In the piece School Opening - Not a Binary Choice, I made an offhand comment, “To reduce the number of nodes is to create an exponential reduction in contacts.“ I also asserted that disadvantaged kids are safer in school because of the reduced attendance of non special needs affluent kids participating remotely. This was in response
to an absolutist statement by Emily Oster in The Atlantic, “If school isn’t safe for everyone, why is it safe for low-income students? And if school is safe for low-income students, why isn’t it safe for everyone?” At which I responded, “This is to say that low-income students are safer at learning centers because the other students
aren’t there. Here’s Dr. Carroll MD again, ‘Too many view protective measures as all or nothing: Either we do everything, or we might as well do none. That’s wrong. Instead, we need to see that all our behavior adds up.’ And, ‘Instead of asking why we can’t do certain activities, we might consider what we’re willing to give up to do them more safely. Even better, we might even
consider what we’re willing to give up so others can do them, too.’”
We should explore this. Assume each person is a node in a system. Every node can connect to all other nodes. Each connection is an opportunity for disease to spread. Granted, this
model won’t hold as numbers of persons increases as not everyone interacts with everyone else each day. Not every kid likes to play with each of the other kids. It will work up to a classroom size, however, as in a class, each person is exposed to the presence of everyone else in that class. We can see reducing persons has
significant reduction in contact connections and therefore spread opportunities. In this we’ll find Nodes (N) relate to Connections (C) as:
N’ = N +1
C’ = C + N
You might more easily think of this as for every new node added, there is a new connection for each of the previously existing nodes. Total connections sum old connections and new connections.
With this, we see 300 connections for a room of 25 persons. Cut the room to 15 and you have 105 connections. Think about that for your Thanksgiving Dinner! Now trim the class size to 10, we’ll see 55 opportunities including the teacher (11 nodes). Meanwhile 5 total persons gives you a mere 10 connections. Any given connection has a low possibility of transmission as you don’t know who is hot but background prevalence is still a small though increasing percentage. Even a hot connection doesn’t always transmit. Pile on the connections, pile on the chances. Pile on the prevalence, pile on the chances. Pile on connections and prevalence, really pile on the chances.
Individually, in a room of 10 you’re likely only worried about 9 connections personally, but administration needs to worry about the 45 connections. Indirectly, you probably should too. As not all hot connections may heat a node, you might not get hit by one of your 9, but another could through the others which means 2 of your 9 become hot; with more hot vectors heating you, you’re more likely to get hot. If these 2 don’t get you, they may get others such that you’re now exposed to a greater number heating connections. As exposure concentration goes up, risk goes up not only to catch but for a higher dosage at your potential catching, which carries more risk of being a harder case.
Back to the administration considerations, again they need to worry about all connections. Any node could be hot. Now, if a node can be observed as cold via rapid testing, we remove that node’s connections as if the node were temporarily removed. Mass rapid surveillance testing removes most nodes.
In case you were wondering, the relaxed low prevalence maximum occupancy setting of 50 would give 1225 connections.
With this, please note, just because a location may be doing well doesn’t mean they’re doing it right. Assertions that Rice and Duke are Universities that have it figured out may be wrong. They may just be lucky thus far. Random spread means clusters with density and vacuum will be found. If you find yourself in a vacuum, you do have a little more leeway to play going forward in the short term, however. Such should relate to community prevalence and benchmarks for openness. Having said all this, Rice and Duke could in fact be doing things right with full impact or partially right with partial impact. In Duke’s case, they have a mass surveillance testing program with cell phone contact tracing. Rice has been enforcing reduced connections within the student body. This sort of thinking drove Liars and Fires.
Per my School Opening piece, if you want open schools, you need mask mandates and mass rapid surveillance testing - or - you need a sufficiently low community prevalence and a mask mandate. Differing levels of community prevalence could be tied to differing levels of school openness.
Saturday, Nov 28, 2020 · 6:22:32 AM +00:00 · Fffflats
In my original post, I mentioned I had made an offhand remark to the effect of something exploding exponentially. Then I took us to explore what is actually happening looking to inspire what I considered more practical concerns. Mainly I was trying to give a simple illustrative means to inspire fear appropriate to our situation. I also made the discussion in reference to schools though hoped others would take it to any and all interactive social situations. I never actually named what was happening mathematically. I didn’t conduct a full exploration. It is quadratic, not exponential. I stated that N’=N+1 and C’=C+N, which models connections relative to nodes but did not qualify the relationship. The logic was for every new node added, a new connection to each previous node is also added. Others in the comments used an alternate method seeing each node must connect to all other nodes while each connection takes two nodes, so you only need to count connections for half the nodes. This gives C’=N’(N’-1)/2. If you prefer, you can use C=N(N-1)/2; I’m merely keeping the prime to match primes with primes from my method. Their method is nice as you need not know previous counts, instead you only require knowing the current number of nodes. I used mine for better illustration following the progression in the drawings in the above article. Both methods work and should illustrate rapid growth potentials. My leaving an impression that such was exponential was wrong, however. Above are three depictions showing rather drastic differences between quadratic and exponential. The two are simplified A=B2 versus A=2B. You’ll note the quadratic has a symmetry and actually touches zero while the exponential only approaches zero as a limit.
The piece’s title is appropriate, however, as the situation is explosive though with the quadratic nature, it is mere black powder explosive. An exponential growth would be more like that of a nitroglycerin or dynamite.
An alternate analogy for increasing connections could be a comparison of roads. Infection, on the other hand, may be more like a study of traffic. Initially and until a sufficient portion of population has either become infected or been immunized or recovered with immunity, such growth can appear exponential. Then again, with super-clustering, such may look like random traffic jams. Another alternative for modeling could be made by drawing egg-shape “spheroids” about each person with concentric radii for heavier exhaled output to lighter to none. One could then look for overlaps of the spheroid of a potentially hot person to other persons. One could also use multiple such spheroids with increasing density or color/transparency plot to show increasing numbers of potentially hot persons and their combined increasing hot areas. This sort of modeling may be more representative to any room with any occupancy though is also a pain in the ass to produce. In any case, “if you want open schools, you need mask mandates and mass rapid surveillance testing - or - you need a sufficiently low community prevalence and a mask mandate. Differing levels of community prevalence could be tied to differing levels of school openness.” And you really shouldn’t be holding large religious gatherings in the midst of hot zones. Assisted suicide is a contentious issue. Religions tend to be against it. Yet some think it ok to pull extra cards from the stack hunting for a Joker.
Let’s add a conclusion to the exploration. Reducing nodes creates quadratic reduction. Whether or not you have quadratic or exponential, I would still advise backing off by orders of magnitude when running into system failure. As previously mentioned, as numbers go up, the model loses value as not all connections will regularly be exercised.