This is my third post on the press conference Rich Fitzgerald and Dr. Bogen gave on July 7, 2020. The first can be found here, and the second here.
When Dr. Bogen was asked about closures other than restaurants and bars, she stated that there are no other activities associated with people who have tested positive that compare in magnitude to bars and restaurants. She said that contact tracing results showed that out of 347 people who tested positive only 13 had been to gyms. This is 3.7%.
3.7% of infections may not seem like a lot, but because COVID-19 is contagious preventing 13 people from being infected at a gym (or from any other source) prevents a much larger number of infection in subsequent infection cycles.
The number of people that each infected person infects is represented by $R_0$. This paper estimates the $R_0$ for SARS-COV-2 to be 3.28 with no preventative measures in place. The exact value of $R_0$ with the preventative measures in place in Allegheny County is unknown. If it is greater than 1, additional measures to prevent transmission are needed or the disease will spread exponentially and the hospital system will eventually be overwhelmed.
If $R_0<1$, then the formula for the sum of a geometric series can be used to calculate the total number of people that will be infected from an initial infected population $n$.
$\textrm{Total}=\frac{n}{1-R_0}$
For the sake of example, if $R_0=0.9$, and there are 13 people infected at gyms, they will go on to infect a total of 130 people.
$\frac{13}{1-0.9}=130$
Similarly, if $n$ cases are seeded from outside the county they will be multiplied by community spread using the same formula. If $R_0=0.9$ and 3.7% of local transmissions are at gyms, closing gyms would reduce $R_0$ to 0.8667. The community spread multiplier has been reduced from 10 to 7.5.
It is not known how many of those 13 actually contracted the virus at the gym. The main point of this post is that actions that prevent a small number of infections now will prevent a large number of infections later on. It is for this reason that a contact tracing program that is only able to directly prevent a small number of infections indirectly stops a lot more and makes a big difference.
There is not enough data to accurately estimate what $R_0$ is with restaurants closed. On difficulty comes from delays in obtaining test results. For example, test results reported on July 9 included tests taken as early as June 25. Secondly, due to the incubation period of the virus we won’t start to see the affects of the bar and restaurant closures in the data until two weeks after the respective policies were announced.
The good news is that based on the case counts in the last few days the growth in new cases may be slowing. This suggests that even before the restaurant closure $R_0$ was less than 1.
The following plot compares the number of new reported cases per day with what the expected trends would be for different values of $R_0$ after an increase in the number of externally seeded cases per day. If $R_0<1$ the number of cases would rise and asymptotically approach $\frac{n}{1-R_0}$ as shown in the red curve. If $R_0>1$ the cases would rise exponentially as shown in the yellow curve.
In Allegheny County the rise starting around June 18 was caused by a combination of an increase in the number of cases seeded from other states and a rise in $R_0$ caused by the transition to the green phase. We don’t have enough data to know if $R_0<1$ prior to the bar and restaurant closures, but the data trend suggests it may have been.
If $R_0$ is significantly less than 1 after the bar and restaurant closures, infections from gyms and other comparatively small sources of transmission will still be amplified by subsequent infections, but to a much lesser degree than the calculations in this post that assume $R_0=0.9$. However, given uncertainties a conservative approach is appropriate.
A final thought is that it is very important that we reduce case prevalence to the point that schools be able to reopen safely in the fall. Steps that reduce a small number of infections now will have a big impact on case prevalence 8 weeks from now. If a closure has the potential to reduce the number of cases, we should ask is it more important to be able to do this activity than have our children able to attend school.
