Your math is not making sense to me.
First a DTR (Daily Transmission Rate) of 1.004 is way too low.
And I’m not sure what you are doing putting both the DTR and the DaysOut in the exponent. That might work in your formula, but I don’t think the formula makes logical sense.
Work it through one day at a time and see if you get the same result.
Start with 100 cases and let’s do just 3 days.
If you have 100 cases on day 0 and the DTR is 1.004 then you would have 100*1.004 = 100.4 on day 1, and 100.4x 1.004 = 100.8 on day 2, and 101.2 on day 3.
But if you plug it into your formula for 3 days, then you get:
100^(((1.004-1)*3)+1=
100^(.004*3)+1 =
100^(.012+1) =
100^1.012 = 105.6, but it should equal the same result that we walked through day by day. So either your formula is wrong, or I don’t understand exactly what you are doing.
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.