Modeling personal risk of contracting COVID19 while attending university classes

Using a new model of exposure risk to estimate the chances of contracting COVID19 if attending in-person classes this fall

If you’re a student or instructor facing the prospect of in-person classes in the fall, and worrying about what the risks are of being infected by COVID19 as a result, you’re not alone.

As a university professor and academic administrator, I’ve been grappling with the potential risks of in-person teaching in the light of COVID119, and wondering just how effective measures being discussed are going to be. And my background in public health and aerosol exposure has just served to sharpen my concerns here.

Most universities are currently working hard to reduce the risks of becoming infected on campus through measures like temperature screening, mask-usage, reduced occupancy and hybrid in-person/online teaching models. Yet without a clear sense of where these measures are backed up by evidence, I find myself finding it hard to get a good feel for what the personal risks might be.

And that’s speaking as someone who studies risk for a living!

Paying attention to ventilation in classrooms

One factor in particular that has been concerning me is the rate at which potentially contaminated air in enclosed spaces is replaced with clean air, and how this in turn impacts potential risk. As a result, I’ve been pleased to see a growing body of preliminary research looking at just this — including a recent pre-print on medRxiv from Dr.Shelly Miller at the University of Colorado, Boulder, and her colleagues, on COVID19 transmission associated with the Skagit Valley Chorale superspreading event.

Although Dr. Miller et al.’s paper still needs to undergo peer review, the underlying aerosol science and modeling are sound, and it does appear to offer new insights into the importance of effective ventilation in indoor spaces where there are potentially infected people.

Building on this work, Dr. Jose Jimenez — a co-author on the Skagit paper, and colleague of Miller’s at the University of Colorado Boulder — has created a useful spreadsheet-based model that estimates the probability of becoming infected with COVID19 while attending class, if someone else in the room is infectious.

The model — which is freely accessible as a Google Sheet — relies on a number of assumptions, and is still in the process of being tested and developed. Yet it provides an intriguing way of getting a first-order handle on the potential risks of attending class while COVID19 remains a risk.

As I’m particularly interested in the risks to students and instructors in my current role, I was keen to dig into the model, and play around with it, while getting a sense of how it might inform decisions around the risks associated with in-person classes.

Modeling in-class COVID infection rates

The model is based on the spread of fine airborne microdroplets (aerosols) containing COVID19 in enclosed spaces, and the accumulation and inhalation of these over time. It assumes that these aerosols are so fine that they rapidly mix and spread through the whole volume of a room (which is reasonable), and so the physical distance between people in the room becomes less important than how rapidly the contaminated air is replaced by clean air.

This assumption obviously doesn’t account for exposure to spatter from coughs and sneezes, although over time (and where masks are worn) it is likely to be the finer exhaled and inhaled aerosol microdroplets that are the greatest worry.

The model also allows assumptions to be made about the effectiveness of masks where they are worn. This is critical as, important as mask-wearing is, while they reduce exhalation and inhalation of infectious aerosols, they don’t eliminate these.

In its supplied form, the model provides estimates of the the number of new infections in a single class if either a single student or the instructor are shedding the virus. It also allows a crude estimate of campus-wide cases of infection.

I was interested in another estimate of risk though — the probability of someone attending the class becoming infected over the course of the semester if, at some point, someone else attending the class was infectious.

It’s this shift in focus from population health to personal health that many people are looking for information on, yet not finding it easy to find.

Modeling of Personal Risk of Becoming Infected

Fortunately, the model allows a crude estimate of personal risk of developing COVID19 if one of your class mates is a spreader, with the addition of a few lines of calculations.

Of course, even if personal risk can be estimated, it’s not that straight forward knowing how to interpret it. One approach though is to start by assuming an acceptable threshold of a one in a hundred risk of being infected by COVID19 in a class over the duration of a semester.

This is far from negligible, and I suspect that some people would baulk at it. It is, however, a useful starting point for weighing up the relative risks and benefits of attending class.

So with this threshold in mind, I started playing with the model.

Preparing the Model

Dr. Jimenez’s model allows users to alter a number of key parameters including:

  • Room area and height;
  • Number of instructors and students;
  • Class duration;
  • Air exchange rate (or clean air supply rate per person); and
  • Mask efficiency.

To this, I added:

  • The number of classes per semester; and
  • The individual risk of becoming infected if the course instructor or another student are infectious.

This calculation requires some unpacking — especially as, if I’ve missed something here, it would be good to know! (Update: turns out I did — it doesn’t change the estimates here, but see below for details)

Feel free to skip this bit, but for completeness, the calculation made is:

Personal risk = [(a1*b1*Ns)+(a2*b2*Nc)]*Nclass

where the following apply:

a1: Probability of becoming infected with COVID19 if there’s an infectious student in the class;

a2: Probability of becoming infected with COVID19 if there’s an infectious instructor in the class;

b1: Fraction of students who are infectious but come to class anyway;

b2: Fraction of instructors who are infectious but come to class anyway;

Ns: Number of students in the class;

Nc: Number of instructors in the class; and

Nclass: Number of classes in a semester.

This essentially estimates the probability of virus-shedding people being in each class and the probability of them infecting you, summed up over a semester-long course. It’s a simplification, and is only valid where class infection rates and class numbers are relatively low. But it is adequate for this analysis.

(For a more accurate approach to estimating risk with higher infection rates and class numbers, see the notes at the end of this article, which were added after feedback on the article.)

Like any model, this one needs to be treated with a great deal of caution, and the recognition that garbage in will always give you garbage out. And many of the parameters it uses, while based on best possible estimates from the literature and elsewhere, will probably shift over time. Nevertheless, it does provide an extremely useful tool for exploring multiple exposure scenarios.

In running the model, I was especially interested in the potential risks of contracting COVID19 from other students or the instructor in a semester-long 3 credit course (15 weeks in all), where the class was held in-person in either a large (900 square foot) or small (250 square foot) room (both with an assumed height of 10 foot in the model).

The model requires an estimate of the room air exchange rate — the rate at clean air is brought in and potentially contaminated air is removed. The American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) has a standard for this (ASHRAE Standard 62.1–2019, Ventilation for Acceptable Indoor Air Quality — available from this site). To keep things simple, I kept the air exchange rate at 5.7 exchanges per hour most of the time — based on a 900 square foot auditorium designed to seat 100 people.

The model also requires an estimate of the probability of either students or the instructor being infectious and still attending class. This is clearly a highly variable figure, and not one that’s easy to pin down. On one hand, it can be argued that personal and institutional screening is likely to keep it low. On the other hand, the possibility of people attending class who have COVID19 but who are asymptomatic is likely to elevate the value used.

In the end, I opted for a compromise of a 0.3% probability of either a student or an instructor being infectious yet still attending the class.


Risk versus room occupancy

First off, I ran the model to explore personal COVID19 infection risk in a large auditorium (900 square foot) and a small classroom (250 square foot), each with a range of students physically present in the class (figures 1 and 2 below).

The simulations here assume in-person students taking a 3-credit class (165 minutes of contact time) over 15 weeks, with an air exchange rate of 5.7 exchanges per hour, and a probability of people in the class being infectious as they come in of 0.3%.

Figure 1. Personal risk estimates for a 900 square foot class room as a function of occupancy.
Figure 2 Personal risk estimates for a 250 square foot class room as a function of occupancy.

As would be expected, risk increases with increasing numbers of students in the room. This is not directly related to reduced distancing in the room, but to the increased likelihood of one of them already having COVID19 and being infectious.

What surprised me running these first two simulations is just how high the risk is. For the 900 square foot auditorium it’s possible to get below the 1:100 risk threshold (although how significant this is, I don’t know). But under the modeled conditions, the risk of infection in the smaller room is above this threshold, even with only 5 students in it.

Masks versus no-masks

I included risk estimates with and without masks here, as I’m pretty sure that, despite current requirements on most campuses, students and instructors are going to discover very rapidly that teaching large groups (especially where there is a lot of discussion) for 2–3 hours where everyone is masked, is going to be extremely challenging. And the more challenging it gets, the more temptation there’ll be to remove the masks.

The data in the figures above are pretty conclusive though that, even while masks are not 100% effective (the model assumes masks are 50% effective at blocking exhaled aerosol particles, and 30% effective at blocking inhaled particles), not wearing masks in class substantially elevates the risk.

Increasing room size while keeping occupancy fixed

I next ran the simulation for a fixed number of students (15) taking classes in rooms of varying sizes, to explore how room size affects risk. In all cases I stuck to the ventilation rate of 5.7 air exchanges per hour (remembering that this assumes clean air coming into the room).

Figure 3 Personal risk estimates while varying room size for a fixed number of occupants

As expected in figure 3, the larger the room, the lower the risk if you keep the number of people the same — which is a nice confirmation that reducing room occupancy is an important risk mitigation strategy. Again though, quite drastic measures are needed to get the risk below 1:100, and not wearing a mask has a quite dramatic impact on elevating the risk.

Varying Ventilation Rates

An alternative to increasing room size or reducing occupancy, is changing the air exchange rate. This will not always be possible, or even advisable as rooms and HVAC systems are designed and set to specific standards. But just to see the potential impact, figures 4 and 5 show the potential impact of changing the ventilation rate in a large auditorium and a small room.

Figure 4. Estimating personal risk in a 900 square foot classroom for a range of air exchange rates.
Figure 5. Estimating personal risk in a 250 square foot classroom for a range of air exchange rates.

As would be expected, increasing ventilation rate decreases personal infection risk. What is perhaps less expected is just how high the air exchange rate needs to be in a smaller room to get the risk closer to 1:100 — in this simulation with 15 students in the smaller room, the risk at 10 air exchanges per hour is still above this threshold.

Of course, this is a simplification, and social distancing will still reduce spread through coughs and sneezes (especially if people aren’t wearing masks) — but where people are exhaling airborne aerosol through breathing, talking and, occasionally, shouting, how far away you are from others is less important than how rapidly the air around you is cleaned.

These plots also assume that the HVAC systems used by universities have the capacity to introduce clean air to rooms, either by using outside air, or by completely stripping recirculated air of viable viruses.

Expanding Risk Reduction Scenarios

Beyond the scenarios above, the model is also useful for exploring other risk reduction scenarios. For instance, what if classes were split so that 50% of students attended in person and 50% attended online, or in-person classes were halved in length with additional instruction occurring online and/or asynchronously, or in-person classes were only held every other week.

Figure 6 below gives an indication of the personal risk impact of each of these scenarios, assuming a 900 square foot auditorium and a 25 person class, with an air exchange rate of 5.7 changes per hour.

Figure 6. Estimating personal risk in a 900 square foot classroom for four different scenarios.

The results here are relatively intuitive, although the nature of aerosol release and behavior in the modeled room does lead to some small differences between the three risk reduction scenarios.

What is helpful here though is the clear indication that relatively simple interventions such as reducing class attendance, reducing class duration, or reducing the number of times in-person classes are held, are effective ways to substantially reduce personal risk of infection.

This suggests that are are innovative ways to combine online/asynchronous teaching with in-person that effectively reduce personal risk. For instance, I suspect that the possibility of shorter classes followed by increased asynchronous/online teaching (a version of the flipped classroom) would appeal to a lot of instructors — probably more so that trying to simultaneously juggle in-person and online participants in each class! Similarly, I suspect there’s a lot of appeal in a one week in-person, one week online approach to teaching in a time of COVID19.

However, these scenarios also suggests that teaching strategies should be based on data and analysis, and not just what feels right.

Variable community infection rates

Finally, I wanted to get back to the somewhat arbitrary use of 0.3% as the estimated percentage of students and instructors who unwittingly come to class while infectious.

Figure 7 explores the relevance of this number by first assuming that the instructor is clean (a stretch I know, but it’s a useful simplification), and then estimating risk for a range of infection rates amongst students attending class.

The analysis is based on 25 students in a 900 square foot class (relatively low occupancy), and estimates risk of contracting COVID19 as a function of percentage of students on campus who are infectious while attending class. And so, if the percentage of students who are infectious is, on average, 0.2%, and everyone’s wearing masks, the model estimates the risk of being infected over the course of a semester if you’re attending the class is a little over 0.5%. On the other hand, if 1% of students are infectious (which sounds high, but maybe not that high if we’re still struggling to contain the virus), that risk goes up to 2.7%.

Figure 7. Personal risk estimates as a function of the percentage of students attending class who are infectious

As there is so much uncertainty around this number, figure 7 should be treated as indicative only. But it does give a sense of how ensuring a low infection rate amongst students attending class can help reduce the risks of infection. It even indicates that, if this number can be kept low enough, there is a decision-point where masks in class may no longer be required.

The question is, how can this number be reduced? Living and working in Arizona, we’re seeing a worrying rise on COVID19 cases, suggesting that we’ll be facing a high percentage of infected students in the fall. However, there is the possibility that, with institution monitoring and self-monitoring, the number of infected students coming to class will be substantially lower than the infection rate in the community more broadly.

That is, if we can also help students avoid becoming infected outside class through socializing and mixing with others, without taking due precautions!

Proceed With Caution

Of course, this assessment should be taken with a large pinch of skepticism. As researchers and risk managers get a better handle on the necessary parameters, estimates of risk may go up as well as down. Yet it does indicate that the risks of infection through in-person classes is not insignificant, and care is going to need to be taken as students come back to university. It also suggests that innovative approaches are still needed to reduce exposure through the creative use of distance learning, and that simply assuming that wearing masks and keeping distant from others in class will do the trick.

This latter point is especially important, as the model and scenarios suggest that keeping students distant from one another is far less important than ensuring appropriate room occupancy and ventilation rates, while reducing class duration and frequency.

This is good news for instructors who are deeply worried about how to use teaching skills that depend on nuanced student-student and student-instructor interactions (I’m one of them — many of my classes depend on fast-paced class discussions and considerable movement/animation). But it does mean that institutions need to take emerging indications of good risk reduction practices on board.

Update on calculations (July 3, 2020)

As was noted in the comments on the original post, the expression for estimating semester-wide risk used above is not correct, but is rather a somewhat naive simplification that falls apart when in-class risks are high, and the number of classes taken is high.

Fortunately for this analysis, it works, as the risks are relatively low, and the data above are still valid. However, a more rigorous form of the risk estimate is as follows (based on Ben’s comments):

The risk of becoming infected with COVID19 in a single class is (a1*b1*Ns)+(a2*b2*Nc). Correspondingly, the probability of attending a class and NOT becoming infected is [1-(a1*b1*Ns)+(a2*b2*Nc)].

This is important, as it allows the probability of reaching the end of the semester without becoming infected to be calculated, assuming that the risks associated with each class are the same.

Each time an additional class is take, the total probability of NOT becoming infected is the probability of not becoming infected up to that point, multiplied by the single-class probability of not becoming infected. And so, over the course of a semester, the total probability of getting through unscathed is


And from here, the the risk of being infected is simply 1 minus this:

Personal risk = 1 — [1-(a1*b1*Ns)+(a2*b2*Nc)]^Nclass

The spreadsheet I used for the calculations above has been updated to include this. You can run the figures in it (see the original post for the link) and see that in the analyses above the simplification I used first time round holds. However, running the model for higher in-class risk rates, or with longer classes and more classes per semester, this more rigorous expression should be used.

Originally published on Andrew Maynard’s personal blog

Update July 2, 2020: Added explanatory text around figure 7, and updated the figure axes and description for clarity.

Author of Future Rising: A Journey from the Past to the Edge of Tomorrow, and Associate Dean in the ASU College of Global Futures

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