A Risky Visit
Today I tried to write about the risk of passing the coronavirus when going to visit someone. It looked nice enough and I'm copying an edited version here, using KaTeX instead of the original LaTeX version I wrote. Also, I was experimenting with unicode symbol support, for which I used XeTeX.
I didn't make up the numbers that appear, but I haven't included the references either. It is just a simple note with no big pretension.
You would like to go and visit your family. But we have an ongoing pandemic ☣ and there are risks. In particular, you risk infecting them.
Given that, is it worthwhile to go?
What Is Reasonable?
What is the acceptable threshold for the probability of passing the virus to your family? Is it one in a thousand? One in a million?
We could first see what we would accept as a reasonable chance. Then, make a rough estimation of what the actual probability is, and see if it is reasonable.
Current Conditions
We assume some facts. You:
- Live in Madrid and have gone out / socialized very little. We know that the incidence in Madrid is less than 1%, so $\Pr(\text{infected}) \lesssim 1 \%$.
- Have no symptoms. We know $\Pr(\text{no symptoms} \mid \text{infected}) \lesssim 50 \%$.
- Are going to get a PCR. If the result is negative, we have to consider that it has a false negative rate $= \Pr(\text{negative result} \mid \text{infected}) \lesssim 20 \%$.
- Use mask and hand sanitizer regularly, and keep interpersonal distance. In such cases $\Pr(\text{passing virus} \mid \text{infected}) \lesssim 10 \%$.
Estimating The Probability
Giving the current conditions, we have
$$ \Pr(\text{infected with no symptoms & PCR negative}) \approx I \times S \times F $$
where $I \lesssim 1 \%$ is the incidence, $S \lesssim 50 \%$ the probability of not having symptoms if infected, and $F \approx 20 \%$ the probability of a false negative in the PCR test.
Thus, the probability of passing the virus to your family is
$$ \begin{align} \Pr(\text{passing virus}) & = \Pr(\text{passing virus} \mid \text{infected})\Pr(\text{infected}) \\ & = P \times I \times S \times F \\ & \lesssim 10 \% \times 1 \% \times 50 \% \times 20 \% = 1 / 10000 \end{align} $$
That is, the probability is around one in ten thousand.
Limitations
What about the chances of you getting infected? Or of indirectly increasing the chances of infecting your family because they would go out more during your visit? And even if you don't pass the infection, what about the stress for you and your family of living under such threats? How is it going to affect all your interactions and feelings? How much frustration and worry can it all generate?
All those are important considerations too, and they go beyond the mere estimation of the probability of infecting that we did. So this is by no means a full account of all what's going on. It is just done in the spirit of helping inform the decision by providing some insight to one of the pieces.
And for fun ☺
Appendices
Notation
We write $\Pr(A)$ to represent the probability that the event $A$ has happened, and $\Pr(A \mid B)$ to represent the probability that $A$ has happened given that $B$ has happened.
Deriving Probabilities
When we want to estimate numbers such as $\Pr(\text{infected with no symptoms & PCR negative})$ we use Bayes' theorem:
$$ \Pr(A \mid B) = \frac{\Pr(B \mid A) \Pr(A)}{\Pr(B)} $$
For example, calling ☣ being infected, ⚕ being healthy, 🤒 having symptoms and ☺ having no symptoms, the probability of being infected given that we have no symptoms is
$$ \begin{align} \Pr(☣ \mid ☺) & = \frac{\Pr(☺ \mid ☣) \Pr(☣)}{\Pr(☺)} \\ & = \frac{\Pr(☺ \mid ☣) \Pr(☣)}{\Pr(☺ \mid ☣) \Pr(☣) + \Pr(☺ \mid ⚕) \Pr(⚕)} \\ & \approx \frac{\Pr(☺ \mid ☣) \Pr(☣)}{1 \times \Pr(⚕)} \\ & \approx \Pr(☺ \mid ☣) \Pr(☣) \end{align} $$
where we have used that $\Pr(☺ \mid ⚕) = 1$ and $\Pr(⚕) \approx 1 \gg \Pr(☣)$.
Finally, when we estimate the joint probability of not having symptoms (☺) and giving a negative result in the PCR, we assume they are independent:
$$ \Pr(\text{☺ & negative PCR} \mid ☣) = \Pr(☺ \mid ☣) \Pr(\text{negative PCR} \mid ☣) $$