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Early detection and prediction of the size and the peak time of an epidemic outbreak (malicious or natural) is of crucial importance for a timely medical response (quarantine, vaccination, etc). A conventional approach to this problem is based on large scale agent-based computer simulations. This paper proposes an alternative framework formulated in the context of stochastic nonlinear filtering. The framework is based on the stochastic SIR epidemiological model of infection dynamics, with syndromic (often non-medical) observations of the number of infected people (e.g. visits to pharmacies, sale of certain products, absenteeism from work/study etc.). The unknown parameters of the SIR epidemic model are estimated via the sequential Monte Carlo method, with the prediction based on the dynamic model. The numerical results indicate that the proposed framework can provide useful early prediction of the epidemic peak if the uncertainty in prior knowledge of model parameters is not excessive.