  Missing data is a recurrent issue in epidemiology where the infection process
may be partially observed. Approximate Bayesian Computation, an alternative to
data imputation methods such as Markov Chain Monte Carlo integration, is
proposed for making inference in epidemiological models. It is a
likelihood-free method that relies exclusively on numerical simulations. ABC
consists in computing a distance between simulated and observed summary
statistics and weighting the simulations according to this distance. We propose
an original extension of ABC to path-valued summary statistics, corresponding
to the cumulated number of detections as a function of time. For a standard
compartmental model with Suceptible, Infectious and Recovered individuals
(SIR), we show that the posterior distributions obtained with ABC and MCMC are
similar. In a refined SIR model well-suited to the HIV contact-tracing data in
Cuba, we perform a comparison between ABC with full and binned detection times.
For the Cuban data, we evaluate the efficiency of the detection system and
predict the evolution of the HIV-AIDS disease. In particular, the percentage of
undetected infectious individuals is found to be of the order of 40%.
