Stranger danger: are we more Covid-safe around friends and family?

The more that people spend time with one another, the more likely they are to share infection status. This implies that spreading the disease among those with whom we are in frequent in-person contact – friends, family and colleagues – becomes less and less likely.

Facemask mandates have been introduced in many countries to help stem the spread of Covid-19. But while masks have been shown to reduce the spread of the disease (Hansen and Mano, 2021), imposing obligations to wear them has been controversial, and there has long been significant political pressure on governments to scrap mandates altogether.

Now that the UK vaccine rollout is well underway, the government has found it possible to withdraw mask requirements. Yet wearing masks is still encouraged, especially in enclosed spaces. It was against this background that a recent picture of the new cabinet, which showed unmasked ministers and advisers huddled around a table, prompted fierce criticism.

Confronted with the criticism on 15 September, the new secretary of state for health Sajid Javid controversially stated: ‘At the right time, people should wear masks and we set out guidelines around that. A lot of people do, particularly if you're in a crowded place with lots of strangers.’

Speaker of the House Jacob Rees-Mogg further raised the temperature by suggesting: ‘If you are in a crowded indoor space where you come into contact with people you don't normally meet, wearing a face covering can help reduce the spread of Covid.’ He continued: ‘If they [critics of Javid] came a bit more, they worked a bit harder, if they put their elbows to the grindstone, wherever you put your elbows, elbows to the wheel, they might not need to wear face coverings either because they would meet members of Parliament more regularly.’

On social media, the pushback was immediate, with prominent epidemiologist Christina Pagel tweeting of Rees-Mogg’s comments: ‘This makes no sense at all. None. Not a thread of logic’.

Let us examine more carefully whether and how familiarity and frequent contact between people may influence how the disease spreads.

First, it is clear that the virus can spread whenever there is an opportunity to do so, and the virus will not distinguish between friends and strangers. So it is not immediately obvious why the relationship between people should determine whether the virus spreads. Surely physical proximity is all that matters?

While it is true that close contact is necessary for the virus to spread, it is not sufficient. For disease transmission to take place, it must be the case that one individual is infected and another is susceptible. For infected people do not infect other infected people; and healthy people do not infect other healthy people. In other words, the disease can only spread when people’s health statuses differ.

In short, for the disease to spread there must necessarily be both the opportunity for the disease to be transmitted and the people involved must have different infection statuses. And it is precisely here that familiarity may play a role. For most people, the bulk of their contacts are with people they know, either within their homes (family), socially (friends) or in the workplace (colleagues).

How should we measure an individual’s exposure to infection?

There are two ways to measure a person’s exposure to infection, namely how many different people the person comes into contact with – which we can call the extensive margin – and how frequent or transmissive contacts with those people are – which we can call the intensive margin. Both of these margins are important, but not in the same way.

Let us take two extreme cases. Consider a cashier in a supermarket checkout. A person in this employment may come into contact with hundreds of people each day, but each contact is relatively brief and unlikely to be repeated on any given day. At the other extreme, a dentist may have only a dozen patients a day, but each appointment involves close proximity with each other person for an extended period of time.

To limit infection, both margins can be influenced and different interventions do this to varying degrees. In principle, facemask mandates are intended to decrease contacts at the intensive margin, because they limit the possibility of disease transmission, given some fixed number of contacts.

But it has been argued that while facemasks do indeed decrease contacts at the intensive margin, they may increase exposure at the extensive margin by encouraging people to reduce their social distancing efforts. In other words, restricting one margin can inadvertently influence the other margin.

Another instance of this happening is when the government imposes restrictions on social interaction and requires people to stay at home. Such measures effectively reduce contacts at the extensive margin, but may in fact increase them at the intensive margin.

This is because people who would usually spend the day at work, outside the household, now meet fewer other people overall. But while confined to their homes, they now spend much more time with their family, thereby increasing contacts at the intensive margin. As both the intensive and extensive margins matter for the spread of the disease, why should we expect stay-at-home orders to decrease infection overall?

To understand this, we must return to the idea that infection can only happen between people with different health statuses. While it is true that more intensive contacts do increase the chances of transmission (conditional on people having different health statuses), the probability that they in fact have different health statuses may be relatively modest compared with people in the general population.

Simply put, people who interact frequently or who engage in highly transmissive activities become more alike over time in terms of their health status. This means that transmission between such people becomes increasingly unlikely.

How does infection status change with more frequent meetings?

Let us consider a thought experiment (taken from Toxvaerd, 2017). Suppose that two people, let us call them Alice and Bob, may each be infected but do not (yet) show any symptoms. Alice and Bob meet again and again and on each meeting, the disease may spread if one turns out to be infected and the other is healthy. What can we say about how the infection status of Alice and Bob are related and about how this relation changes over time, as meetings become more frequent?

Let us start by considering the easy case when one person, say Alice, has shown clear symptoms of being infected but Bob has not. In this case, it is clear that the more meetings they have, the more likely is it that Bob will end up being infected too. Even if Bob was originally healthy, the repeated exposure over time means that Alice is likely eventually to infect Bob (unless she recovers), and therefore that they will both end up infected. We can say that the health statuses of Alice and Bob become more correlated over time.

Now consider the case where neither shows any symptoms but keep meeting each other. There are four possibilities in each period, namely that (i) both Alice and Bob are infected; (ii) both Alice and Bob are healthy; (iii) Alice is healthy and Bob is infected; and (iv) Bob is healthy and Alice is infected. If each continues to have no symptoms, then the probabilities of these different cases are as seen in Figure 1.

Figure 1: Probability of different health states of A and B with more meetings

Source: Author's calculation

Figure 1 shows that over time, the absence of symptoms means that the probability that either is infected alone decreases (cases iii and iv), while the probability that neither is infected is increasing (case ii). This makes sense because neither person has shown any symptoms.

More interestingly, the probability that both Alice and Bob are infected (case i) first increases and then decreases. The reason is that at the outset, we are very uncertain about whether Alice or Bob are infected. As they start meeting, it becomes more and more likely that one will infect the other but has simply not shown any symptoms yet. This explains the initial increase in probability.

As time passes and neither Alice nor Bob show symptoms, it becomes increasingly likely that neither were in fact infected at the outset. This means that the lack of symptoms now carries more weight and the probability starts decreasing.

Eventually, it becomes overwhelmingly likely that both are healthy and thereby share the same infection status (neither is infected). This can be seen from Figure 2, which shows how the correlation of the health states of Alice and Bob increases over time.

Figure 2: Correlation between the health states of A and B with more meetings

Source: Author's calculation

Conclusion

The more that people meet one another, the more likely they are to share infection status. But this also implies that the probability that they do not share infection status must decrease, and so spreading the disease within this group becomes less and less likely.

What does all this imply for the wisdom of not using facemasks around people you know? While there is indeed some logic to the statements of the health secretary, one should be very careful about basing policy on these ideas.

For starters, people inhabit intersecting social circles, which means that even people you meet frequently may suddenly change infection status while you remain healthy, thereby putting you at risk.

Second, there are lessons to be learned from the HIV/AIDS epidemic, where strategies like serosorting have been considered. These are strategies in which people only expose themselves to others with whom they share health status – what is known as seroconcordant. In practice, because infection is often asymptomatic, such strategies are very difficult to implement in practice.

Where can I find out more?

For more on the economic analysis of disease spread and the relation to the health states of individuals, see two early contributions by William Dow and Tomas Philipson and, more recently, by Flavio Toxvaerd:

Dow, WH, and TJ Philipson (1996) ‘The Implications of Assortative Matching for the Incidence of HIV’, Journal of Health Economics 15(6): 735-52.
Philipson, TJ, and WH Dow (1998) ‘Infectious Disease Transmission and Infection-Dependent Matching’, Mathematical Biosciences 148(2): 161-80.
Toxvaerd, F (2017) ‘On the Dynamics of Beliefs and Risky Sexual Behavior’, mimeo.
Toxvaerd, F (2021) ‘Contacts, Altruism and Competing Externalities’, CEPR Discussion Paper No. 15903.