Saturday, August 21, 2021

Vaccine efficacy at preventing Covid hospitalizations can't differ much from the efficacy at preventing positive Covid tests

...which is why the Pfizer jab is basically useless in the age of delta (and it's been lethal for tens of thousands)...



Many people (including epidemiologists and other doctors whom I respect and Dr Beran and Dr Pollert happen to belong to this set) don't get this simple yet math-loaded point and prefer to believe all sorts of voodoo science, especially the proposition that "while the Covid vaccine only has a ~40% efficacy against the positive PCR tests, it heavily [almost always, ~90%] prevents a serious disease, hospitalization, and death". It isn't true and it's straightforward to see that this unusual combination of softly but (numerically) heavily conflicting propositions can't be true.

Vaccine efficacy is defined as \[ VE = (1-RR) \times 100\% = \frac{ARU-ARV}{ARU} \times 100\%. \] Up to the "plain English" factor of 100 percent, vaccine efficacy is either equal to one minus the relative risk (RR) of developing the disease among the vaccinated over unvaccinated ones; or the relative reduction of the rate of the disease (note that \(RR=ARV/ARU\)). It's simple.



Delta (the politically correct name for the "Indian variant") is totally dominating in Israel which will be the country used for my simple empirical calculations and comments. OK, four weeks ago, we were told that Pfizer was only 39% effective against delta in Israel. That figure had gone down from 64% reported two weeks earlier, conflicted with pro-vaccine results from the U.K., but the Israeli data still argued that the vaccine was 88% effective at preventing hospitalization; and 91% at preventing serious hospitalization.



It just couldn't be true. When a vaccine is effective for you, it means that you don't even develop symptoms – and you probably don't develop any viral dose that is comparable to the doses that cause symptoms. I have discussed the positive form of this statement in January,
When they are effective, vaccines prevent transmission.
You may formulate this sentence (an implication) in an equivalent way: you negate both propositions and flip the arrow in between them:
When vaccines don't prevent transmission, they are not effective.
It is really common sense because the very purpose of every vaccine is to prevent a disease. If some solution doesn't prevent a disease but it is promised to make the disease better, it is a miraculous elixir (a drug to alleviate the disease) sold according to the rules of pseudoscience, not a proper vaccine working according to the general principles of all vaccines, as I will argue in detail.

First, let us complete the Israeli empirical argument. My assertion is that it is clearly nonsense that the vaccine prevents positive tests at the 39% or 42% efficacy (the figure 42 is from the Mayo-Nference study/preprint, see e.g. Fox News for a popular review); but it prevents the hospitalization or a serious disease at roughly 90%. Such a huge difference between these two numbers is impossible (wait for better proofs) and indeed, you may empirically see that this statement about the huge gap of efficacies was a ludicrous fairy-tale spread by Pfizer and its fanatical pro-vaccination advocates (which contains many people who should know better; but they become mindless defenders of the indefensible).

Let us just look at the updated Israeli figure. Search for Israel percentage hospitalized vaccinated, it's simple. You will quickly get to a Science Magazine article that is five days old and that says that Pfizer blunts but doesn't defeat delta. From that article, I will use the assertion that 78% of the Israeli above 12 years of age were fully vaccinated, almost all of them by Pfizer.

The same article said that among 514 currently hospitalized Covid patients in Israel, 59% were fully vaccinated. OK, if the vaccination "almost always" (90%) worked against hospitalization, the unvaccinated would dominate the hospitalizations, right? Instead, the vaccinated ones still have a majority there. OK, let's do the simple math. In the general population (Israeli above 12 years), the vaccinated and not-fully vaccinated are 78-vs-22 percent, OK? Among the hospitalized patients, the ratio is 59-vs-41, OK? So the rates of hospitalization (with some other factors that are common for the vaccinated and unvaccinated Jews), the rates of the disease are 59/78 and 41/22 for the vaccinated and unvaccinated people, respectively. Numerically, these numbers are about 0.756 and 1.864. The aforementioned ratio of risks RR is 0.756/1.864 = 0.41 and the implied vaccine efficacy is 0.59 or 59 percent, not close to 90%.

In reality, I feel confident that even this 59% is a significant overestimate and the true number really is closer to those 39% or 42%. Why? It's because some hospitalizations are still pre-delta (mainly alpha for which Pfizer worked at ~90 percent). The new cases are totally dominated by delta but because alpha is a more serious disease, a disease that is more likely to lead to hospitalizations, the percentage of the alpha patients in the Israeli hospitals is not quite negligible yet. Consequently, our final result 59% for the efficacy is not really a vaccine efficacy against delta but a less well-defined vaccine effectiveness against a mixture of alpha and delta. As the percentage of alpha decreases among the positive tests and then among the patients, the calculated Pfizer/delta efficacy is getting and will be getting closer to those 40%, whatever method of quantification you use.

Note that the Wikipedia article on vaccine efficacy sensibly recommends you to use the word "efficacy" for the relative reduction when measured on a well-controlled set of people that are representative of the overall population; vaccine effectiveness is some real-world fraction that you obtain from a less well-controlled ensemble of people. But again, the main problem is that the variant of the virus of the hospitalized patients isn't pure delta so the Israeli result isn't producing the efficacy on the pure delta variant.

So the most straightforward calculation from the Israeli hospitals indicates the Pfizer vaccine efficacy against hospitalization to be 59% but that number itself has been decreasing from over-80% figures as well and I tell you that when alpha disappears completely, this efficacy will get very close to those 39-42 percent.

Much more theoretical arguments

OK, let me get to the more theoretical underpinnings. In general, a vaccine that works prevents the disease completely or almost completely; it prevents the growth of the pathogen to doses that are comparable to the doses when symptoms emerge. It is not infinitely hard to create vaccines against various diseases that work in this sense; and indeed, the standard vaccines that kids have been getting do work in this way. Flu vaccines are a different matter; the efficacy in the U.S. is around 35%, an insignificant reduction of the risk of the flu (mostly because the right strains are often absent; the efficacy would be much higher against a specific strain). But even in that case, we just don't distinguish "being sick with a vaccine" and "being badly sick", both are considered a failure of the vaccine because the decision whether the vaccine works is decided much earlier than at the moment when the disease may get really serious. A working vaccine just shouldn't allow you to get this far in the progression of the disease.

To start, it is useful to have some ideas about the number of individual viral particles (virions). Take this cute June 2021 PNAS article and you will learn that a fully infected person near the peak has 1-100 billion SARS-CoV-2 virions. I find it silly to use these different acronyms for things that are known to be equivalent (even in the case of HIV and AIDS); I will call the virus and virions Covid-19, too.

These 1 billion to 100 billion virions in an infected human body weigh at most 0.1 milligrams, we learn. Multiply by some ~100 million current infected hosts and you will get at most 10 kilograms. The PNAS paper returns between 0.1 and 10 kilograms for the total mass of all Covid-19 virions inside humans right now! Imagine, roughly 1 kilogram of stuff has (mostly very indirectly) forced us to sacrifice over $10 trillion. Gold looks like a precious metal but one kilogram only costs $57,000 or so. The Covid-19 virion stuff has been about minus 175 million times more precious than gold! ;-)

Fine. The infection starts with inhaling a small number of virions, let us say it is 1,000 of them. Within 7-8 days, the patient reaches the peak which is 10 billion, as we saw, plus minus one order of magnitude. That is an increase by roughly 7 orders of magnitude in 7 days. By fitting an exponential function, you see that the number of virions gets multiplied by a factor of 10 every day (during the initial growing phase of the disease). If you met a person with Covid-19, you may think about the number of virions that you own. 1,000 on Day 1, 10,000 on Day 2, 100,000 on Day 3, and so on. I hope you will enjoy the counting.

Now, you may want to quickread this page about the PCR tests. These devices run X cycles and each of them doubles the number of copies of the DNA from the sample. So it is the nice geometric series that you get from a doubling, 1,2,4,8,16 and so on. The Covid-19 positivity has been seen after as few as 14 cycles but sometimes also needed as many as 45 cycles; most normal countries use 30-35 cycles to identify a Covid-positive person (not necessarily a patient because the Covid positiveness is not necessarily a disease).

Fine. We want to understand the role of a vaccine in taming the exponential growth, from 1,000 virions to 10 billion virions, and whether it is important to distinguish the taming needed to avoid a positive test; and the taming needed to avoid hospitalization.

An important quantity we haven't discussed is how many days after the infection does it take for you to test positive. Take e.g. this page and you will learn that positive tests (with a choice of the number of cycles that doesn't produce too many false positives) only start to emerge 5-7 days after the infection. It's really roughly the same duration as the duration needed for symptoms to emerge (5 days) or even a bit longer than that. You may rather easily have a person with symptoms who still tests negative! In fact, most people will test negative at the very beginning of their symptoms. While the PCR technology may be said to be high-tech, it just isn't capable of beating (at least not "safe beating" of) the simplest method to guess whether someone is sick – by seeing the symptoms. So people often do become contagious (for 2 days) before they obtain a positive test; even if all humans were tested thrice a day, the PCR tests combined with the isolation of the positive ones just couldn't prevent all infections (and with the delta's high \(R_0\), you would still get an exponential growth). You may increase the number of cycles and get the positive results earlier but then you heavily increase the number of false positives; the false negatives won't quite diseappear, either. Increasing the number of PCR cycles to insane levels isn't a wise method to deal with the problems. You must just live with the fact that we don't have reliable methods that determine who can spread a disease and who can't. Results of PCR tests are at most "proxies" to something that matters; clearly, viewing these bits of information as "religiously important messages from Heaven and Hell" (that may be used to decide about the quality of life of the people for a very long time) is one of the most important traits that define unhinged Covid fanatics. Many Western nations have been totally overrun by these unhinged fanatics, the worshiping of the PCR tests is unquestionably one of the most characteristic symptoms of the disease of the Covidah Witnesses.

Great. I have said that during the initial week after the infection, the number of virions grows from 1,000 to 10 billion, by a factor of ten million. There are other things in the blood whose concentration is important but I won't give you specific integers. The most elementary unit on the "defense" side is an antibody; a rank-and-file soldier. A molecule that fights against a particular antigen (as a key-and-lock pair), the latter is a molecule on the surface of a pathogen (the Covid-19 virus); see my Intro to Immunology that has over 13,000 views now, mostly from revolver.news. When your concentration of antibodies is sufficiently high, you are immediately immune against the corresponding disease. That's a situation that is mathematically equivalent to "herd immunity" at the level of the population. The Covid-19 virions want to grow by a factor of 10 per day but the antibodies in the blood destroy a certain fraction of the virions during the day, too. Once the concentration of the antibodies is high enough, the "\(R_0\)" for the virions in one body drops below one and the disease starts to exponentially drop instead of growing. The difference from the normal \(R_0\) of the populations is that at the level of the populations, the virus ceases to spread because a sufficiently high fraction of the people is already immune; at the level of an individual body, the virion number stops growing because the antibodies destroy a sufficient fraction of the virions during a chosen period of time (so that the killing of virions by antigens beats the exponential reproduction of the virions).

A high enough concentration of antibodies (a single-human counterpart of herd immunity) is the "final, happy result" of the single patient's fight against the disease that he or she (with different odds of death!) encountered for the first time. In other words, it is the final outcome of the primary immunity response. How does it happen that the high level of antibodies is achieved? Well, the antibodies are produced by someone. The producer is always an activated B cell which is obtained from a regular, neutral B cell (which biologists call a naive B cell, and we will use this adjective for the virginity, too) by activation, some exposure to a particular pathogen (the Covid-19 virion in this case). The naive B cells (some kind of lymphocytes or white blood cells) are produced in bone marrow, that is where the letter B comes from.

Now, it is obvious that most of the Pfizer-vaccinated people don't have a high enough concentration of antibodies because that would be enough to make the number of virions exponentially drop from the very moment of infection. If the PCR-test-based efficacy is just 42%, it means that at least in 58% of the people, the concentration of antibodies is below the threshold that is needed to prevent the exponential growth of the virions. But there is again an analogy with the national policies. Some nations (like Australia) think that it's a great idea to slow down the exponential growth even if you can't bend it to a decrease.

I have explained many times that it is utterly worthless to slow down the exponential growth of the number of infected people, especially when the number is still (or again, already) low enough for hospitals to be problem-free. Why is it worthless? Because the speed doesn't mean anything for the question how many people will die; and at the single-human level, the speed of the exponential growth doesn't mean anything for his or her fate. Why? Because what decides about "many deaths" or the "chances of one man to die" isn't some random rate of a growth; the peak number of virions is much more important. And you don't change anything about the qualitative fact of a continuing growth by slowing it! The only valuable moment is when you start to be able to bend the growth and flip it to a decrease (also an approximately exponential decrease, for some periods of time) because that is how you prevent very high (and lethal) numbers of virions. If the peak number of virions were kept the same, a slower disease would arguably be more dangerous because the body has to struggle with the near-maximal number of virions for a longer time! The peak load is what matters and the "force" that makes sure that this is the maximum is the force that should be credited with the rosy future; it is the antibodies (and their high enough concentration) at the individual patient's level and it is the herd immunity at the population level. Everything else is really useless for the arrival of the rosy future!

The existing antibodies can't do anything else beyond destroying some virions, and thus reducing the rate at which the virions exponentially increase, effectively their single-human \(R_0\). So that closes the discussion at the level of antibodies. Most Pfizer-vaccinated people (at least 58%) just don't have a sufficient level of antibodies that is needed to bend the increase of the number of virions and turn it into a decrease. That is also why Pfizer and its lackeys (and the ideological servants of the Covid fascism in any form) refuse to measure the antibodies in the vaccinated people. They just know that they are not high enough (at least in most cases), relatively to the levels seen in true survivors of the disease. Again, we have clearly seen that most of the Pfizer-vaccinated people don't have a high enough concentration of antibodies after the vaccination. Because there is no pathogen left 2 weeks after the vaccination, the amount of antibodies no longer grows. It is actually fading away and the antibody-based part of the immunity gets weaker.

Wonderful. But the direct suppression of the growth of virions by antibodies in the blood isn't the only possible immunity response, is it? The bodies may use memory cells. So maybe the patients who get a positive PCR test after a vaccine don't have enough antibodies but they have some memory B cells or memory T cells that speed up the production of antibodies and the immunity response. The memory cells remember a fingerprint of the pathogen (of Covid-19 in this case); these memory cells were created by a process while the (non-memory) B cells were producing the antibodies. When these memory cells have been previously created by the first encounter with the disease, this scenario is called the secondary immunity response. So maybe the Pfizer-vaccinated people aren't immediately immune (by having a high enough level of antibodies, like in the old classical vaccines with a weakened disease) but they have enough memory cells of the right type that kickstart a much faster reaction ultimately ending with the production of many antibodies.

If the vaccines work like the secondary immunity response, it means that the number of virions may grow for a little while after the vaccinated person is infected; but this increase will be bent and replaced with a decrease a bit earlier than in an unvaccinated person, and that advantage of the vaccinated person in the timing is the reason why the vaccinated person will avoid hospitalization (or a serious disease) despite his positive test.

It sounds like a great, optimistic, pro-vaccine story for gullible sheep to parrot. But if you understand the meaning of the words and their combinations, ideally at the quantitative level, you may see that the story just doesn't add up – or, more precisely, a scenario like this is extremely unlikely and can't apply to a great majority of the 58% of the vaccinated people who will still get a positive PCR test. Why?

An important reason is that the positive PCR tests and "the apparent need for some hospital care" occur at almost exactly the same time after the infection. 5-7 days was the period at which the PCR tests start to be meanintfully positive for a person who is at an elevated risk; but it is also the period at which it may be meaningfully decided that the person needs to be hospitalized. The timing is always the same. Because the PCR tests only start to be meaningfully positive those 5-7 days after the infection and those 58% of the Pfizer-vaccinated people exposed to Covid-19 still get a positive test, it means that there were no antibodies that would prevent the growth that is seen in the first 5-7 days after the infection. It means that if the immunity reaction caused by the vaccine came at all, it came too late! When it matters near the peak of the disease, a big majority of the antibodies are being produced by the natural processes that aren't affected by the vaccination status.

Just the fact that the timing is about the same for the PCR tests and the need for hospitalization, about 7 days, makes it clear that the extra "additions" to the immunity system that are caused by the Pfizer vaccine in the 58% of the people just don't come in time. If you knew something about the secondary immunity response caused by the memory cells, you know that it should kick in in 2-7 days. For a secondary immunity response to work, the speed simply has to be significantly shorter than 7 days, the period when the number of virions grows to the hospitalization or serious hospitalizations level, anyway!

As far as I can say, the very scenario that "the vaccine gives you immunity but without many antibodies, just with memory cells" is fishy in general. In the normal interactions with diseases, the memory cells are only created by the interaction with the antigens and the same processes that created the memory cells have also created a sufficient number of antibodies that exist at least for a little while. So "the immunity with memory cells and totally insufficient antibodies, even right after the vaccination" sounds like an extraordinary assertion to me. But my argument doesn't really depend on the assumption that the antibodies always accompany the creation of memory cells.

What my main theoretical argument does rely upon is the fact that the number of virions in the 58% of the Pfizer-vaccinated people (who showed a positive test) had to be increasing for 5-7 days or, more precisely, it has to grow by 5-7 orders of magnitude from the initial dose (we know that because the test is only positive after this period of growth and they got the positive test, proving this much of the growth) and that is simply slow (and too late) to be too helpful. But we know that for those 42% of the patients, the vaccine did prevent the positive PCR test (a reduction of the rate). What can we say about those 42%? Well, those were the people in which the vaccine did indeed create some community of memory B cells, memory T cells, and antibodies that was enough to kickstart a secondary immunity reaction significantly earlier, before those 5-7 days, perhaps after 2-3 days.

When you look at the spectrum of the Pfizer-vaccinated people, you may plot (imagine!) the distribution of the time that the "enhanced" immunity system needed to develop everything that is needed to stop the exponential growth of the number of virions. And for 42% of the vaccinated folks, that period was simply below 5-7 days. For the remaining 58%, it was longer than 5-7 days, whatever it means. The width of the interval 5-7 days in the previous sentences indicates some uncertainty in all my assertions and indeed, the numerical values of the efficacies at preventing positive PCR tests; and at preventing hospitalization may live within this uncertainty.

But at least 58% of the vaccinated people surely allowed the exponential growth of the number of virions for at least 5 days (lower bound on the duration after which the PCR tests get positive) and this is already long. Symptoms emerge 5 days after the infection. The peak of the viral loads appears 2-3 days after the onset of symptoms.

The point is that the "number of days when the vaccine-inspired immunity response kicks in and bans the exponential growth" has a high spread, sometimes it is shorter than 5 days, sometimes it is longer, and when the lag is over 8 days, the response is useless because the antibodies and other cells are already being produced massively as a response due to the high number of virions (the person is very ill and the immunity system fights) and the natural response to the pathogen is what should get the credit for getting OK. The hypothesis that a great majority of the 58% will get a positive test but the vaccine-caused response will still come soon enough to contribute a significant fraction of the antibodies near the peak (ideally a clear majority) amounts to a very implausible fine-tuning. If a strong enough response doesn't arrive in 5 or 6 days, it will not arrive even in 9 days in a significant portion (and almost certainly a majority) of such people (vaccinated people with a positive PCR test) simply because the distribution of the lags has no good reason to abruptly fall off near 5-7 days. This homework has its analogy at the national level, too. Some countries that managed to stop the growth of Covid in Spring 2020 did it within a few weeks, others needed a few months, others didn't stop the growth at all. When there is a wiggle room, there will be fat tails and the "time needed for stopping the growth before the herd immunity" may be very long or "greater than infinity", too.

Replacing the two efficacies with a more theoretically well-defined

I am saying the same thing which I consider almost trivial in many slightly different ways. Let me add another one. You know, the problems with the "efficacy at preventing PCR tests" and "efficacy at preventing hospitalizations" are two different but vague conditions that must be refined to be quantitative. In particular, the number of cycles of the PCR tests (and other details) has to be specified; similarly, the criteria for "how serious condition is needed for hospitalization" must be outlined, too. There is too much space for subjective choices and they're subjective choices in 2 different things. So in effect, it seems like we are comparing apples and oranges and many people end up believing that the two efficacies (against positive PCR tests; and against hospitalization) may be very different. They don't seem to care that the same vagueness that allows them to believe that "they may be very different" also guarantees that these numbers are "too vaguely defined to be useful".

But we may cure the problem with the vagueness by becoming quantitative scientists instead of biased marketers of vaccines who always produce BS going in the desired direction and we may ask:
What is the efficacy of the vaccine at preventing the further growth of the number of virions after at most \(X\) days?
Such an efficacy is defined as a function \(f(X)\) of the variable \(X\). Great. In principle, we may talk about the number of virions (although they are hard to find) and this efficacy is therefore more well-defined and convention-independent from a theorist's viewpoint. Now, the funny thing is that this "Motl efficacy" (which is all about the moment when the exponential growth stops; or perhaps about the number of orders of magnitude that are added to the initial viral load) is a good enough refinement of both efficacies above and both of them may ideally use a value of \(X\) which is something like 5-7 days. For this reason, the difference between the "test-based efficacy" and "hospitalization-based efficacy" can be at most as large as the difference of \(f(5)\) and \(f(7)\) and because this function \(f(X)\) is rather smooth and the values five and seven are in no way "ultimate limits where we could expect the steep drop of the distributions or a corresponding steep slope of \(f(X)\)", it follows that the values of \(f(5)\) and \(f(7)\) differ at most by dozens of percent, and it is therefore impossible for one of them to be 42% and for the other to be 90%. I am confident that I have been extremely generous in this paragraph because \(f(7)\) is the best refinement of both the "PCR test efficacy" and the "hospitalization efficacy" which is why they should be considered as being tautologically equal to each other!

And that's the memo: for a given patient and a specified variant of a disease, a vaccine either works or it doesn't work, and because "a vaccine that works" changes the dynamics really qualitatively (and the peak loads differ by many orders of magnitude in the "works" and "doesn't work" situations), the detailed information about what we mean by "it works" are not affecting the numerical values much. It's really just like a NASA question whether a rocket works and can deliver capsules to the outer space. It doesn't really matter whether the target is near the Moon or Mars. What matters is whether the rocket beats the gravitational field of Earth; and whether the vaccine beats the virions before they grow big.

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