Delta Doubles Down

It seems like yesterday that we were planning on meetings in person, dinner in a restaurant, shopping at the mall. Those simple activities are all the adventure that we crave. We did think that we had made it, vaccinations checked off of our list of things to do and now we are entitled to move on. Going back to last April, even, I had been talking to others about what I should do next. And here we are, a new variant and new precautions.

It has taken some firm messaging to convince others - especially businesses that had held back and now were inviting customers back. How did this happen so fast? Did it truly happen over a week's time? To answer these questions, one has to appreciate the concept of exponential growth: the spread of infected people is a function of the density (i.e. concentration) of people already infected.

X Rate of Growth = a * Concentration of X

It turns out that exponential growth occurs in many fields besides epidemiology. In chemistry solutions reacts faster the more concentrated they are in reactant, in finance - the bottom line grows faster the higher the initial investment, in traffic - cars in an accident pile up more quickly the more cars on the road. The units on either side of the equal sign are number of new infections-per-day-per-capita (or per-week-per-100,000 people or per month-per 1M people,....per unit time per population density * area of county, region, state). This suggests a dynamic situation where there is either a surge or a decline depending on which side of the reaction (light-off vs. quench, rising market vs. sell-off, accident occurring vs. accident cleared, vaccinations start vs.. vaccinations completed). The dynamics are a function of a variety of features (including weather, access to infrastructure, and length of time that an infected person is contagious). These are captured by the factor, "a". The plot of X depends on time-zero or the amount of x at the "initial condition".

Typically, we measure X over time to see whether it is increasing or decreasing and making note of any changes in the various important features, we can plot the results to see whether we can predict (or validate) the size of a. This factor can be compared from one variant to the next (one catalyst to the next, one freeway to the next, one time of year to another, one county to another, ....). This methodology is based on rewriting the above equation in terms of its integrated solution

(X/Xo) = Exp(a * time) Equation(1)

If the number of patients doubles over a period of time t-double, then

X = 2Xo for time = t-double Equation(2)

Ln (2) = a * t-double Equation(3)

a = Ln(2)/doubling time Equation(4)

The same approach applies if, instead of doubling, the increase goes up by a multiplier of 3, 5, or 10. The point is that an exponential function is part of the solution - and so we say that cases (or Covid-19 Hospitalizations) are increasing exponentially

The discussion below uses Covid-19 Hospitalization Data for Texas TSAs to examine Delta Doubling Rates (July-August) vs. those corresponding to the Winter Surge, October to January. The purpose of this evaluation is to answer the question, "Is the Delta Variant more transmissible than last Winter's Covid-19 strain?"

Figure 1 is the a spreadsheet with Covid-19 Hospitalization data for a list of TSAs as a function of date. This is our "base case", the surge during late Fall, going into the Winter Holiday Season. The entries fall between September 20, 2020 and January 30, 2021 even though most of the columns in advance of the January run-up of patients are hidden. The comparison of September 28 and January numbers provides an indication of the extent to which doubling occurred during the Winter, 2020 Surge.

The ratio between the Hospitalized Covid-19 Patients on a given date and the "initial"value, 9/28, for the TSA corresponding to that row is the Left-Hand-Side of Equation(1). These are the Entries presented in Calculation Table Figure 2. Highlighted cells identify dates and rows that have achieved another factor of 2: Yellow for 2*Initial Condition,

Pumpkin for 4*Initial Condition, Magenta for 8*Initial Condition. Note, despite the steady increases during the holiday season, masking was in place along with other business restrictions - therefore, only two of the TSA's experienced three doubling cycles (2*2*2 = 8 time more patients than the number reported on 9/28).

The same methodology is used with the recent data Delta Variant or Summer,2021 Surge. Figure 3 has these results through 8/16/21. 6/26 was selected as the Initial Condition Column. In contrast to the Winter Surge, Covid-19 Restrictions were lifted back in mid-March. And,despite evidence showing that the Delta Variant is

more transmissible, even including vaccinated individuals with the spread, the Governor

has implemented a strict ban on mask mandates. Figure 4 is the counterpart to Figure 2, the Calculated Table which shows the extent of doubling over a 3 week

period (as opposed to a 3 month period above). Here, 9 out of 10 TSA's saw a factor of 4 ( 2*2 = 2 rounds of doubling) and of these, half have then doubled again yielding Covid-19 patient loads that are 8-times higher than they were in June when vaccination rates had reduced the Covid-19 population significantly below where they had been last Summer.

Figure 5 is the summary of the comparison. These two tables report the number of days required to achieve successive doubling. On the left is the base case with numbers on

the order of months (pre-variant, mask and business preventative restrictions pre vaccination); on the right is our test case (Delta Variant, no spread preventative controls at 50% vaccination). While only two TSA's were able to achieve a factor of 8 increase over the 3-4 month period at the end of 2020, all but two were this much higher during 3 weeks of the Delta Variant. The first doubling rate was achieved between 15 to 94 days during the Winter compared with 12 to 18 days for the Summer, 2021. In both cases there was a reduction in the doubling time for those TSA's which went through a second round. This result was also obtained for the summer. It is notable that TSA-H went from 18 to 39 Covid-19 Hospitalizations in the space of 6 days - a factor of 4 higher than its initial value of 9 on June 26.

The standard way to track doubling rates, or exponential growth in general, is to plot the log of data rated to its initial condition (Calculated Table entries) vs. linear time. The slope, based on the coefficient "a" in Equation (1), indicates the growth in terms of a factor multiplied by time. Figure 6 is that plot. The larger the slope, the greater the ratio between Covid-19 patients at time-t and the occupancy at time-zero. Note that a comparison of Summer, 2021 vs. Winter, 2020 Surge Log plot slopes indicate that:

(1) data from both sets give positive exponential growth; (2) the Summer slope is higher than the Winter values consistent with a shorter doubling time.

As with any analysis, these results are a function of current conditions: school has been in progress for less than one week, no hurricanes or major storms affecting any part of the state, vaccination of the 12+ age bracket in progress, and hospitals are still open despite staffing that is spread very thin. If one or more of these change, then the doubling rate will be affected. For example, if school (elementary and college) turns into additional interactions which facilitate spread, then doubling time will go down. Alternatively, if vaccination rates take a step change up based on all manner of incentives offered to reward people to step forward, these will help "flatten the curve" keeping individuals out of the hospital and thereby increasing doubling time.