In this research paper we modify a classical SIR model to better adapt to the dynamics of COVID-19, that is we propose the heterogeneous SQAIRD model where COVID-19 spreads over a population of economic agents, namely: the elderly, adults and young people. We then compute and simulate an optimal control problem faced by a Government, where its objective is to minimize the costs generated by the pandemics using as control a compulsory quarantine measure (that is, a lockdown). We first analyze the problem from a theoretical perspective, claiming that different lockdown policies (total lockdown, no lockdown or partial lockdown) may justified by different cost (concave or convex) structures of the economies. We then focus on a particular cost structure (convex costs) and we simulate a targeted optimal policy vs. a uniform optimal policy, by dividing the whole population in three demographic groups (young, adults and old). We also simulate the dynamic of the pandemic with no policy implemented. Simulations highlighted the fact that: a) a policy of lockdown is always better than the \emph{laissez faire} policy, because it limits the costs that the pandemic generates in an uncontrolled situation
b) a targeted policy based on age of the individuals outperforms a uniform policy in terms of costs that it generates, being a targeted policy less costly and equally effective in the control of the pandemic.