Perhaps it helps to highlight some of the basic ideas behind the models underlying many of the strategies currently in play. A simple, but frequently used one are SEIR models or their variations. These are simple differential equations calculating how the rate at which folks move from Susceptible to Exposed to Infected and finally to Recovery status. You can think it as a simple compartment model in which the population is part of either of these compartments. Values such as effective reproduction number (how many people an infected person infects), length for which a person is infectious, initial number of infectious people and so on then determine the movement of the population through these compartments.
From there, one looks at the expected number of infected folks at any given time and often using age-adjusted models (as well as other info, if available) one can estimate how many folks at any given time might need hospitalization. As the capacities can vary significantly, especially in larger countries, usually more local modelling is conducted to understand the health impact of different infection rates.
The major factor that measures such as social distancing and shutdowns are affecting are therefore the expected number of folks being infected in a given time frame (essentially by adjusting the effective reproduction number). This is pretty much an established effect by now, with different estimates for different measures (and areas). So basically the major strategies revolve around these measures, as well as increasing health care capacities, if possible. So one might allow the transmission to increase a little, provided that the health care facilities are free, but may want to shut down again, once they go up.
Two critical things we need to know, but don't know yet is how efficient immunization is going to be (either via vaccine or via infection) as well as a how long potential immunity will last. There are models for that, which in the simplest case just has a flow back from Recovered populations back to Susceptible after a certain time.