# Using rotation to manage school contagion

### Some school districts are splitting students into two groups and having them come in separately. How much will this help with Covid-19 spread?

Welcome to **Plugging the Gap** (my email newsletter mainly about Covid-19 and its economics). My goal is for several posts a week explaining economic research and the economic approach to understanding the pandemic. I may also deviate to other topics as well. (In case you don’t know me, I’m an economist and professor at the University of Toronto. I have written lots of books including most recently on Covid-19. You can follow me on twitter (@joshgans) or subscribe to this email newsletter at the link below).

**Today I am going to look at practices many places — notably, schools — are deploying to reduce Covid-19 infection risk: that is, using a rotation schedule to reduce the number of people in a place at any one time. The advantage of rotation is to reduce the time people might be exposed but those advantages are reduced if other practices such as mask-wearing and ventilation are used. Thus, places should consider not taking a ‘kitchen sink’ approach to contagion management and relying on a few, more effective interventions.**

When public schools in Toronto open up again a few weeks, things will be very different. While the younger kids won’t see much change, older ones will have to wear masks all day. For high school, things will be more complicated still. They will only go to school on alternating days with half of the class on one day and the other half on the next. The teachers will be the same but, instead of the usual way of presenting lessons, they will repeat the lessons to smaller groups (15 students per class is the aim). That is an interesting experiment in itself. Maybe we will find out if it is better for educational outcomes to have half the number of classes with half the number of students. It isn’t obvious to me that it will be worse. But that is not what I want to focus on here.

My thoughts are about contagion risk from Covid-19. You can see the potential benefits of having two separate groups of people who regularly interact with one another in the group but not outside of their group. If one person becomes infected and the virus spreads, it will be contained to a single group. But there are some interesting other choices too. First, there is the issue of cadence. Is it better to have different groups on alternating days or a longer period such as alternating weeks? Second, there is some crossover — the teachers and staff. Might that be enough to remove the benefits of group separation and rotation entirely?

### How to rotate

Since we don’t have much in the way of experience with these measures for Covid-19, predicting what might happen requires some mathematics. Fortunately, that task has been done in a new paper by Jeff Ely, Andrea Galeotti and Jakub Steiner. And what they find could give many places guidance as to how to use rotation as a tool to mitigate contagion.

The first thing to note is that the effectiveness of different rotation strategies depends on how much and how quickly the organisation reacts in dealing with potential infections. If you wait too long, it is likely that all of the groups will become infected. In that case, rotation doesn’t buy you much in terms of reducing the scale of that (although it may slow it down). If you can react quickly, on the other hand, you can isolate one group and keep the other going.

What is the likely reaction time for a Toronto school? With Covid-19, we know that people — especially younger people — tend to asymptomatic. That means that it may be some weeks before you can tell that there is an outbreak. Add to that getting tested and confirmation and it is plausible that a month could pass. Thus, it is likely that it would take, on average, about 30 days for a school to notice and react to a potential outbreak in a group.

Now let’s consider a school with 500 students. I think a school rather than a class is the right unit here as during a rotation they share common bathrooms and hallways. Covid-19 has a reproduction rate of about 2.2 to 2.5 (that is, every infected person is likely to infect 2 or so others if there isn’t any interventions going on). That means that once one person is infected, the number of infected people will double every 5 days or so (as this is how long it takes for an infected person to start infecting others). Wait 30 days before dealing with the problem (by a lockdown or mass testing) and you would have 16 people infected; (that is, 2 to the power of 4 where 4 is the number of doubling cycles in 30 days if you subtract 8 days of weekend).

All of this and what I am about to talk about assumes that in a school of 500 students, someone will eventually be infected. Thus, the 30 days we talk about are the time from that infection to a reaction from the school to deal with any outbreak. As we will see later on, if you can reduce this ‘reaction time’ things can change significantly.

Option 1 (One active half): Suppose we took half of the students and asked them to learn remotely with the other half coming in as per usual. The isolated students would be safe but the other students would likely become infected at some point. In this case, we potentially get fewer infections. Specifically, if prevalence is low in the general population, the chances that the 'seed’ or ‘Student 0’ infection is in the group who are active is 1/2. Thus, the total expected number of infections is 8.5 (= (1 + 16)/2).

Option 2 (One month on and one month off): Suppose we rotated the students in two groups with one month on and one month off. In this case, chance will determine what happens and that is in our favour. If the infected student was present at the beginning of the month, there would be 6 doubling cycles. If they were present, mid-month, there would be 3 doubling cycles and, if the infection happened late there may be no doubling cycles. In other words, the forced rotation is ‘naturally’ forcing isolation which reduces the potential exposure time. Do the maths on this and you would expect 4-5 students to be infected.

Option 3 (One day on and one day off): Now suppose we increase the frequency of rotations to every other day. In this case, there are 11 days where a group with an infected student exposes others to contagion. That happens for sure so the total number of infections over those 2 maximum doubling cycles is 4 (or 2 to the power of 2).

You’ll notice that Option 3 is the best outcome because, with two separate groups, over the course of a month, students in one infected group are exposed to others half of the time (or 11 school days over a month). Thus, if we had one week on and one week off, the same outcome would arise. This means there is some flexibility in terms of rotation frequency beyond a point.

This analysis also shows the importance of weekends. If students had activities that brought them together (but still in their separate groups) for weekends, then without rotations there would be 64 infections, option 1 would have 32.5, option 2 would have 15.1 and option 3 would have 8 infections. The weekends are like a natural break on contagion (at least to some degree).

On that score, if schools are selecting groups who are not going to interact, then if you want to take advantage of weekends or outside of school time to reduce contagion, then you had best allow friends to be part of the same in-school group. Otherwise, you will have them become mixers between groups which, as I will show in a bit, can partially undermine the whole point of this.

### What about the teachers?

In any rotation plan, the students spend time at school and then time at home but the teachers are there all of the time. What does that do to these calculations since those teachers mean that a fraction of the school population is mixing?

How many teachers are we talking about? If class sizes usually average 30 students, then those 500 students need 16-20 teachers. Add to that administrators, custodial and other staff and you might get 40 people or 8% of the population.

Such mixing does undermine the potential benefits of rotation. But the good news is that it doesn’t do that by much. If there is an outbreak in one group of 250 students, then the probability of that infecting a teacher is not very high which means the probability of an outbreak jumping between groups is also low. This is certainly true if the frequency of rotations is high which is another reason in their favour.

There are some wrinkles. The first is that each teacher is seeing 60 students whereas each student is likely only interacting with 15 to 20 students if classes don’t mix. Thus, teachers may be more likely to become infected than a typical student. If classes mix, then those factors even out.

Second, the teachers may take more care — more on this below. In that case, there is even less reason to worry about mixing from them.

This is all certainly good news but it is contingent on an outbreak being recognised within 30 days and then being mitigated at that point. Extend the time at which an outbreak can grow without reaction then, as we saw in nursing homes, mixing can lead to problems.

### Some other wrinkles

The goal of rotations is to reduce exposure risk. This is done by limiting the number of days interactions can take place before there is a reaction to deal with an outbreak and by reducing population density by having asynchronous groups. What happens, however, when we change things like school size, the flow of interactions and there are potentially superspreaders?

**School size**: Suppose there were 1000 students instead of 500. In this case, because the reaction time is 30 days, then there is no real change in the number of infected students from a given seed or ‘Student One.’ But there is a change in the probability of there being a seed. Those double. In other words, there is a greater probability that more than one group might be infected. This increases the value of having rotations that reduce exposure risk. Similarly, the value of those rotations goes up if the virus is more prevalent in the general population.

**Managing flow**: Thus far, the discussion has assumed that the school is the relevant unit to consider potential outbreaks. But some schools are going to greater lengths to have separate groups down to the ‘floor',’ ‘building’ or even ‘class’ unit. The effect of this is to divide the school into more separate groups which achieve a similar outcome to rotation. However, if you can do this without having to have half of the school at home at any given time, you can achieve many of the benefits of rotation itself and have everyone at school all of the time. The point being, separation into more groups reduces the chance of cross-group spread and thus, effectively reduces exposure.

It is useful to emphasise that rotation has limited value if you are thinking of just taking a separated class of 30 students and rotating them with 15 on and 15 off. In this situation, the total numbers of infected students do not change when the infection reaches a class. Thus, before there is a reaction, a virus may have spread throughout the class and they may have achieved a form of herd immunity. At this point, there may be little cost to putting the groups together again as enough have been exposed that there is not much additional benefit to the continued rotation.

**Superspreaders**: The analysis here assumes a fairly smooth infection process. But the problem with places is that the smooth processes we see at a regional level may hide some events that cause many infections. In particular, what might happen is that a particularly infectious person enters a place and the infections spread wide and quickly after that.

Your intuition might be that such events make taking actions such as rotations more important as at least half of the population might be protected in that situation. This is true but that also means that finer grades of rotations at a higher frequency are going to be less useful. In other words, superspreaders are a reason to manage flow but less of a reason to have rotations on top of that.

### Are other interventions complementary to rotation?

Many schools are not just doing rotations. Instead, they are taking other forms of potentially protective actions.

**Masks**: In Toronto, older kids will wear masks. You also get significant gains from practices such as mask-wearing in schools. Masks potentially reduce the spread of the virus which would extend the time it took to double the number of infected people in a population. Suppose this increased the doubling time by 100%. In this case, over 30 days with weekends, there would be 2.5 doubling cycles. In this case, without rotations there would be 5-6 infections, option 1 would have 3 infections, option 2 would have 2.7 infections and option 3 would have 2 infections. In other words, **mask-wearing really reduces the need to have rotations.** Thus, one wonders whether the Toronto policy of no masks/no rotations for younger kids and masks/frequent rotations for high schoolers is really the optimal approach.

**Testing and surveillance**: A key parameter in the analysis of rotations is ‘reaction time.’ How short is the time between the first infection at a school and the isolation of exposed students? You can reduce this time by engaging in practices that monitor students for infections. This means testing — perhaps the fast point of entry tests or group testing at schools — and surveillance such as the monitoring of sewage waste from schools. Suppose that doing this allowed you to cut the reaction time in half to 15 days. In this case, there would be only 2 doubling periods before an infection was detected. Without rotation, 4 students would be infected before an outbreak was recognised. With daily rotation, you are down to 2-3 students. In other words, rotation likely is not necessary if you have better surveillance.

For schools, we don’t know what the reaction time might be. A high school in Atlanta that opened last Monday discovered 9 cases with symptoms who tested positive for Covid-19. The school decided to suspend classes and go online for the first two days of this week — a four-day break overall. The good news was that the reaction time was quick. However, this is weakened by the fact that this school was the subject of controversy when a student photographed crowded hallways and was briefly suspended for posting those pictures online. The reaction may have been driven by this attention. The bad news is that for most of those cases, they were likely present as soon as school opened and not the result of a single seed at school.

**Barriers**: What about installations such as plastic barriers to protect teachers from students or classroom design to assist in social distancing? If these are effective then they are likely more so if there are fewer students in a class. Are they effective? This, from one University of Georgia classroom instructor hints at a problem.

**Ventilation**: What is potentially more effective is proper ventilation. Keeping kids outdoors when the weather is right. Keeping the windows to classrooms open. In Australia, the winters are relatively mild so one can imagine that this will be possible. For Canada, the tolerance for cold is better and so maybe this can be done for a month longer. What better ventilation will do is again reduce the length of time it takes for infections to double. Thus, it has a similar impact as the discussion on masks above.

In summary, what we see here is rotation is more useful when you do not have other actions you can take to mitigate viral spread or improve the speed at which infections are detected. This is because what rotation is really doing is forcing you to take regular breaks in exposure which is valuable precisely when you don’t have other ways of preventing exposure. That said, rotation itself could reduce viral spread (that is, the reproduction number) because it forces a reduction in population density in a place. Again, it is precisely because these other interventions can reduce viral spread that they are substitute options to rotation.

Thus, **while it is tempting to take a ‘kitchen sink’ approach and take every intervention at your disposal, those interventions have costs. Masks must be procured and worn. Tests require infrastructure. Finally, rotations themselves lead to kids at home which has its own costs above the potential education costs. Instead, there is a good case to be made that the optimal strategy is to either rely on rotation to reduce exposure risk or to invest in other interventions like mask-wearing and testing. Doing all of them may give fewer benefits relative to cost.**