Skip to content

How much do you risk

Often and unless some rules have imposed you to stay home, you find yourself in the situation to decide whether meeting other people is too risky. This little post will help you compute the risk that you incur.

The calculation that I present here will overestimate the risk. The details and background of the calculation are presented below.

Probability that at least one is infected

Let us start by considering the following situation. In your town there are currently (say today evening) N_0 cases of infected people. Let us suppose that the number of infected people grows exponentially. The number of infected people walking around but still undetected is unknown but we can make the assumption that they will show up in the statistics of the number of infected in about ten days. If the growth is exponential with rate 0.4, the total number N_{10} of people infected in ten days from now will be

    \[ N_{10}\, =\, N_0 e^{0.4\cdot 10} \, =\, N_0 e^4\, . \]

Therefore, the number of infected people walking around in town tomorrow is given by

    \[ N_1\, =\, N_{10}-N_0\, . \]

Suppose now that the number of inhabitants in your town is M, which we also assume to be much larger than N_{10}. Therefore, the proportion of infected people tomorrow will be p=N_1/M. It turns out that p is also the probability that any one person that you will meet tomorrow is infected.

Suppose now that tomorrow you will encounter, or be in contact with, n people. Then, the probability Q_n that at least one of them is infected is given by

    \[Q_n\, =\, 1-(1-p)^n\, .\]

At this point, one could assume that Q_n is also the probability to get infected. If you make the calculation and you get Q_n < 0.001 you can decide on your own if it is low enough to go out and meet those n people.

Background material

In this model. I have assumed that people that are infected now will show up their symptoms within the next 10 days. If you believe that 14 days is better, you should change the first formula by substituting the 10 with a 14.

I have also assumed that the growth of the population is exponential. This is not always the case, especially towards the end of the expansion of the virus. In case the expansion of the infection is slower than exponential, this calculation may very badly overestimate the risk.

I have assumed that the exponential growth occurs with a rate \alpha equal to 0.4. In fact, we find that for many countries and regions, during the exponential growth the growth rate obeys 0.3 \le \alpha \le 0.4. I have no explanation for these values (and there are also exceptions). To avoid an underestimation of the risk I have chosen the upper value \alpha = 0.4.

When p is the probability that any randomly chosen person is infected, then the probability that m out of n people that you will meet tomorrow are infected, is given by the binomial distribution

    \[ \Pr\{X = m\}\, =\, \left(\begin{array}{c} n \\ m \end{array}\right) p^m (1-p)^{n-m} \, .\]

You do not need to understand everything of this formula. What is important is that the probability that no-one is infected is given by

    \[\Pr\{X=0\} = (1-p)^n\, .\]

Thus, the probability Q_n that at least one is infected is just Q_n = 1 - \Pr\{X=0\}, which leads to the formula for Q_n as shown above.

Thank you for reading.

Leave a Reply

Your email address will not be published. Required fields are marked *