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We address the problem of assessing the fit of stochastic epidemic models to data. Two novel model assessment methods are developed, based on disease progression curves, namely the distance method and the position-time method. The methods are illustrated using SIR (susceptible-infective-removed) models. We assume a typical data observation setting in which case-detection times are observed while infection times are not. Both methods involve Bayesian posterior predictive checking, in which the observed data are compared to data generated from the posterior predictive distribution. The distance method does this by calculating distances between disease progression curves, while the position-time method does this pointwise at suitably selected time points. Both methods provide visual and quantitative outputs with meaningful interpretations. The performance of the methods benefits from the development and application of a time-shifting method that accounts for the random time delay until an epidemic takes off. Extensive simulation studies show that both methods can successfully be used to assess the choice of infectious period distribution and the choice of infection rate function. [ABSTRACT FROM AUTHOR]