Excellent Question!!!
I think part of the problem is that I call it a Daily Transmission Rate. I chose a bad name for it. It's not really a rate in the sense we're used to, e.g., "rate of speed", where you have "miles per hour", or "new cases per day". Rather, it's really a rate of increase in the rate of new cases per day. In other words, it's a measure of acceleration of transmission, based on the fact that as the number of patients increases each day, there will be correspondingly more patients infected by them.
Using a true "rate", we would say, for example, that "each day there are 10 new cases of Ebola". So if there are 100 cases on Day 0, then on Day 1 there will be 110 cases, and on Day 2 there will be 120 cases, and on Day 3 there will be 130 cases. Every day we would simply add 10 new cases to the total.
But that's not the way it works. In addition to each of the 100 cases on Day 0, each new case also becomes capable of transmitting Ebola. Therefore, to project into the future, we need to measure how many cases there will be on Day 2, given that there are 110 cases on day one. Or, to put it another way, given that 100 cases became 110 cases on Day 1, how many cases will 110 cases become on Day 2? It's kind of like compounded interest.
But wait... it gets even more complicated. In most diseases, once a person recovers or dies, they no longer transmit the disease. So some of those original 100 are no longer able to infect others. So the question becomes "given that 100 cases became 110 cases on Day 1, and that some unknown number of those original 100 are no longer infectious, how many cases will 110 cases become on Day 2?"
But wait... there's more... With Ebola, even those who have recovered or died can transmit the disease.
But wait... there's even more... Those who have recovered, and those who have died can only transmit the disease under certain circumstances, and for each circumstance, there is both a different transmission rate, and a different length of time they can transmit it.
To go back to the compounded interest analogy, it's kind of like saying "How much money will I have in 90 days if I start with $100, but x dollars don't generate any interest, xx dollars generate interest at yy APR, for zz days, and and xxx dollars generate interest at yyy APR for zzz days, and I don't know what the values of x, xx, yy, zz, xxx, yyy, zzz are?"
So you see it gets very complicated very quickly. So to simplify all of this, my model looks at what actually happened over a period of time to calculate a rate of increase in the transmission rate (the DTR), from day to day. Because it is based on what actually happened, it takes all those factors described above into account, as well as the effect of the weather, the number of people per acre, the effect of quarantine and treatment, the deaths of healthcare workers, and on and on, including factors we would never think of.
So the question gets reduced to this: What exponent can I put on the number of cases, such that, if I apply it x number of times, will estimate the number of cases I can expect to see after x number of days?
The Microsoft Excel formula for the exponent is: (LOG(EndingCases,StartingCases)-1)/DaysOut + 1, where EndingCases is the number of cases at the end of the time period being considered, StartingCases is the number of cases at the start of the same period, and DaysOut is the number days in the period. I call this exponent, somewhat inaccurately, the Daily Transmission Rate (DTR).
The DTR measures the rate of increase in the rate of transmission over a previous, known period of time. To project into the future, I apply the DTR in the formula: FutureCases "X" Days from Now = CasesToday^(((DTR-1) * X) + 1).
But wait... there's still more... The DTR fluctuates, depending on the specific StartDate and EndDate examined. And even a small fluctuation can alter the projections dramatically. So, to minmize the fluctuations, I chose a period of time where the DTR was fairly stable over a significant period of time. That time period is June 1 through September 10.
Research Model Projects 1.1 Million to 2.3 Million Ebola Deaths By September 2015
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 what I would predict acceleration doing 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.
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.
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.