![\"\"](\"https:\/\/wp-media.patheos.com\/blogs\/sites\/533\/2020\/10\/covid-from-cdc-2.jpg\")
So I\u2019ve been doing some COVID math \u2014 the sort that isn\u2019t really good enough to put myself forth as an expert but which I want to write down, in a bloggy fashion.<\/p>\n
It starts with this:\u00a0 two weeks ago, in a CNN interview, CDC director Rochelle Walensky said<\/a> that for \u201cevery 20 vaccinated people, one or two of them could get a breakthrough infection.\u201d\u00a0 \u00a0A White House staffer said, \u201cIf 10 vaccinated people walk into a room full of COVID, about 9 of them would walk out of the room WITH NO COVID. Nine of them.\u201d\u00a0 And the New York Times<\/em> tweeted, based on an internal CDC report, that \u201cThe Delta variant is as contagious as chickenpox and may be spread by vaccinated people as easily as the unvaccinated, an internal C.D.C. report said.\u201d<\/p>\n Shortly afterwards, I attended a school board meeting and talked to parents beforehand \u2014 specifically, skeptical parents, Polish-immigrant parents conditioned, I suspect, to distrust the political authorities \u2014 and they expressed the belief that vaccine mandates were to be distrusted because the CDC itself said that the vaccine doesn\u2019t work<\/em>.\u00a0 I tried to redirect their frustration \u2014 yes, the CDC\u2019s spokespeople do a bad job and stoke fear, the entire point is that the vaccines work so we shouldn\u2019t be locking down again, and, besides, the school board should be taking responsibility for its decisions rather than just claiming that the superintendent can do whatever he wants<\/em> \u2014\u00a0 but it was clear to me that they were not using the CDC messaging as an excuse for being anti-vaccine, but that they were sincere that they believed the CDC said, \u201cyou\u2019re just as much at risk if you\u2019ve been vaccinated as if not.\u201d<\/p>\n Let\u2019s start with the basic math:\u00a0 the Pfizer vaccine is 95% effective.<\/p>\n That does not mean that you have a 1 in 20 chance of getting infected if you have been Pfizer-vaxxed, let alone a 1 in 10 chance of getting infected after a single instance of exposure.<\/p>\n In simplest terms, it means that your risk of COVID is reduced by 95% from what it otherwise would be<\/strong>.<\/p>\n But it\u2019s not actually that simple.<\/p>\n Before I get into my math (both the reasonable parts and the whoa! that\u2019s a stretch parts), let me add that I\u2019m asking myself what the eventual outcome of the pandemic, the variants, the vaccination rates, and so on.<\/p>\n Will we eventually boost our vaccination rates high enough that we can beat back the breakthrough infections once and for all, and our mistake was simply believing that a 70% vaccination prevalence was \u201cgood enough\u201d?<\/p>\n Will we have to resign ourselves to everyone, vaccinated or not, getting covid, and it\u2019s just a matter of time \u2014 but that getting vaccinated nonetheless is vitally-important for almost everyone, except, perhaps, for those at low enough risk of serious consequences from COVID that they\u2019d prefer to take their chances?<\/p>\n And on the one hand, the CDC\u2019s reaction to the Provincetown outbreak \u2014 in the midst of what is apparently called \u201cBear Week,\u201d when large numbers of people engage in all sorts of risky activities \u2014 was clearly overblown, since any such statistical distribution will inevitably have outliers, but, on the other hand, well, the memory care community that my mother lives at, is in the midst of a similar outbreak \u2014 and to my knowledge, there is no available data on the likelihood of breakthrough infections among people of different ages or health statuses.<\/p>\n And on the third hand \u2014 well, there\u2019s Iceland.\u00a0 They have \u201cone of the highest [vaccination rates] in the world<\/a>,\u201d but they are now seeing soaring case counts attributed to the Delta mutation<\/a>.\u00a0 As of just last week, on August 6, the country had a rolling-average case rate (32.1 per 100,000) pretty much equal to that of the US (though theirs has dropped with new mitigations imposed, and the US rate is still climbing), after having previously had such incredibly small rates that they were considered one of the prime examples of successes, with, indeed, virtually no deaths at all.<\/p>\n Now, to be sure, virtually none of the infected vaccinated Icelanders have been sickened to any serious degree.\u00a0 But it\u2019s enough to raise concerns.<\/p>\n Now, the math:<\/p>\n The Pfizer study<\/a> consisted of 46,320 participants.\u00a0 During the course of the trial, there were 162 infections among those who got the placebo, and 8 among those who got the real vaccine.\u00a0 That translates to a rate of 175 (OK, 174.9)\u00a0 per 100,000 for the placebo group and 8.6 per 100,000 for those with the vaccine.\u00a0 Divide 8.6 by 174.9 and subtract 1 and you get a 95% reduction in infection for those with the vaccine vs. the placebo group.<\/p>\n That\u2019s the math.<\/p>\n Is your personal risk reduced by 95% if you\u2019ve been vaccinated?\u00a0 I\u2019m not really sure.\u00a0 The press release says that \u201cefficacy was consistent across age, gender, race and ethnicity demographics,\u201d and that efficacy was 94% for those over 65 years.\u00a0 Was it consistent across health status, barring the rare issues such as being immunocompromised, or were certain groups more likely to be infected?\u00a0 With only 8 infections among the vaccinated, how is it even possible to say that \u201cefficacy was consistent\u201d?<\/p>\n Similarly, for the Moderna trial<\/a>, there was 28,207 participants; 185 of the placebo recipients and 11 of the vaccine recipients became infected during a follow-up period of 9 weeks after the second dose.\u00a0 (I couldn\u2019t find the follow-up period duration for Pfizer).\u00a0 This translates to rates of 328 per 100,000, unvaxxed, for the placebo group, and 19.5 per 100,000, for the vaxxed group, for the 7 weeks from full effectiveness.\u00a0 Again, the math says that the risk of infection was reduced by 94%.\u00a0 But, again, how do you interpret this, especially over the long term?<\/p>\n Here\u2019s one question which I haven\u2019t yet seen addressed:\u00a0 who are the 5% (or 6%)?\u00a0 Are there particular categories of people who are at greater risk of a breakthrough infection (not counting, again, the immunocompromised, who the Moderna materials specifically state were excluded from the trial)?\u00a0 After all, in the case of birth control pills, we know that women who are obese experience lower effectiveness, as well as women who are taking antibiotics.\u00a0 Are there similar instances for vaccines?<\/p>\n (According to a Washington Post<\/em> article<\/a>, immuncompromised-ness is rare, affecting people with \u201cspecific blood cancers or organ transplants\u201d and prevalent, affecting 3 \u2013 4 million people, or 1%-ish of the population.\u00a0 I can\u2019t imagine that the specific causes of immunocompromise listed in the article truly affect this many people, and that there\u2019s both a narrow and a broad definition of the term.\u00a0 But, again, the Moderna trial specifically states that they were excluded, and I presume that was the case for Pfizer as well.\u00a0 Oh, and that same article said that people over 65 generally get a higher-dose flu shot because they produce fewer antibodies, but the Pfizer info specifically says that oldsters were well-protected.)<\/p>\n In any case, it would be reasonable, if there were certain types of people who would be less protected, for this information to be made available to people.\u00a0 Is this another \u201cnoble lie,\u201d and the CDC has decided that \u201cfor their own good,\u201d people can\u2019t be told this?<\/p>\n Or is it simply the case that, by and large, the breakthrough infections are caused by \u201csuperspreaders\u201d \u2014 a level of infectiousness so substantial that it pushes past the vaccine, and that the reason why the vaccine is so effective nonetheless is that this level of infectiousness is fairly rare?\u00a0 This seems to be the case from the available reporting, though it\u2019s poorly reported on, for example in a New York Times<\/em> article that promises to explain \u201cWhy Vaccinated People Are Getting \u2018Breakthrough\u2019 Infections<\/a>,\u201d but, honestly, doesn\u2019t really do so.<\/p>\n In any case, having said that, what is striking about the two trials is, honestly, how few infections there were in the placebo group.<\/p>\n Again, in the Pfizer group, unless my math is wrong, there were 175 infections per 100,000 people, for the entire duration of the study.<\/p>\n In the Moderna group, there were 328 infections per 100,000 people.<\/p>\n Why these rates would be different for the two groups is not entirely clear to me.\u00a0 The Moderna materials, again, state that they measured infections for a period of 7 weeks following the \u201cfully vaccinated\u201d date.\u00a0 I don\u2019t see anything in the Pfizer materials that specifies this; I have the impression, instead, that the study ended when they deemed the data to be sufficient.\u00a0 (Maybe that was also true for the Moderna trial \u2013 that is, that the 7 weeks was not predetermined?)\u00a0 So perhaps it\u2019s as simple as the Moderna study, which had only 3\/5ths as many participants, needed to run longer to collect the necessary data.\u00a0 On the other hand, it may also be possible that the parts of the country or the other countries that Moderna recruited study participants in, had a spike in cases that didn\u2019t occur in the Pfizer regions.<\/p>\n In any case, again, consider that at the moment, Florida\u2019s daily per 100,000 case count stands at 89.6.\u00a0 That\u2019s according to the Financial Times<\/em> covid tracker<\/a>.\u00a0 What\u2019s more, in Florida, half the population (49.4%) is fully vaccinated.\u00a0 On the basis that a vaccinated person has a 95% reduced risk of infection, I did a little bit of math (again, my math could be wrong) to calculate that, adjusted for vaccine protection, Florida now has a baseline infection rate of 169 per 100,000 per day.<\/p>\n Yes, my math says that the number of infections occurring per day in Florida is equal to the number of infections (for the placebo group) that occurred for the entire course of the trial, for the Pfizer group.<\/p>\n Or, to take a smaller example, the current rate in Illinois is 18.9 per 100,000.\u00a0 The vaccination rate is nearly identical to that of Florida (yeah, I know, Florida\u2019s high case rate is being blamed on the low vaccination numbers but that doesn\u2019t seem to match up with the data itself, per the New York Times<\/em><\/a>).\u00a0 Doing more math says that with a vaccination-protection-adjusted rate of 35.4, Illinois has as many infections in 5 days and the Pfizer study did over its entire duration.<\/p>\n Let\u2019s do some more bad math:<\/p>\n Let\u2019s assume that the category of immunocompromised people are statistically irrelevant, because there\u2019s no real way to factor them into the story.<\/p>\n Then let\u2019s define an \u201cexposure period\u201d as \u201cthe length of time, under a given set of conditions, it takes for there to be 175 new infections per 100,000 for an unvaccinated group.\u201d\u00a0 And let\u2019s say that, for any given \u201cexposure period,\u201d a given person has a 0.175% probability of becoming infected, if not vaccinated or, if vaccinated, 5% of that risk, or 0.0087%.\u00a0 This translates into a 99.825% risk of remaining uninfected in the first case, or 99.9913% in the second.<\/p>\n But if getting infected is mostly a matter of bad luck, rather than a personally-elevated risk, then the risk of being infected in any given \u201cexposure period\u201d is independent from any other one, and the risk of remaining uninfected is calculated in an exponential manner:\u00a0 for example, for five periods, the probability of remaining uninfected is 99.825% x 99.825% x 99.825% x 99.825% x 99.825% (or 99.9913% x 99.9913% x 99.9913% x 99.9913% x 99.9913% for the vaccinated case).<\/p>\n And, still, as long as the \u201cexposure periods\u201d remain fairly low, because we determine that each such period of time is fairly substantial, the overall probability of remaining uninfected is still fairly high, and the risk of becoming infected is small.\u00a0 But if you consider that, in Florida, an unvaccinated person has just as much likelihood of being infected in a single day, as was the case for the entire duration of the Pfizer trial, could we say that an \u201cexposure period\u201d is simply a day, and that the risk compounds so quickly that over the course of a year, an unvaccinated person has a 50-50 chance of getting covid?\u00a0 In that scenario, a vaccinated person would have a 3% chance of becoming infected over the same timeframe.\u00a0 \u00a0This illustrates the combination of the effectiveness of the vaccine and the low prevalence of covid overall.<\/p>\n But let\u2019s cook up some more scenarios.\u00a0 We\u2019re being told that new variants are far more infectious.\u00a0 What if a new variant were twice as infectious?\u00a0 \u00a0My Bad Math says that over a year\u2019s time, an unvaccinated person would have a 72% chance of becoming infected, and a vaccinated person, a 6% chance.\u00a0 (Yes, that\u2019s the math of compounding.)\u00a0 You really have to push the numbers a fair bit to get significantly-elevated risks of covid for a vaccinated person \u2014 for example, a virus that\u2019s 10 times as virulent, over 2 years, would produce that same 50-50 chance for infection for a vaccinated person, was we previously had for an unvaccinated person over one year.<\/p>\n But what if the vaccine becomes less effective against a new variant?\u00a0 At 90% effectiveness, our 3% one-year \u201cvaccination failure rate\u201d becomes 6%.\u00a0 At 80%, it\u2019s 12%.\u00a0 And so on.<\/p>\n So the first thing to note is that my calculations were based on a hypothetical that the only people with reduced risk of infection were the vaccinated.\u00a0 Obviously, a fair share of the population, in addition to the vaccinated, have already been infected.<\/p>\n So the first thing to note is that this means that our \u201cadjusted infection rate among the fully-exposed\u201d increases significantly.\u00a0 If we use a 65% \u201cvaccinated\/immune\u201d rate, then that boosts the infection rate among the full-exposed to 234 in Florida, or a 33% increase, and means that the \u201cexposure period\u201d is 75% of a day.\u00a0 So this means that the 50-50 probability of being infected is that of a 9 month period; for a full year, that rate climbs to a 57% chance of infection for the non-protected, and 4.2% for the protected.<\/p>\n At the same time, yes, the cumulative effect of each person who becomes sick meaning that they won\u2019t get sick and expose others later, does make a difference.\u00a0 But when I take the same calculation, and adjust the risk of infection to reflect the decreases due to previously-infected people reducing by some small fraction, that initial probability of infection \u2014 well, there\u2019s not that much of a difference, really.<\/p>\n After a year (365 \u201cexposure periods\u201d), the unvaccinated have a 39% probability of having gotten covid, instead of a 47% chance.\u00a0 The vaccinated, a 3.09% chance rather than a 3.14% chance.\u00a0 But recall that my adjustments are that, taking into account the previously-sickened, we have to assume that the prevalence of infection among the not-protected population increases; and then, well, it\u2019s a wash.<\/p>\n Now, again, my math is shaky.\u00a0 The biggest issue is whether, even without being in a category of \u201cimmunocompromised,\u201d some people are more or less likely to be infected, even given the same level of exposure.\u00a0 This means that each \u201cexposure period\u201d is not independent (in the statistical use of the term) and all of the above math falls apart.\u00a0 But this, in turn, suggests that if, for whatever reason, you are more susceptible to covid than the average person (not severity of infection but exposure to a person to a given degree of closeness and infectiousness), these numbers will understate your risk and that if not, these numbers overstate your risk.<\/p>\n But, as a final bit of math: how does this affect effectiveness, not just infection rates?\u00a0 3.1\/39 \u2013 1 = a 92% reduction.\u00a0 We can calculate a likelihood of getting infected, in an absolute sense, but the ultimate effect of the compounding is still fairly small.<\/p>\n Remember chicken pox?\u00a0 A childhood disease, a rite of passage.\u00a0 It was inevitable that you\u2019d get it, so some moms had chicken pox parties for their kids if someone they knew had chicken pox and if it was a convenient time for a kid (or all kids in the household) to be sick, both in terms of family activities (no upcoming vacations) or because chicken pox is milder for elementary school kids than teens or adults.<\/p>\n A vaccine first became available in 1996, though there were questions about just how necessary a wide vaccination program was because serious cases were so rare, and it was mostly a matter of kids being sick inconveniencing parents.\u00a0 It took a decade for the vaccine to be officially recommended, which in turn meant that the vaccination rates for children grew slowly, from about 15% in 1996 to 40% in 1997, 65% in 1998, then increasing more gradually to 85% in 2000 and 90%-ish thereafter.\u00a0 (This is from a graph, eyeballing the percentages, for a specific geographical area<\/a>.)\u00a0 How long did it take \u201cherd immunity\u201d to develop?\u00a0 That is, how soon were infants, too young to be vaccinated, well-protected?\u00a0 In 1995, 1.6% of infants got chicken pox; in 1998, when vaccination was at a rate of 65%, that infection rate dropped by half.\u00a0 Only in 2002, when we\u2019re at rates of 85% or more, did the rate of chicken pox among babies settle in at the low rate of 0.2% that held true, more or less, for the rest of this study\u2019s data.<\/p>\n Now, to be sure, it\u2019s not clear whether, once the toddlers began to be vaccinated, the vaccine was likewise offered to older children who hadn\u2019t yet gotten chicken pox, so it\u2019s not clear whether this data over- or understates the prevalence of the vaccine.\u00a0 It\u2019s just an interesting comparison. Does this show a speedy decline in prevalence, or a slow one?\u00a0 It depends on expectations.<\/p>\n Reports suggest that the Pfizer vaccine is only about 80% effective against delta.\u00a0 So let\u2019s redo the final numbers once more:\u00a0 instead of a 3% chance of infection after a year-sort-of, you have an 11% chance.<\/p>\n Why is Iceland\u2019s case rate so high?\u00a0 In the first place, they very recently relaxed their requirements for visitors \u2014 the explosion in case rates began on July 20th (or so; my data source is a chart<\/a> so it\u2019s hard to say exactly).\u00a0 Is this due to the elimination of travel restrictions as of July 1st<\/a>?\u00a0 (Strictly speaking, these were relaxed, not eliminated \u2014 there was still a requirement to have been vaccinated.)\u00a0 There\u2019s nothing in the data that suggests this.\u00a0 For that matter, there\u2019s not really anything in the way of analysis as to the cause of the outbreak, though it appears that the Janssen vaccine, which was used for 21% of those vaccinated<\/a>, is not as effective against the Delta variant, given that there is a proposal to re-vaccinate this group<\/a>.\u00a0 (This is apparently their term for the J&J vaccine, which is 67% effective against Delta according to a study<\/a>.)<\/p>\n Again, to repeat my math:<\/p>\n For the vaccine to be 95% effective does not mean that you have a 1 in 20 chance of getting sick.\u00a0 It means that your chances of getting infected are 95% less than an unvaccinated person, over a specific length of time \u2014 or, rather, that under the study conditions of relatively low prevalence rates and short time periods, the chances of infection for the study group were reduced by 95%.<\/p>\n How this plays out in the real world, over longer time periods and with different underlying covid prevalence rates, can differ substantially, when you look at your actual chances of getting infected, rather than the reduction in the likelihood of infection.<\/p>\n On any given day, your risk of getting covid is small, with or without a vaccination.\u00a0 But at the high rates of prevalence we\u2019re seeing in Florida right now (and could see in the fall, in the North, when people stay in their houses now, as much as people in the South do in the summer), mean that over time, your risk of infection grows:\u00a0 to a Bad Math risk of 50:50 if unvaccinated, but still only less than 5% if vaccinated, or 11% under \u201cdelta\u201d conditions.<\/p>\n One obvious bottom line is that statements like \u201cherd immunity at 70%\u201d was nonsense \u2014 and that the feds, and community groups need to do a better job of marketing the vaccine, not as your civic duty but \u201cyour chance of a side effect is X, and you\u2019re worried about that, and we get it, but your chance of a serious effect from covid is worse\u201d \u2014 with actual numbers.<\/p>\n But I was hoping that in working through these numbers and reading the links, I would acquire some insight into what the government should be doing in the meantime, with respect to precautions against breakthrough infections, which currently affect my family in two ways:\u00a0 first, my son is required to wear a face mask all day for a second year, with no known information on how much of a difference it really makes to kids who have been vaxxed; and, second, the memory care community mom lives at, has just had an outbreak, and I\u2019m asking myself, statistically, does this mean that we\u2019re in trouble, and, beyond that, what are the appropriate level of precautions to take, even with everyone vaccinated?<\/p>\n So I\u2019m going to click publish and walk away from this, and apologize for the 3,000 word length here, and move on to something else (Illinois pensions, anyone?).\u00a0 Sigh.<\/p>\n \u00a0<\/p>\n \u00a0<\/p>\n \u00a0<\/p>\n \u00a0<\/p>\n \u00a0<\/p>\n \u00a0<\/p>\n \u00a0<\/p>\n<\/body><\/html>\n","protected":false},"excerpt":{"rendered":" So I\u2019ve been doing some COVID math \u2014 the sort that isn\u2019t really good enough to put myself forth as an expert but which I want to write down, in a bloggy fashion. It starts with this:\u00a0 two weeks ago, in a CNN interview, CDC director Rochelle Walensky said that for \u201cevery 20 vaccinated people, […]<\/p>\n","protected":false},"author":2209,"featured_media":17630,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-18097","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"yoast_head":"\nAbout the studies.<\/h3>\n
But what does it mean?<\/h3>\n
What about \u201cherd immunity\u201d?<\/h3>\n
A comparison to chickenpox.<\/h3>\n
Repeating the above, with delta effectiveness.<\/h3>\n
Back to Iceland.<\/h3>\n
Your mileage may vary.<\/h3>\n