No I didn't misunderstand that. The curves start off looking like a simple exponential (E^t/tau) which increases slope forever. That is the scariest part by far. But then they start to lose slope, so they are no longer that simple exponential. True they never were, but they behaved like one in the beginning. A constant slope in rate of death or infection may be a horrible thing, but it is not nearly as scary as a constant increase in slope in the rate of death or infection. That is what I was saying, and that is simply true. I didn't say everything was ok, I said I stopped shitting my pants.
There are exponentials that asymptote to some fixed value and no slope, (1 - E^-t/tau) but those start out with a finite slope and then continuously decrease slope, not start with zero slope then increase slope such as our curves. If these curves where well behaved and symmetric, they might model ok as sigmoidal curves which are related to sign waves and so can probably be represented as complex exponentials (function of E^ix). But that is the only kind of exponential that resembles these curves for their entirety. These are not exponentials. Just portions of them are shaped like exponentials. This stuff is not my area, I would agree with that. I have never spent any time on bio / disease spread modeling, and if someone on the board has, and that could very well be, I hope I haven't taken up any of their oxygen.
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In response to:
"I don't think you fully understand the difference between a "linear" increase and an "exponential" increase. When you say "now we are seeing a very small linear increase" -- it's just patently false. Even if it flattened out now, we're looking at 20K new cases per day. That's a *huge* linear increase, even if it's not growing exponentially. You're looking at the difference in changes, rather than the changes themselves.
To go back to high school calculus -- you're looking at the derivative, when we need to be looking at the integral. We care about the area under the curve (i.e., new cases), not whether the curve is flat (an increase in the daily new cases). Even one new case shows that transmission is not under control and the problem continues to grow."
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