# How safe is it to socialize?¶

OK, we are all tired of hearing about social distancing. Me too. I live completely alone, with but a trusty hedgehog and an untrustworthy dog to keep me company. Recently, I found myself in a room that I felt was more crowded than I felt comfortable with given the November COVID surge. However, I rapidly realized I don't have a good sense for how bad it is to be in a crowded room. How crowded is too crowded?

So, I decided to answer this question. For this blogpost, I assume that:

a) All COVID cases are randomly uniformly distributed. b) A person only comes into contact with a person if they are within a well-defined distance (say, the same room for more than 45 minutes) c) The period of time a COVID positive individual is infectious is 10 days. d) Nobody knows they have COVID. e) Except for me. I know I don't have COVID.

Given these assumptions, we can calculate the probability that a random person has COVID by summing over the number of cases reported in a state (Mass.) normalized to that states population for the last 10 days:

$$P(\text{random person has COVID at time, }t) = \sum_{i = t-10}^{t}\text{Reported Cases per person}_i$$

Then, if you are in a room with $N$ people, we are interested in the probability that none of those $N$ individuals have COVID:

$$P(\text{None of N individuals have COVID at time } t) = [1 - P(\text{random person has COVID at time, } t)]^N.$$

With the above equation, we can now begin to think about our safety. Suppose that every time I want to enter a room, I wish to be 95% confident that nobody has COVID. Using the equation above, I can simply find the largest $N$ such that the probability that nobody in that room has COVID is greater than 95%. In the code below, I simulate exactly that question:

## How has my safety fluctuated this week, if every day I interact with two new people?¶

Notice how safe each individual day is on average! Even at the height of the pandemic, each day is, relatively speaking, quite safe!

# How safe is "safe"?¶

Well, so far we've established that from the beginning of the pandemic to date, I have met 2 people a day every day, how safe have I been?

The answer for this is also easier to compute. Assuming I meet new people every day, then each day is independent from each other. Independent probabilities multiply. So I can take the probabilities above and multiply them to get my answer:

The number above is the probability that I have not come into contact with a single infectious individual this year if I hung out with 2 new people every day of the pandemic to date. Clearly, my safety to date if I met 2 people each day is quite far from a 95% "safety" profile!

How safe would I have to be to be sure that I had a 95% safety profile over the year? Then, we wish to solve the equation:

$$0.95 = p ^ {52}$$

I would have to keep a 99.9% safety profile. That means, if I wanted to avoid any infectious exposure, most days of 2020 I would have to... socially distance! Most days, we should NOT be meeting new people! Remember, here, new people means the people that you do not spend 100% of your life with.

Stay safe, everyone.