Monday, April 13, 2020

Herd immunity assuming the co-existence of different R0 regions

Most of the blog posts in recent two months were about Covid-19; I am sure that not everyone likes it. A month or two before the scientific public (and an even longer time before the general public) has figured it out, the TRF folks understood that Covid-19 is effectively just another variation of flu (and should be treated as such; this remains the most important divisive point that separates the sensible people from the hysterical demagogues), it's probably significantly older than from December 2019, flattening of curves doesn't save any lives, almost all similar restrictions are just wastes of money, what is really effective is to isolate the vulnerable and leave the free world for the resilient, the economic suicide and the enthusiasm around it is shocking, lockdowns seem harmful by the empirical data, viral doses matter for fatality rates and probably are increased by lockdowns, masks are more effective than restrictions of people's movement for the reduction of the basic reproduction number R0, and dozens of other key points.

Zoom in by the mousewheel, the map carries the information about the U.S. states and other things, too. Full screen.

In recent days, it becomes empirically clear that numerous countries have peaked as well – and they did so because they have reached the herd immunity calculated for the value of R0 coming from their recent policies, see e.g. this claim about Lombardy where the drop of the deaths in recent days was staggering (from 458 to 110 if you cherry-pick days a little bit).

OK, what is my mental model for the propagation of the virus? If you open some graphs for individual countries at the Worldmeters website, you will see that the early dynamics often represents an exponential growth with the doubling in some 3 days. It is very fast.

We have had doubts whether it's real or an artifact of the growth of testing. After some time, I think it was real, after all. The testing rates also went exponentially up at many places but I am not aware of any place where the increase of testing "safely" outpaced the increase of the disease in the early stages (a situation that would be seen as a dramatic drop of the percentage of tests that end up as positive).

OK, if the disease (the total number of cases) doubles in 3 days, it quadruples and 6 days and quintuples (5.04) in 7 days i.e. in one week. Five to the power of 8 is 400,000. If you start with 25 seeds, 8 weeks are enough to propagate the disease to a nation or city of 10 million. In two months, you simply approach the saturation. In two months, the fastest growing hot spots of the epidemics run out of vulnerable people. So because the disease may spread so quickly, the peak is almost guaranteed in all places that feel threatened at all within some two months!

It's obviously nonsensical to try to slow down this process by expensive policies if you can't turn the increase to a decrease. Asian countries may have managed to get a real decrease but no European or North American countries did. Free whites are simply not ready to establish the same restrictions on face masks, human movement, and on being spied upon as the East Asians. If you can only get a slowdown – e.g. doubling in 4 days instead of 3 days – all the policies are absolutely worthless because you at most delay the convergence to saturation from 2 months to 2.5 months or something like that. It's so clearly pointless.

Now, the funny thing is that the R0 – or, equivalently, the time needed for the doubling of cases – isn't the same in all regions. It is generally higher (the virus spreads faster) in dense regions and cities, socially interacting places, at lower temperatures (the warm spring hurts the propagation), and R0 depends on the restrictive policies that are at place. (Independently of that, the case fatality rate grows with the PM2.5 pollution, percentage of old people etc.)

To understand the dynamics of individual countries and regions, you simply cannot imagine that there is a one-size-fits-all, single, universal value of the basic reproduction number R0. That value is lower in Singapore, somewhat higher but still low in Czechia with face masks, it is much higher in Sweden, and it is even higher in New York.

Zoom in up to Czech districts. Full screen.

Imagine that R0 is the multiplicative number in the "number of active cases" that you get by waiting for 3 days. I think it is a reasonable time for this R0. 3 days is shorter than a typical "generation" of the virus. But I think that people already get infectious 3 days after they get infected themselves – some 2 days in average before they show the first symptoms (those appear 5 days after the infection: 5 days is the median) – and this stage is actually critical for the propagation of the virus simply because it's faster than the slower regimes (and because the asymptomatic people are less careful, and so are the people around them).

The rules of the game are simple. Locally, the number of active cases gets multiplied by R0 after 3 days. If R0 is smaller than 1, the disease exponentially fades away in the region. For R0 above 1, it increases.

An important "detail" – mentioned in the title – is that the regions with different values of R0 co-exist and interact with each other. People from regions with a higher R0 (and higher or lower current density of active cases) travel to regions with a lower R0 and vice versa. So most of the first 100 positive cases in Czechia were skiers who got infected in the Italian Dolomites. The "foreign infections" were a majority up to the 150th Czech case or so. Now, with 6,000 cases, only some 10% of the known positive cases were infected abroad, the rest already got it in Czechia.

Even if your region manages to have R0 below 1, so that the disease would be shrinking and it would go extinct within a month or two, you won't be permanently protected against the disease simply because some people – either your people going outside, or foreigners visiting you – will sometimes bring the disease. If the disease is currently very active (high number of active cases) in the source of your tourists, or in your people's favorite tourist destinations, you will get some new cases, anyway.

An important simple formula I must mention is\[

f_{\rm herd} = \zav{ 1 - \frac{1}{R_0} } \times 100\%

\] I have just upgraded Mathjax 2.7.7 to a much faster 3.0.5 in both templates. Enjoy the fast equations such as one above!

One minus the inverse basic reproduction number (multiplied by 100 percent) gives you the peak percentage of the immune people at which the herd immunity is reached. It's simple to see why the formula above is right. The virus normally spreads to R0 people in those three days. But if f times R0 of them are immune, it doesn't work. Only the not yet immune fraction \(1-f_{\rm herd}\) may be used by the virus to spread the disease. It means that the immunity of a big part of the population effectively reduces R0 by the factor of \(1-f_{\rm herd}\). And when \(f_{\rm herd}\) gets high enough, this difference is small enough and the reduced R0 becomes one – stagnation instead of the growth for the first time. When it happens, the condition says \[

R_0 \times \zav{ 1 - f_{\rm herd} } = 1

\] which is equivalent to the previous displayed equation. \(1-1/R_0\) is an excellent easy formula. If R0 is equal to 2, this fraction is 50%. If R0 is \(3\), this fraction is \(2/3\). When R0 is \(1.5\), the fraction needed for herd immunity is \(1/3\). And if R0 is \(1+\epsilon\) where \(\epsilon\) is small, like two percent, the fraction of the immune population needed for herd immunity is (a bit below) \(\epsilon\).

I believe that R0 of Czechia happened to be something like \(1.1\) with the face masks and other things which means that herd immunity is reached when 10% of Czechs are immune. And that's where we probably are. If we tried a little bit harder, we could get a straight decrease but I think that we didn't quite get it. We are only getting a decrease now after we reached the 10% herd immunity.

Immunologist Karel Drbal had a nice argument that I believe, but with different numbers. He said that we have gone through "five or so" periods of a major infection, the testing with the 5% (average) positive rate should be interpreted as having 5% of the Czech nation infected at each moment, and because we've had 5 such groups (weeks of March plus April, roughly speaking; one week is his period in which one patient remains tested as positive, you may dispute this figure, too), he got 5 times 5% equals 25% of immune Czechs. I think it is a clever simple estimate. I just think that the number of relevant weeks is a bit smaller than 5 and the average relevant infection rate in the nation is also lower than 5%: the percentage of positive tests is 3% these days and even weeks ago when it was 6% in average, it was only elevated because the tests focused on groups of people who were more likely to be infected than the general population. These two reductions lead me to reduce his 25% estimate to something close to 10%, too.

If and when we relax our Czech restrictions, R0 may grow from 1.1 to 1.5, and the required percentage for herd immunity will go up from 1/10 to 1/3. The number of deaths would triple, too. It will really happen at some moment, assuming that we get tired of the masks and closed restaurants, and there's no real advantage of keeping the restrictions, unless we really want to keep them up to the availability of the vaccine or a great treatment (which may however come in days, favipiravir).

Meanwhile, the white dense places without the face mask culture, like Stockholm or New York, have R0 at 2 or higher. The exponential increase is fast and guarantees the saturation within two months after the first infections. It's futile to fight this very fact which really follows from the laws of Nature.

Like in the climate change debates, people shouldn't "mitigate" something that is basically unavoidable, assuming at least the basic preservation of the people's freedom and lifestyles. Instead, we should cleverly adapt to the unavoidable natural processes. Herd immunity is reached in two months or so, whether someone likes it or not. But it may be achieved with different people. Sweden is simply selectively offering "young and healthy" people to the virus to create the herd immunity fraction, while the old ones are protected. That's what I recommended a month ago and it works. It is really the only "epidemiological" policy that works to reduce the fatality count. Some 50% of Stockholm and some other large cities have been exposed to the virus so far and the nation has reached the herd immunity without too huge losses of lives. Some 1,000 have died in Sweden, that is 0.01% of the population.

It's well below the 0.1% death rate of flu (including complications) but it's really because of the 1) selective protection of the vulnerable ones during the process, 2) the fact that there are still regions in Sweden with R0 below 1 where the disease internally fades away, and can only be imported if you want many deaths. Sweden is beyond the peak, or will be very soon, and the total deaths won't be significantly above 0.01% of the population. That's what you get in Western nations assuming a clever "let it be" policy of the Swedish type. If you extrapolated to the whole world, 0.01% of mankind is some 800,000 people, comparable to one flu season.

White countries and states with the lockdown have made things worse in every conceivable way. They reach the herd immunity, anyway. But the lockdown means that people of all age groups (and health condition groups) are being exploited "equally" as new building blocks to create the herd immune fraction – which comes in those 2 months or so, in one way or another. So the lockdown of the young and healthy people really means that they are not protecting the old ones at all! That comes on top of the insane, unprecedented, suicidal multi-trillion-dollar economic losses, the destruction of individual freedoms, and of the psychological well-being of tens of millions.

Regions, cities, states, and countries with different values of R0 are sending people to each other. But the capacity of the disease is highly restricted in time, due to the aforementioned impressive power of the exponential growth (with a doubling in 3 days). So you are typically afraid of tourists who come from the regions with a high R0 and a high current number of active cases. However, because we determined that the highest R0 regions reach the herd immunity in less than 2 months, then the disease starts to drop.

That's why there is no point in protecting your countries against China or North Korea these days. I am still uncertain about the question whether China has actually defeated the virus internally, and/or what is the percentage of the Chinese who are currently immune to the virus. It is possible that their R0 was sent below 1 and they made the disease fade away which could mean that the percentage of the Chinese who are currently immune is tiny. On the other hand, it's also possible that their R0 is higher than one and they have simply reached the herd immunity much like others and realized that the disease isn't really a serious problem that needs to be wrestled with. (China's leaders probably want to damage others, so they want others to believe that it is a serious problem that should make you panic and do irrational, self-harming things.)

So the early "hottest spots" with the disease are usually those where the R0 was highest, i.e. the exponential growth was the fastest, but those are also the regions that "run out of the non-immune people" early. In effect, it makes no sense to focus on them as threats. When you evaluate which countries may export their virus to your "nearly virus clean" country, you should simply look at the current number of active cases as the percentage of population. China seems safe, whatever is the exact reason. But within a week or two, even Lombardy will be safe. Just like Madrid. New York. Many others.

To predict the spreading of the disease, you would need to assign realistic values of R0 to various cities; and you would need to add the information about the rate of travelling between all relevant pairs of regions. Some regions with a low R0, especially those with R0 below 1 where the disease would fade away without viral imports, may see the disease as an issue for many months, but they will also see it with a lower intensity.

I have mentioned that Czechia may have R0 about 1.1 – our R0 is closer to 1 than the R0 of any other nation because no one has managed to get this nearly perfect real world model of "stagnation" as we got in recent two weeks. As particle physicists know from the considerations of naturalness vs fine-tuning, it is extremely unlikely that R0 is extremely close to 1 (the "number", in the sense of measure theory, of numbers that are very close to 1 is much smaller than the number of positive numbers that are sufficiently far from one). It means that the regions and communities where the disease grows but it takes a very long time should be rare.

SARS-1 was arguably eliminated from the world. Covid-19 could have been close enough to SARS-1 and it was possible to imagine in February that Covid-19 would go extinct, like SARS-1 – I surely considered that a possibility two months ago. But it didn't happen. Sometime during March, there is no precise date, sensible people saw that Covid-19 really spreads faster, just like a common cold (well, 18% of common cold cases are caused by a milder type of a coronavirus), and the plan to globally eliminate it within months is utterly unrealistic. It's futile to try to totally stop diseases that spread like a cold.

OK, I am pretty sure that the number of cases and even fatalities will be dropping already in this week and it will become even clearer in the next week, and so on. I hope that people regain their common sense and most countries will loosen or cancel their insane lockdowns. The virus will be with us, the probability that the number of actively infected humans will drop below 1 before a vaccine seems close to zero to me. The disease may be eliminated as "domestic dynamics" on the Northern Hemisphere during the summer but some cold places in Southern Chile or Argentina will protect the survival of the virus during our Northern winter season.

The immunity will almost certainly last for a year or more; an experiment was done with four monkeys and they couldn't be re-infected after they recovered from Covid-19. So when it gets cold on the Northern Hemisphere sometime in October again, we will see another wave in which the propagation of the virus may accelerate. But the previous hot spots will already be largely immune and therefore not good enough for the promotion of the disease. So instead of cities like New York, Stockholm, and Prague, we may get a wave of the disease in Pilsen and similar towns in late 2020. That wave will be less dramatic because we have a lower natural R0 than Prague under the same policies.

There are lots of clever ideas and facts that lots of people deny – most of people are both scientifically illiterate and brainwashed by the fake news media. But the primary point – that was turned into a near heresy by so many incompetent or lying bullies – is that there is nothing qualitatively different about this disease in comparison with flu or the common cold. It was always insane to establish policies that resemble the "regime change" to fight something that may be biologically different from flu but that has extremely similar implications for the humans.

It's extremely important to force the people who were hysterical and completely wrong about these matters to admit that they were completely wrong. The mistake that they have done was so incredibly costly (and it has increased the number of fatalities by tens of thousands) that they shouldn't influence the evolution of the nations ever again.

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