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Teacher asks: What are the Covid odds?

James W. Swan, associate professor of chemical engineering at MIT, "plays" some numbers and checks the odds of Arlington students getting Covid-19.

Covid-19 testing for students.

It is essential to get infection rates lower in Arlington. Making schools safe enough for kids and teachers should be a top priority for our community.

I am a town parent and also a teacher. Quantifying the risks to kids and teachers during this stage of the pandemic in Massachusetts has had my attention recently. I want to take the opportunity to illustrate here how we can come to understand these risks through some simple statistical calculations. I also want to show how we can change those risks substantially by continuing to lower the infection rate in town. I will summarize the analysis first and then discuss it in detail for the interested reader.

In the 20 weeks since the outbreak began, there have been five documented cases of Covid in the 0-19 age range in Arlington. By my estimation, there is a 30-percent chance that no school-aged children have Covid in town, right now. If we had a smaller infection rate, as in Belmont, there would be a 78-percent chance that no kid has Covid. Additionally, at our current infection rate, or at the rate in Belmont, it is a virtual certainty that at least one Arlington kid will be infected with Covid during a 40-week school year.

Simply put, there will be a small number of school-aged children infected with Covid over the next year. But we can control that number. It is of the utmost importance to model public hygiene for our kids and our neighbors: wear a mask, wash your hands, avoid high-density gatherings. It is also very important to establish effective safety protocols at schools to reduce the risk of transmission when these rare Covid cases occur.

While a Covid vaccine is sure to arrive within a year, there is no guarantee of its effectiveness or widespread distribution and adoption. We cannot rely on this medical intervention alone. However, we are on the cusp of mitigating the risks to children through our actions as a community. By committing ourselves to minimizing our own infection risks and by preparing our kids and their schools to minimize Covid transmission, we can make in-person school safe.

Following the stats

The statistical methods are described below.

In Arlington, there have been five documented infections per 100,000 residents over the last two weeks. Because Covid infections last about two weeks, this means that there is a 0.005-percent chance that any given resident has a documented Covid infection. The total number of infections is thought to be anywhere from two to 10 times higher than the number of documented infections.

I will choose a factor of four to estimate the total number of infections in this article. In that case, there is a 0.02 percent chance that any given resident has a Covid infection presently. That’s two in 10,000 residents infected with Covid currently. Our schools host about 6,000 students every year. These odds of infection suggest that right now there is about one infected kid in our district (6,000 x 2 / 10,000 = 1.2).

What are the odds?

We can ask a deeper statistical question with this data: What are the odds that no kid is infected right now? To answer this question, imagine first a more familiar problem – calculating the odds of successive coin flips. Consider flipping a coin two times and determining the chances of not getting a head (or only getting tails). It’s a 50-percent chance to get tails on the first flip and a 50-percent chance to get tails again on the second flip. The combined odds of getting two tails is 50 percent x 50 percent = 25 percent.

Now, let’s apply that same approach to calculate the probability that no Arlington kid is infected with Covid. The chance that a town resident is not infected with Covid is 99.98 percent (100 percent - 0.02 percent). The odds that no kid is infected can be computed in the same way as the coin flip odds, by multiplying the probability of not being infected for each and every student: 99.98 percent x 99.98 percent x 99.98 percent x … x 99.98 percent = 30 percent. We have to repeat the multiplication in this calculation 6,000 times!

Right now, the simplest statistical analysis says that there is a 30-percent chance no kid in town is infected with Covid. Equivalently, there is a 70-percent chance that at least one kid in town has Covid.

Repeat calculations

To illustrate how sensitive these odds are to the infection rate, I encourage you to repeat the calculation with five times the infection rate: one in 1,000, as in Somerville. I will demonstrate what happens if the rate of infection in town were just five times smaller: two in 50,000. In that case, the chance that a resident is not infected is 99.996 percent, and the probability that no kid in town has Covid is 78 percent.

This demonstrates why we need to get the infection rate lower and keep it there. If the infection rate were just five times smaller, as in neighboring Belmont, we could feel confident that there is a four in five chance that no kid in our town has Covid during any two-week period.

Another question

Let’s use this same statistical analysis to answer another question: What are the chances that no kid has a Covid infection over an entire school year?

A school year is 40 weeks long and comprises 20, two-week periods. To compute the probability that no kid has Covid in each of these periods, we repeat the coin-flip-odds calculation again. We simply multiply the probability that no kid has Covid in a given period 20 times. For a two-in-10,000 infection rate, that calculation is 30 percent x 30 percent x … x 30 percent, with a result that is indistinguishable from a zero-percent chance. For a two-in-50,000 infection rate, that calculation is 78 percent x 78 percent x … x 78 percent = 0.7 percent.

Dropping the infection rate by another factor of five – meaning Arlington has one total Covid case every two weeks – brings the odds of no child being infected over the entire school year to 38 percent. 

Aug. 3, 2020, and after: Tracking cases, deaths in town

This viewpoint was published Wednesday, Aug. 5, 2020.

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