Interesting in the abstract but we know that even simpler model works well in the case of the epidemic we're sadly watching.

In this model, change in the number of people is function of the number of people infect, that's it. The number of potentially infectable people is so large that changes in it's value won't enter into consideration until the US is at WWII-invasion level of fatalities.

dS/dt = Sr, for some value of r.

A simple differential equation indeed, solved by S = e^(t*R); exponential growth as we see (note, I'm not claiming my r here is R0 or whatever).