Thread by @Max_Deforet

Random graph and epidemiology people, I have a question for you: what is the marginal probability of indirectly killing someone by going out without a mask? There is a small probability I get infected, and then a small probability I infect other people, which in turn...

The impact of my choice looks exponential, but the likelihood is small. Is there a way to calculate it?

I have too many friends who are getting ready for apero from May 11. I want to tell them how many people they will indirectly kill.

(I feel like a mean field approach is easier to use, but it doesn't quantify the consequences of one given person.)

Allright, I calculated it myself, with back of the enveloppe simulations. I'll present it tomorrow. Spoiler: stay at home.

Alright, so let’s go. (TW: armchair epidemiologist here). Here is my attempt to calculate the marginal effects of being infected. I’ll present all the steps of the model, so you can clearly see all the assumptions (mean-field, non spatial, non structured, etc).

I start with the SIR model en.wikipedia.org/wiki/Compartme…

With parameters reasonable for Covid-19: R0 = 2 (Reproduction number), γ = 0.2 (Recovering rate), λ=0.01 (1% Death rate). I consider a population of 100000 people (like a city, or a large neighborhood), starting with N0=1000 infected.

I run SIR simulations, and I consider that the number of death is λ*number of recovered.
SIR gives this trajectory: it ends up with 800 deaths and 78888 recovered (immune).

My initial question was: what is the consequence of going out without a mask? Let’s break up this question in two:
- What if the probability of getting infected?
- How many people will I indirectly kill if I get infected?

At time t=0, there are S=N-N0 susceptibles, I=N0 infected, R=0 recovered. At time t=1 day, there will be βIS/N fewer susceptibles, βIS/N-γI more infected, and γI more recovered, including more λ*γ*I deaths. (That's the SIR model)

For a susceptible, the probability of getting infected is the number of newly infected divided by the number of susceptibles: βN0/N. It looks good: the more infected people(=contagious), the more likely to get infected.

Now what about the consequence of being infected? If I get infected, I will transmit the disease to R0 people, by definition of the model. These people will in turn infect others, etc. A λ-fraction of these people will die.

If the epidemic starts with N0=1000 infected, there will be X deaths after T days. But if the epidemic starts with N0=101 infected, there will be X’ deaths after T days. I want to measure X’-X, the number of deaths that are related the one extra initial infected person.

I choose T to be as such X’-X is maximal.
X’-X at 0 (because X’=X=0 at first), peaks, and goes back to zero. The final state of SIR is nearly independent of initial conditions. The same number of people will die eventually. The question is how high it peaks #flattenthecurve.

Now the results. I decided to present the results as a function of N0, the number of initial infected. The effect of an extra infected individual is not the same if there are very few cases or if there are infected people everywhere. First, the likelihood of getting infected:

The number of extra deaths at the peak:

Now the combined result (the product of the two last graphs): the expected number of deaths caused by someone getting out without a mask.

Conclusion: This is a very basic model (again, #armchairepidemiologist), I wish I could make it more realistic. I understand one needs to go out to work/buy groceries/etc. That’s fine as long everyone is making their best to protect each other.

If not protected, with R0=2, the effect of going out is (indirectly) killing 0.0006 people. Again, that's the marginal effect, for one person, going out for one day. Now you know.

I believe it would be useful to publicize more widely the impact of going out without a mask, and more generally the marginal expectancy of measured quantities, to raise awareness about individual responsibility. Reopening does NOT mean "going back to normal". #keepstayinghome

One extra note:
- individual consequence of going out: probability of getting infected (1% for N0=1000) x death rate of the disease (1%). That 0.01% of one death (of oneself).
- collective consequence of going out: 0.1% of a death.

Interestingly, the former increases with N0, while the latter remains nearly constant with N0. People will stop going out if the number of cases outside is too high because the individual risk is too high. But the collective risk starts already at a very low number of cases.