**Why to Keep Supermarkets Open for 24 Hours During a Pandemic**

# Recommendations

For those that are impatient to go to the theory borrowed from wireless communication protocols, here are the proposal for the policy makers as well as some behavioral recommendations:

*Policy makers:*Allow the shops to stay open as long as possible, even in the unusual hours. During the strict lockdown allow supermarkets and pharmacies to stay open for 24 hours. When the society starts to reopen, allow all the other businesses to be open longer than normal.*Shop owners:*During the first days of the pandemic, when there is a panic shopping, randomize the supply and the replenishments of the goods on the shelves. Employ extra staff to clean and disinfect continuously. (The staff for doing this can come from other areas, such as hospitality, where workers are being laid off.)*General population:*StayHome most of the time. Do not follow your usual schedule for shopping, try to choose the time to visit the shop randomly and arbitrarily whenever there is a need to buy something. If there are too many people in the shop, come back at a random time. Organize smaller group of households that are coordinated in a way that one person does the shopping for the whole group of households at a given time.

**Background**

During the ongoing COVID-19 pandemic, a number of governments have imposed measures that attempt to enforce social distancing and minimize the spreading of the virus. One of the first measures was to cancel the events that have more than *K* people, where *K* is, say, 1000, but as the pandemic progresses, this number *K* has been decreased to 100 or similar. While it is obvious why this idea is beneficial, it is less obvious what should the proper value of *K* be.

However, there are gathering places where *K *cannot be controlled, such as supermarkets or shops. Due to this, we have, in fact, witnessed behaviors that severely undermine the policies that aim to mitigate the spreading of the virus. For example, after the announcement of new restrictive measures by the government, people storm supermarkets for gathering food and home supplies (here I must remark that I am not aware of any SciFi author, even not Douglas Adams, guessing correctly that the toilet paper is the main product bought when people fear that doomsday is approaching). This is because the signal by which the government puts restrictions or a new policy creates correlation in the demand from the population: People gather in large numbers simultaneously, thus countering the strategy of minimization of number of people that come in close proximity.

Interestingly, the problem of access to shared resources (for example, goods in a supermarket) is at the heart of the design of wireless access protocols. For example, if multiple laptops connected to the same WiFi router (access point) want to send data simultaneously to the router, then they need to find a way to do it one after another, otherwise they collide and interfere with each other and, as a result, none of them manages to send its data to the router. The (partial) analogy with the supermarket is the following. There are two persons X and Y that want to go to the supermarket and none of them is aware whether she or he has the virus. Then these two persons want, ideally, to go to the supermarket at different times.

Let us explore what can we learn from wireless access protocols and how that can be turned into a policy as well as behavioral recommendations for the people.

**How spreading in time helps**

Let us assume that there are *K* persons that buy their home supplies from a supermarket. For simplicity, assume that shopping lasts for 30 minutes. As an extreme case, assume that the supermarket is opened only 30 minutes a day. Then all *K *people will come simultaneously to the supermarket and will get in touch. Assume that the probability that a healthy person *X* among those *K* persons gets the virus from another person *Y* present in the supermarket is *v*. Then the probability that it will stay healthy is (1-*v*). Note that here the probability *v* accounts both for the fact that *Y* may not have the virus or, even if *Y* has it, *X* does not in touch with Y. Then the probability of *X* getting the virus is:

However, if the supermarket is opened for 1 hour per day and we somehow convince half of the people to come in the first half hour and half in the second half hour, then the probability that one gets infected is:

Finally, if the supermarket is opened 24 hours, then there are 48 shopping slots and if we somehow manage to organize the people to spread equally across the slots, then the probability that *X* gets infected is:

To illustrate with some numbers, assume that the chance to get infected by a person that carries the virus is 1%, that is, *v*=0.01 and assume there are *K*=480 people shopping in that supermarket. Then, if the supermarket is opened for only 30 minutes, a healthy person comes in contact with 479 potentially infected persons and P1=0.99, that is, the chance of getting infected is 99%. If the supermarket is opened for 1 hour, then P2=0.91, i.e. the chance is 91%. Finally, if the supermarket is opened 24 hours, there are 48 shopping slots, in each slot a person can meet with 47 other people and the chances of getting infected fall below 10%.

The simple conclusion from this is that the supermarket should be opened as long as possible, that is, 24 hours a day. But then there are two key questions:

1. How do we ensure that people can be divided into smaller, ideally equal-sized groups?

2. How to remove the incentive for the people to go to markets immediately after an important event, e.g. announcement by the government, fearing that there will be no sufficient supplies if they wait?

**Some ideas taken from wireless communication protocols**

Let us at first look at question 1.

A seemingly efficient solution would be centralized: Each family gets a time slot and a supermarket where to shop, while the supermarkets adjust their supplies to the demand posed by the families. In communication protocols this is called centralized scheduling. The technical benefit of this approach is that it de-randomizes the times at which the demands of the people to the supermarket arrive and manages to guarantee ideal division into smaller groups. The problem with this approach is that it relies on a correct assumption about the demands posed by the families, which is unpredictable and very diverse. Furthermore, the human aspect of this solution is horrible, as it is irresistibly reminiscent of Orwell’s 1984 and similar historical setups.

However, the idea of de-randomizing the demand can work at a smaller scale. For example, a group of neighbors can make a decision that only one of them goes shopping on behalf of all. Just to make a point about the analogy, In wireless communication protocols for supporting Internet-of-Things, this would be called user aggregation through capillary networks.

A protocol that is much more respectful to the individual freedom is the random access protocol, such as ALOHA. If a person would follow this protocol to go to a supermarket, the person *X *would decide to do so at a random time. Upon arriving to the supermarket, *X *would count the number of cars on the parking and, if the number of cars is large (e.g. larger than 50), then *X* woudl decide to come back at a random time in the future. Here the random choice of the time to come back is crucial, since if *X* and *Y *come at the same time, see more than 50 cars, and both decide to come back in an hour, then they will meet each other again, while the point here is to minimize the meetings. Even if this randomized approach does not guarantee that *X* and *Y* will not meet again in the supermarket, it does decrease the probability of encounter and can make use of the extra opening hours of the supermarket.

Regarding the question 2 and how to reassure the people that there will be supplies even if they come later, the policy makers and the supermarkets can use randomized strategy for replenishment of the goods. In other words, instead of keeping the shelf as full as possible all the time, the supermarkets can randomize the supply quantities and the times of replenishment. This will discourage the use of any “strategy” by the buyers, such as hoarding immediately after an announcement of new restrictions from the government.

**Conclusion**

As every model, the analogy with wireless communication protocols is not full and it has its drawbacks. For example, if an infected person comes to a shop, he can touch an object that stays infected even when that person is not there. The model described above does not capture this case. However, it is sufficient to provide some ideas both for the side of the policy makers as well as behavioral recommendations, stated at the beginning of this article.