Here in California, we’re finishing our first week of shelter in place. Governments around the world have implemented a variety of similar policies, from complete quarantines to simple travel advisories. As I wrote Tuesday, I believe swift quarantine measures are the only way to stay the virus’s ascent. This isn’t just opinion. The math says isolation is the fastest way to make this pandemic disappear. Of course, like every optimization problem, there are constraints. People need to shop for food. Families live under the same roof. Society depends on interactions, however minimal. If that’s the case, how dangerous is it for us to cross paths?
More directly: in a group of people, what are the chances that no one has Covid-19? Answering this allows us to understand how safe we are when we’re in groups.
Some Quick Math
First, let’s answer an easier question: in the course of your daily interactions, what are the chances that any one person you interact with has Covid-19?
If you divide the number of active cases in your location by the population in your location you obtain a rough estimate. This is ‘rough’ because most people stay home if they’re sick. However, let’s assume there’s an equally sized group of individuals that are infectious but asymptomatic roaming out there. At the beginning stages of the infection this a reasonable simplification. The beauty of modeling is that you can decide to adjust this number based on your own beliefs if you disagree.
Let I be the number of local active cases and N be the local population:
To answer our original question, we take the complement (i.e. what’s the probability someone is not infected?):
If you want to know the probability that multiple (independent) events are true at the same time, you have to multiply their individual probabilities. Therefore the probability that no one in a group of k has Covid-19 is:
Coast to Coast
Let’s look at San Francisco and New York City as examples. As of March 25th, they had 178 and 20,011 cases and populations of 884k and 8.6M respectively. Taking the equations above, the probability of no one being positive in a group of k follows these curves:
Said differently, if you were in a room of 250 people, the chances that everyone is negative is only 56% in NYC and 95% in San Francisco. Although only two in every 1000 people have coronavirus in NYC, probability works in such a way that your chances of encountering at least one person in 250 are staggering. This is the math of why groups are so dangerous, the chances compound as you add people even though individual probabilities are low.
You can also interpret these numbers in a different way. For purposes of illustration, say the average person in NYC has about 250 people in their personal network. This implies that 44% of people know at least one person with the virus today.
If you flip the equation around and solve for group k, you can ask how many cases there has to be in NYC for people to have a 90% chance of knowing one person in their network who has it. The answer is 78,800 cases. At the current rate, NYC will cross this threshold in the next few days.
How many people can I safely see in a day?
Safe and legal are two different things. First, there are laws – read yours. Second, these equations are helpful, but they shouldn’t be interpreted too precisely, so stay safe.
No matter your situation, it’s not safe to see lots of people right now. Although you have a small chance of interacting with someone positive, there are many people taking that chance every day. Some of those people will be unlucky. Those unlucky few infections compound quickly through this process.
Working the Numbers
Let’s assume you want to have some ‘margin of comfort’ (probability) of knowing you won’t run into anyone with coronavirus in a group of k people. What’s the largest group you could be in?
This is best answered through an example using real data. The following graph shows the maximum safe group size given a margin of comfort using the equations above and real data from NYC cases:
If you wanted to be 90% sure you didn’t run into anyone with the virus you’d stay below the red line. If you were conservative and wanted to be 99% sure, you’d be below the blue line. On March 1st, you could walk freely. Just two weeks later the picture was far more dire.
You can see that as cases grow, the safe group size falls precipitously – no matter your margin of comfort. This is why it’s important to act quickly with social distancing: the safe group went from 100,000s to 100s in a few days. In fact, the graph compresses so quickly, it’s easiest to see in log scale:
An interesting overlay is what New York University (NYU) decided to do with their classes. NYU has a population of about 51,000 students. If you ran the university and your margin of comfort was 90% of not having any students with COVID-19, you moved classes online at precisely the right time.
Now let’s say you run Starbucks, and you have to decide when to close your stores in NYC. The average Starbucks sees roughly 500 customers per day, so in a busy city let’s double that and assume 1000 customers per day. Using the same equations, you can then answer: “What are the chances no customers enter with coronavirus today?”
While it’s unlikely customer count would stay the same for this period, the graph is instructive. In the days following the Starbucks announcement, the chance of a store having all customers be coronavirus-free went from nearly 100% down to 10%. It’s interesting to me how many of these organizations intuitively made the right call at just the right time.
These are just a handful of examples but they support the same point: any individual is unlikely to be infectious, but as you add them to groups, the chances skyrocket that there’s at least one covid19 carrier in the group. This is why social distancing and limiting groups is so critical to stopping the spread.
Side note: this math underscores how heroic it is for any person to step into their job to keep us fed, healthy and safe. A huge thank you to everyone in those roles; doctors, police officers, chefs, and more.