  In this article we propose a compartmental model for the dynamics of
Coronavirus Disease 2019 (COVID-19). We take into account the presence of
asymptomatic infections and the main policies that have been adopted so far to
contain the epidemic: isolation (or social distancing) of a portion of the
population, quarantine for confirmed cases and testing. We model isolation by
separating the population in two groups: one composed by key-workers that keep
working during the pandemic and have a usual contact rate, and a second group
consisting of people that are enforced/recommended to stay at home. We refer to
quarantine as strict isolation, and it is applied to confirmed infected cases.
In the proposed model, the proportion of people in isolation, the level of
contact reduction and the testing rate are control parameters that can vary in
time, representing policies that evolve in different stages. We obtain an
explicit expression for the basic reproduction number $\mathcal{R}_0$ in terms
of the parameters of the disease and of the control policies. In this way we
can quantify the effect that isolation and testing have in the evolution of the
epidemic. We present a series of simulations to illustrate different realistic
scenarios. From the expression of $\mathcal{R}_0$ and the simulations we
conclude that isolation (social distancing) and testing among asymptomatic
cases are fundamental actions to control the epidemic, {and the stricter these
measures are and the sooner they are implemented,} the more lives can be saved.
Additionally, we show that people that remain in isolation significantly reduce
their probability of contagion, so risk groups should be recommended to
maintain a low contact rate during the course of the epidemic.
