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 Monte Carlo Markov chain 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. We apply the Approximate Bayesian Computation framework to calibrate an epidemiological model dedicated to the analysis of the HIV contact-tracing program in Cuba. We first evaluate numerically, using synthetic data sets, the statistical properties of the estimated posterior distributions obtained with different variants of Approximate Bayesian Computation. Then, once the epidemiological model has been calibrated with the Cuban VIH database, we make predictions concerning the efficiency of the detection system, and the evolution of the disease in the forthcoming years.