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Since the start of the COVID-19 pandemic in China many models have appeared in the literature, trying to simulate its dynamics. Focusing on modeling the biological and sociological mechanisms which influence the disease spread, the basic reference example is the SIR model. However, it is too simple to be able to model those mechanisms (including the three main types of control measures: social distancing, contact tracing and health system measures) to fit real data and to simulate possible future scenarios. A question, then, arises: how much and how do we need to complexify a SIR model? We develop a θ -SEIHQRD model, which may be the simplest one satisfying the mentioned requirements for arbitrary territories and can be simplified in particular cases. We show its very good performance in the Italian case and study different future scenarios. • Mathematical model for coronavirus disease (COVID-19). • New θ -SEIHQRD model taking into account quarantine and undetected infections. • Validation of the model with the reported data for Italy. • New formula for the effective reproduction number. • Different scenarios to show the impact of undetected cases and control measures. [ABSTRACT FROM AUTHOR]