Quote:
Originally Posted by Dudd
Because R0 is defined as the initial transmission rate. As more people get it, a higher percentage of the population is immune, and the R at that point is lower than R0. So with the example of R0 of 1.0, after the first person gets it, R is no longer 1.0 but maybe 0.99999999999999 because there's a chance, however small, that the third person to get sick would have been the first person who gave it to the second person, or maybe the fourth person would have given it to either of the first two cases, etc. Eventually the chain gets broken and the disease dies out.
For an R0 of 1.0, that happens pretty quickly. Think about if an outbreak happened in NYC. In order to reach 1% infected, that's 100,000 people if we say the metro population is 10,000,000. Think about the odds of keeping that streak going. You can do the math, but the odds of a third case occurring are 9,999,998/9,999,999, a fourth case 9,999,997/9,999,999, and so on. The odds of the chain never being broken approach zero way before you hit even 1% of the population being infected.
I'm not an epidemiologist so I'm sure that's a super simplistic model of the way diseases actually spread, but it seems like it should be right and, if anything, overestimates the spread of a disease. For example, say a husband is the initial case of a disease with an R0 of 1.0. He gives it to his wife. She then gives it no one and the chain ends. However, if she had been the one to get it first, she would have likely given it to him as her one transmission rather than some random person in the supermarket. So maybe something with an R0 of 1.0 stops even quicker than that model would suggest because it's not pure randomness of who in a population gets infected. And then, of course, there also is pure randomness. R is just an average figure, so maybe the chain dies out because one person gets sick and then gets hit by a bus the next day or just gets lucky and doesn't pass it on to anyone, and that's how the chain breaks.
But then it's not enough just to get the R0 down below 1.0. If you look on the graph, way more than 50% of the population gets sick with an R0 of 2.0 even though you would think, based on that model, that R goes below 1.0 when more than 50% of the population has been infected. But if you think of it like generations, the R0 generation is 1 person. The R1 generation is 2 people. The R2 generation is 4 people, etc, etc. By the time you get up to 50% of the population, there's going to be a lot of currently sick people, and even if the generations that follow R(50%) start getting smaller, it's going to take a while for the transmission chain to completely burn itself out as the number of new cases approach zero, which it looks like it takes an additional 25% to 30% of the population getting sick before that happens.
I appreciate the explanation! It sound like the graph is assuming that only one person starts out infected when the R0 is measured for the purpose of the graph. In which case, the outcome infection % would be close to 0 in a large population when R0=1.
But we now have a situation where R0 might be close to 1 under social distancing restrictions, but a substantial % of the population was already infected before the restrictions were put in place. In this situation, a very large percentage of the population will end up infected even if we get R0 a bit <1. (If R0=.9 and 10% start infected, those 10% will infect another 9%, who in turn will infect another 8.1%, who will then infect 7.29%, etc. The outcome is going to effectively be herd immunity.)
If we got R0 -much- <1 (e.g. .3), we could still eradicated the virus fairly quickly. But this would require much stronger restrictions. It might be easier to implement this sort of thing if we were simultaneously issuing immunity certificates to those who are already recovered, and the could do the work of opening up and running the economy while everyone else (especially vulnerable populations) were under stricter lockdown. Given that it sounds like it is much more feasible to do widespread serology tests than active infection tests, this might also be more realistic than a universal tracing program.