 2003 (pre-campaign), and
    • 2005-2006 (post-campaign)

Research Questions

  • How effective was the school immunization campaign?

  • Which cohorts (subpopulations) remain susceptible or weakly immune after the campaign? (Possible targets for intervention, e.g. catch-up vaccination campaigns)

Mixture Models

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

http://blog.bogatron.net/blog/2014/02/02/visualizing-dirichlet-distributions/,/small.

Bayesian Model

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

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

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

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

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

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

Bayesian Model - Remarks

  • \( Y_{i} \) 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.
    • (Note: Dirichlet distribution is conjugate for the categorical distribution.)
  • 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
  • Penalized expected deviance (PED) used as selection criterion for the best goodness-of-fit among possible models

MCMC Details

  • 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 \( f(a) \)

Results

  • slightly larger prevalence in the youngest age groups (may reflect vaccination campaign)

  • Average antibody levels in adultsare lower in 2005–2006 than in 2003

Results (con't)

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