"If the government implements its plan B, then what impact will that have on event venues?"

This question, voiced here by Kevin Instance of Gull Rock Events, is one that probably occupies the mind of many events organisers this autumn. Currently, large events — for example sports matches and music festivals — can take place without any COVID-19 restrictions whatsoever. If infection numbers rise, however, then plan B may mean COVID passes becoming mandatory, along with face coverings and other social distancing measures.

How would COVID passes impact on sports venues?

Instance is a safety consultant, adviser and practitioner and deals with various types of small to medium sized events. These include music events with a capacity of several thousand, such as music festivals in the summer. In the winter he is safety officer for Bath Rugby and Exeter Football Club.

Instance speaks for an industry that has taken a huge battering over the last year and a half. Its estimated pre-COVID value to the economy is £70 billion, and it provides 700,000 jobs. Helping this industry weather the storms that may still lie ahead is important.

This is why the International Centre for Mathematical Sciences, the Newton Gateway to Mathematics, and the Virtual Forum for Knowledge Exchange in the Mathematical Sciences convened a virtual study group (VSG) in the summer, unleashing the brainpower of mathematicians on the problem of how to manage large events in the face of COVID-19. Kevin Instance took part to provide his expertise, along with representatives of the Church of England and the Edinburgh Fringe, and behavioural scientists and public health experts. The event was part of the Rapid Assistance in Modelling the Pandemic (RAMP) Continuity Network.

### COVID passes and queues

Bath Rugby Club was one of the venues the VSG used as a case study. It has a capacity of just under 15,000, counts as an outdoor venue, and people need to present tickets on entry. If the checking of COVID passes, and possibly people's temperature, becomes mandatory, then how should safety officers manage people's arrival at a match?

It's a question that mathematicians can address with relatively simple tools. The mathematicians at the VSG built a mathematical model to simulate the formation of queues. They assumed that people turn up at the stadium at random, using a probability distribution to describe the chance that someone arrives at a given time before kick-off. Assumptions on the exact nature of this distribution, and how long it takes to check people on entry, then allowed the mathematicians to explore different scenarios.

The toy model suggests that handling people's arrival as you would have done pre-pandemic is not an option. Long queues would be likely to form and some people may not get into the stadium until long after kick-off. One way of avoiding this is to check people's COVID status in advance, possibly off-site. The model also allows you to explore what percentage of people would have to be pre-screened in this way to get the queues down to an acceptable size.

The model is only a simple prototype constructed with Bath Rugby Stadium in mind, but it demonstrates the power of the mathematical tools. Since, as Instance fears, Plan B may be implemented at very short notice, event organisers need to know in advance how their venue will be affected so they can be ready to put measures such as pre-screening into place quickly.

### Vaccinated or unvaccinated?

One thing that can quite easily be checked before a match is people's vaccination status. Assuming people are happy to share this information, they can be asked to do so at the time they buy their tickets. Vaccinated people could then be assigned fast-track queues, and there's also the option of choosing a seat allocation based on people's vaccination status: the aim here would be to seat unvaccinated and vaccinated people in a pattern that prevents "hotspots" at which the risk of infection is particularly high.

This latter possibility was also explored at the VSG. Again, the mathematicians used a very simple mode which assigns different probabilities of being infected to vaccinated and unvaccinated people, and estimates the *viral dose* a person would receive sitting in a particular place with a particular pattern of vaccinated and unvaccinated people around them. The result here is that preferential seat allocation doesn't make much of a difference to the risk of catching infection, at least not according to the simple model. It would be more useful, the mathematicians suggest, to minimise the chance an infected person enters the stadium in the first place.

### Wear your mask, even outdoors

While rugby matches at the Bath stadium served as an example for outdoor ticketed events, the VSG also looked at indoor ticketed events, and outdoor unticketed events. For the latter, you couldn't find a better example than the Edinburgh Fringe. "Outdoor street performances form a major part of the attraction to the Fringe," the mathematicians write in their report, "In a typical year, there might be 130 street performers, 1,200 slots and 400 buskers in 50,000 slots, plus a similar number of promos etc. - they are however an uncontrolled part of event."

Street performers play a hugely important role at the Edinburgh Fringe. This photo was taken in 2014 by Rayonick, CC BY-NC-ND 2.0.

In a normal year, there are only a limited number of stewards to manage the street performances. The crowd may or may not respond to nudges and cues from these stewards, for example regarding mask wearing or social distancing. And, as the mathematicians note, the performers themselves could be agents of control, encouraging people to comply, but also agents of rebellion.

Such a set-up seems impossible to model at first, but the task becomes easier when you break it down into chunks. In this spirit, the VSG asked whether it's possible to quantify the risk of infection within a group of people watching a single street performer. The mathematicians modelled the situation by imagining people standing in a 10m x 10m square at least half a metre apart. At each time-step of the model each person has a 50% chance of wanting to move to another location in a square around them. The model assumes there being one infected person in an audience of around 180 susceptible people, and estimates the amount of virus each individual receives and their risk of infection.

Despite its simplicity, the model suggests an interesting result. Although the event happens outdoors, mask wearing appears to make a difference, more so than reducing the show time. In such a situation everything depends on people's willingness to comply. The VSG suggests to build encouragement into the fun: performers and "pink policemen" could remind people to wear their masks as part of their performance, and "pied pipers" could help direct crowds to avoid congestion.

### A new and useful perspective

The VSG's results on COVID safety at large events are only preliminary. The aim was to identify useful mathematical tools for modelling such large gatherings, which future work will explore further. As with all mathematical models caveats apply: the simulations don't give sure-fire predictions of what is going to happen, rather they allow you to explore possible scenarios based on specific assumptions. To find out more about these caveats, as well as other results of the VSG and suggestions for future work, see the VSG report.

But even this preliminary work is of some use to those tasked with putting on large events. "There is a network of people within the events safety world, but we tend to look at things with the same mind set," says Instance. "Having people look at the same problems with new eyes in a different way was really useful and energising."

Lyndsey Jackson, Deputy Chief Executive of the Edinburgh Festival Fringe Society agrees. "[The results are] going to be hugely helpful in supporting performers to engage with crowds safely, and the key advice is both digestible and useful," she said about the VSG.

One aspect that Instance would like to see explored further is people's behaviour. We have quickly become used to our rediscovered freedoms, and renewed restrictions may not be met with enthusiasm. Effective messaging and communication will be crucial should plan B be instigated. And while integrating such behavioural factors into mathematical models is tricky, help is on the way: the topic is to be explored next month in another event hosted by the Newton Gateway to Mathematics, as part of the RAMP Continuity Network.

*The event on behaviour and epidemiological modelling has now taken place. You can read a report here.*

### About the author

Marianne Freiberger is Editor of *Plus*. She interviewed Kevin Instance of Gull Rock Events in October 2021.

*This article was produced as part of our collaboration with the Isaac Newton Institute for Mathematical Sciences (INI) – you can find all the content from the collaboration here. *

*
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit www.newton.ac.uk to find out more. *