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The initial phase dynamics of an epidemic without containment measures is commonly well modelled using exponential growth models. However, in the presence of containment measures, the exponential model becomes less appropriate. Under the implementation of an isolation measure for detected infectives, we propose to model epidemic dynamics by fitting a flexible growth model curve to reported positive cases, and to infer the overall epidemic dynamics by introducing information on the detection/testing effort and recovery and death rates. The resulting modelling approach is close to the Susceptible-Infectious-Quarantined-Recovered model framework. We focused on predicting the peaks (time and size) in positive cases, active cases and new infections. We applied the approach to data from the COVID-19 outbreak in Italy. Fits on limited data before the observed peaks illustrate the ability of the flexible growth model to approach the estimates from the whole data. [ABSTRACT FROM AUTHOR]