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Well if you have the capability to diagnose 500 patients per month, then you are going to be reporting around 500 new infections each month regardless of whether there are 500 or 500,000. Some of the articles from last week made it sound like most of the blood testing labs are already at capacity, which would make these numbers basically meaningless.

Not all patients are diagnosed in hospital.

I'm not sure what to expect - linear, geometric, exponential. What do statistical models of infection normally look like?

Typically any form of population growth can be modeled as exponential. Any type of percentage growth rate is already an exponential since P=P(0)*e^rt in the most simple case.

The main limitations on this would be the carrying capacity of the environment. Or treatment. Carrying capacity would be as if everyone is sick and there are no new hosts or the population has dropped to such levels chance of infection due to contact decreases. And obviously growth rate would also change if the treatment status quo changes

That's what I thought too. But the curve is already looking pretty scary.