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The rapid spread of SARS-CoV-2/COVID-19 in the first months of 2020 overburdened health systems worldwide. The absence of vaccines led public authorities to respond to the pandemic by adopting nonpharmaceutical interventions, mainly social distancing policies. Yet concerns have been raised on the economic impact of such measures. Considering the impracticability of conducting controlled experiments to assess the effectiveness of such interventions, mathematical models have played an essential role in helping decision makers. Here we present a simple modified SIR (susceptible-infectious-recovered) model that includes social distancing and two extra compartments (hospitalized and dead due to the disease). Our model also incorporates the potential increase in the mortality rate due to the health system saturation. Results from numerical experiments corroborate the striking role of social distancing policies in lowering and delaying the epidemic peak, thus reducing the demand for intensive health care and the overall mortality. We also probed into optimal social distancing policies that avoid the health system saturation and minimize the economic downturn.