The “basic reproduction number,” R_o, characterizes the average number of secondary infections that result from each infection and is related to the early phase (pre-mitigation) exponential growth rate, r_o. R_o depends not only on the characteristics of the disease, but also conditions of the “bath” in which a given local outbreak is occurring, e.g. population density and social customs. (New York’s R_o is almost certainly the largest in the nation.)
As mitigation measures are introduced the basic reproduction number is expected to decrease to smaller values, R, related to smaller (and even, hopefully, negative) growth rates, r.
To determine the growth rate, we need to know not only the R value but the time dependence of the disease’s contagiousness. The simplest assumption is that the contagiousness is constant for a “period of contagion,” T. In that case the rate at which a given contagious person creates new infections is R/T. Moreover, under that assumption, the relationship between R, r, and T is
Finally the doubling time, T_2, is related to r by
T_2 = ln(2)/r
For COVID-19 I’ve seen values of 2 to 2.5 or more quoted for R_o and values of 5 to 8 days for T.
Now, the outbreak that I’ve studied most carefully, California, had an initial doubling time for daily new deaths of 3.3 days running right up until ~March 25. That value corresponds to R_o = 2.07 if T = 8 days and 1.62 if T = 5 days. Because that latter value seems out of range, I have tended toward using T = 8 days for models in which that matters.
Nobody has ever accused me of being an optimist. My own, perhaps pessimistic (!), feeling is that being reality- and fact-based WILL make one more pessimistic than the average person. Nevertheless, I seem to be more optimistic than most about the course of this pandemic and I think that optimism is based on several things 1) my rejection of “confirmed case” data as all but meaningless, 2) my reliance on daily new death data despite the fact that it is a woefully lagging indicator of the effectiveness of mitigation efforts, and 3) my association of changes in the rate of increase of new deaths to policy decisions made some 20 days earlier.
So let's look at some data:
If you look at a graph of log (total deaths) versus time, you will see the slope decreasing as mitigation measures begin to take effect. It is, however, almost impossible to detect the transition from positive to negative growth on such a plot.
The table of doubling times that I chose on April 2 is as follows:
3.3 days (R ~ 2.07) up through March 4, the date of the first death in California and of Governor Newsom’s state-of-emergency declaration
5 days (R ~ 1.65) through March 12, the date that Governor Newsom banned gatherings of 250 or more
8 days (R ~ 1.39) through March 15/16, the dates that Governor Newsom asked for the voluntary closing of bars and restaurants and people over 65 to stay at home, and some counties began issuing stay-at-home directives.
10 days (R ~ 1.3) through March 19, the date of the 19th death in California and of Governor Newsom’s statewide stay-at-home order
Finally a transition to a doubling time of -10 days (R ~ 0.75) over the next few days that is maintained thereafter.
Now, the first two of those periods were based on a fit to the data I had at the time. The transition to 8 days was an educated guess, but it has now been MORE than well borne out by the data of the last five days. I recognize that that last one is a bit of a leap, BUT 1) the statewide order was a definite game changer for most people’s habits and there is little question in my mind that, if anything, our habits have continued to change to lower and lower social contact AND 2) IF we did not go into negative growth after that, we are REALLY screwed.
Tomorrow marks 20 days since Newsom’s statewide stay-at-home order and the day after that will be the first day that it’s effects begin to show up. My model predicts a peak at 54 deaths on April 10. As of right now the data is tracking below my predictions, but there is an VERY stringent test on my model coming over the next week.
In the graphs on that spreadsheet, the blue dots represent the latest reported data, the orange line is the result of the fit and educated guesses at doubling times that I made on April 2, and the dashed line represents the doubling time stalling at 10 days and not moving at all after the statewide stay-at home order was issued. I will be more than flabbergasted if things turn out to be that bad.