If it creeps up to 1.1, you still don't need many rounds of infection to completely overwhelm the system. 7 rounds of infection (~a month) would result in 10.4x more total patients.
That's new cases though, isn't it? I was summing over the entire "tree" to get the total instead.
If R0=2 (each person infects two more), you have 1 case at time zero, two after the first time step, and four after the next, but the total number of cases is 1+2+4=7, not 2^2.
It's true that people are discharged from the hospital eventually (one way or another), so there's a little extra room from that, but not much.
California is somewhere near steady-state (R=1) right now, but that's not a "stable" point, in the dynamical systems sense. A small decrease below R=1 would lead to falling rates, but a small increase above R=1 can make them skyrocket.
I'm not sure if there's good data on the length of hospital stays, but an early preprint from China reported a median stay of 19 days. That's not much help here: On day 20, the intake is 60% larger than the initial cohort--and only half of them (b/c median) will be discharged.
People with more data and expertise!) have worked up full SIR and utilization models, but the general point is that exponential growth is manageable, even boring, until suddenly it suddenly isn't.
If it creeps up to 1.1, you still don't need many rounds of infection to completely overwhelm the system. 7 rounds of infection (~a month) would result in 10.4x more total patients.