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Vaccine effectiveness in "real world"
The CDC did a study to see the effectiveness of vaccines that are being given throughout the county.  They found 
  • 80% effectiveness after 2 weeks after the first shot for the Pfizer & Moderna vaccines
  • 90% effectiveness after 2 weeks after the second shot for the Pfizer & Moderna vaccines
against COVID (incl. asymptomatic COVID).

The clinical studies had shown about 95% effectiveness after 2 weeks after the second shot for detected COVID infection (but may have missed asymptomatic COVID).

The study ran from Dec. 14 to March 13, so the 3,950 subjects were exposed to the mixture of variants across that time.  The subjects took their own nasal swabs weekly, which were tested via RT-PCR.  Thus it could detect asymptomatic cases.  10% of the cases were asymptomatic, but 58% of cases were detected by RT-PCR before symptoms were noticed.

All subjects were essential workers, including health care personnel and first responders from six states: Arizona, Florida, Minnesota, Oregon, Texas and Utah.  (I note that Florida had a number of variant cases, but I'm not sure about the other states.)

I believe this study can help pin down the percent of asymptomatic cases (at least in vaccinated subject) and may clarify how delayed after the infection takes hold before symptoms appear.   Unfortunately, only the press release seems to be out at this time.
Curious as to the severity of the disease for the 10% and 20% that caught it after having both or one shot?
Remember, 10-20% refers to chances of getting it. If on any day, 10 out of 100K unvaccinated people would get COVID, then 1 to 2 of every 100K vaccinated people would get COVID.

Still waiting to find the paper.
Understand. I've heard that the vaccine seems to lower the severity for those that still catch the disease after being vaccinated. Haven't seen a study though. . . .
The study is available now.

They tested subjects weekly (PCR). This is the first I've seen of a real measure of asymptomatic incidence.
Quote:Nonetheless, the majority (87.3%) of PCR-confirmed infections were associated with symptoms consistent with COVID-19–associated illness. The remaining PCR-confirmed infections were associated with other symptoms not part of the COVID-19–associated illness definition (e.g., headache, fatigue, and rhinorrhea) (2.0%) or no symptoms (10.7%). Only 22.9% of PCR-confirmed infections were medically attended, including two hospitalizations; no deaths occurred.

So, about 11% of the infections we asymptomatic; 13% had no typical COVID symptoms (including asymptomatic).
If you counted only cases with COVID symptoms, there were another 14.5% that you missed.

For various features (ethnicity, job, location, ...), the study gives the total number of people, how many were vaccinated or not, and how many got COVID. For each feature, they don't give how many unvaccinated got COVID.

The differences between subgroups could be due to differences in where they worked, what their job was, etc.
But I wanted to see if anything jumped out.

If U is the number of vaccinated in a group and V is the number vaccinated, if there were 90% effectiveness, the COVID cases C for U+V should be about the same as U+0.1V unvaccinated people (I'll call that EU).
So I computed the rate of cases per equivalent unvaccinated (C/EU).

Females, U = 529; V = 1924, C = 109. EU = 721.4 C/EU = 15.11%
Males, U=460; V=1037; C=96; EU=563.7 C/EU = 17.03%

White: EU = 1073..4 C/EU= 16.59%
Other: EU =211.7 C/EU = 12.75%

Hispanic/Latino: EU=279.8 C/EU = 20.37%
Other: EU = 1005.3 C/EU = 14.72%

18-49: EU=945.4 C/EU = 15.44%
50+: EU= 339.7 C/EU = 17.37%

Primary HCP: EU = 142 C/EU = 11.27%
Other allied HCP: EU = 351.3 C/EU = 19.07%
First Responder: EU = 362.4 C/EU = 20.70%
Other: EU = 429.4 C/EU = 10.95%
A study out of England mixes cold water and warm water to get tepid water.

At least as covered by CNBC, the study measures the (combined) efficacy of the AZ and Pfizer vaccinations after a single dose. (That certainly sounds like a politically-motivated decision.)

When no distinction is made between the vaccines, the effectiveness as 65% at 21 days after the first dose (compared to 80% for Pfizer reported by the CDC in a smaller study).
Quote:Symptomatic infections fell by 74% three weeks after a single dose of either vaccine, while asymptomatic cases were down by 57%, the data showed.

A second vaccine dose reduced the overall infection rate by 70%, with symptomatic Covid infections down by 90% and asymptomatic cases of the virus cut by 49%.

If you didn't notice, there were a higher rate of asymptomatic cases after 2 doses than after 1. Presumably this is because a higher percentage of the cases were asymptomatic after two doses than the cases after one dose.
behind a paywall unfortunately . . . my summary take:   makes total sense that vaccinated people "test positive" while being asymptomatic or with minimal symptoms.   That is what vaccines are supposed to do--have us ready to mount an immune reaction to actual infection.
I didn't pay to see more than the intro, where they set up the problem.

I believe there are two reasons for detecting the virus in athletes;
1) For the benefit of the person: Making sure he/she gets the appropriate medical attention.
2) For the prevention of spread of COVID: Avoid putting the case into situations where he may is responsible for others to get the disease.

I believe that "test positive" here really means they have the virus in their system which was detected by the test (and not that the test is picking up the vaccine or evidence of the body's ability to fight the virus).

There are at least two aspects of having the virus that are important:
1) The individual may develop a full-blown infection.
2) The individual may shed virus that can infect others.
3) The individual may develop side-effects of the disease, without developing a full-blown infection.

Regarding #1, Typically, the PCR test picks up the infection before symptoms appear. The patient IS infected, but it is uncertain whether he will develop COVID symptoms or not. The effectiveness of different vaccines is different. Furthermore, each vaccine does have different effectiveness against different variants. It has been awhile since I looked, but the mRNA vaccines were 90% effective against being tested positive for the virus (for healthcare workers, tested weekly), while the J&J vaccine was about 67% effective against developing COVID symptoms. There were reports that some other vaccines were about 50% effective. Roughly speaking, that means that 1 out of 10 with mRNA vaccines, 1 out of 3 with J&J vaccines, and 1 out of 2 with the least-effective vaccines will get COVID versus unvaccinated people in roughly the same situation. I presume, but do not know, that vaccinated individuals that develop COVID symptoms will shed virus that can infect others. It seems to have been shown that vaccinated people that develop COVID have fewer bad outcomes. What I'm not sure about is whether (or how much) that reduction in bad outcomes is simply because of the reduced number of people that develop COVID. That is, if 10% of the 70-74yo males (with no vaccine) with a certain set of comorbidities die if they get COVID, do 10% of the 70-74yo men (with vaccine) with the same set of comorbidities die if they get COVID.

Regarding #2, I have seen little research for detecting shed virus by individuals before they have symptoms. A difficulty is that a person may shed virus under some circumstances (playing basketball vigorously) and not under others (reading a book in bed). I believe that the general belief is that asymptomatic patients shed less virus and for a shorter time than symptomatic patients.

Regarding #3, I have no idea whether this happens. I think I remember seeing some reports of getting later side-effects even though a case was asymptomatic, but I can't quote it currently.
good points, M_T.   

I'll add that this will get increasingly interesting as variants potentially reduce vaccine efficacies over time--and may do so differently for the different vaccines   Of course counter-balanced by new and tweaked vaccines plus boosters.   Still much to learn!
Contra Costa is reporting vaccinated (~7 /100k/day) and unvaccinated (~0.5/100k/day) separately:

They do not break down the vaccinated cases by the specific vaccine. I would speculate that there are more vaccinated cases and that they go undetected because they are symptomatic.

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