But I understand what you are saying. I'm just not convinced yet that the math is right. And I think it's best to start with a day to day formula instead of starting with what exponent we can apply. And then once the day to day formula is developed, work into the exponent.
If your DTR is actually measuring the acceleration of the rate of transmission, given the current slope, and the nature of the disease, I'm not sure how I would predict acceleration in the future. The chart is definitely a curve right now and showing acceleration, whereas my formula assumes a constant rate of transmission and thus a linear progression. I simply pick recent points to calculate the DTR, so I'm picking up the current slope.
But I think I'd prefer to project using a constant transmission rate with no acceleration. I'm not sure what factors would cause it to continue to accelerate.
Plus I'm not comfortable with seeing an acceleration rate in your formula but not an initial transmission rate. That makes me think your formula assumes that the initial spread rate in any projection period is 0 and accelerates from a speed of 0 each time. I'm thinking here of dropping a ball from a building. It starts at 0 and has a constant acceleration to a point. But our Ebolaball is already on the move.
In any event, it appears that my flat rate went from 5% to 3% so I think it's deaccelerating to some extent, though the slope is still insanely scary.
if we knew the missing values, We could expand the formula for Active Cases on Day1 from:
Day1 = Day0*DTR
to:
Day1 = Day0*DTR-ResolvedCases+NewCasesFromResolvedCases
Where
Resolved Cases = Dead_Day1 + Cured_Day1
NewCasesFromResolvedCases = ContagiousDead * ContDead_DTR + Cured * Cured_DTR
But we don't know those numbers.
During the SARS episode, someone had a model of how diseases would spread over the earth, given traffic patterns. I wish I could find that.
That's how I did it. I looked at various time frames and asked "How do we get from this number to that number in so many days?". There are really two complementary pieces to the model: 1) how to calculate the exponent from specific past periods of time; and 2) how to apply that calculated exponent to the future.
I like your terminology of "resolved cases".
But I think I'd prefer to project using a constant transmission rate with no acceleration. I'm not sure what factors would cause it to continue to accelerate.
The point of this whole exercise is to get at the truth. No one model will do that. We need multiple models that look at the problem from various points of view, all of which are imperfect, but each of which reveals something the others don't.
In any event, it appears that my flat rate went from 5% to 3% so I think it's deaccelerating to some extent, though the slope is still insanely scary.
I have noticed a deacceleration of my DTR, too. That's a good thing and will dramatically lower the projections in my model. But like yours, my numbers are still pretty scary.
In college, back in the 1970's, I had a professor of Effective Writing (i.e., Freshman Comp) who told us that "Language is Thought", and if you don't know how to express something clearly, it is because the thought isn't clear in your own mind. This discussion is really helping me clarify my own thinking on the matter.
The more I think about what my DTR really represents, the more I don't like how I've said it. I'm not sure I like the word "acceleration" as I've used it and as others are likely to understand it. I know I don't like the word "rate". It's not a "rate". It is a "rate of change".
So let's try this: the DTR in my model should be renamed "Adjusted Transmission Exponent (ATE)". It is an alternative way of expressing the rate of transmission, adjusted for the variations that have occured over time in the rate of transmission, both positive and negative. It "smooths out" the daily, weekly, and monthly variations in the rate of transmission to provide a single number that can be used as a mathematical exponent to project into the future, based on past transmission rates.