Statistical inference with missing data is a recurrent issue in epidemiology where the infection process is only partially observable. In this paper, Approximate Bayesian Computation, an alternative to data imputation methods such as Markov Chain Monte Carlo integration, is proposed for making inference in epidemiological models. This method of inference is not based on the likelihood function and relies exclusively on numerical simulations of the model. 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 detected individuals as a function of time. In a simple SIR model, we show that the posterior distributions obtained with ABC are similar to those obtained with MCMC. When detection times are binned or noisy, we introduce a vector of summary statistics for which several variants of the ABC can be applied. In a refined SIR model well-suited to the HIV contact-tracing program in Cuba, we perform a comparison between ABC with full and with binned data. The last section deals with the analysis of the Cuban HIV-AIDS data. We evaluate the efficiency of the detection system, and predict the evolution of the HIV-AIDS disease in the forthcoming years. We show in particular that the percentage of undetected infectious individuals among the contaminated population might be of the order of 40%.