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The question is about the methodology/biostatistics of clinical trials (I state this beforehand to avoid accusations of being an anti-vaxxer).

As multiple anti-COVID vaccines are offered on the market, the question naturally arises about their long-term efficacy and possible side effects. As SARS-CoV-2 virus has been known for a bit more than a year, neither of these could be tested directly, so one probably has to rely on indirect knowledge due to the experience in developing other vaccines. Given that multiple vaccines have been approved for a wide use, there should exist well-established procedures to assure their efficacy and safety.

The question is: how does one prove the long-term efficacy and absence of side effects during a short period of time (short in comparison to the expected duration of immunity and absence of side effects).

Update
As @MattDMo have correctly pointed in their answer, the absence of long-term effects is established in the last phase of the clinical trials, while observing cohorts of vaccinated individuals during a long period of time. @MaximilianPress, in their comment, has also correctly brought up the fact that this is done within the framework of survival analysis. In particular, this means that it is sufficient to follow the vaccinated cohort during several years to evaluate the side effects that occur during their lifetime, as the survival analysis keeps track of the sensored data.

Here we are dealing with a situation where such a study could not be done - the best we can guarantee is the absence of the side effects in the next few months. We thus need to rely on a different method, which would account for the similarity between the established and the new vaccines.

I am less concerned about the vaccine efficacy, since the inefficacy of vaccines is usually against different strains of virus, as is in the case of influenza and Dengue.

Update 2 The accepted answer points out two important things:

  • The long-term side effects of a vaccine/medication can be truly understood only in the Phase IV of the clinical trials, i.e., after the vaccine is in widespread use.
  • The threshold for approving this specific vaccine is lower due to the emergency of the current health crisis.

This however still leaves open the question of the safeguards undertaken in the Phase III of the clinical trials, before the vaccine approval. I.e., the question is about ranking the candidate vaccines against the above-mentioned threshold.

Remarks:

  • I understand why this question irritates, but I believe that it is not the asking, but rather refusing to seek the answer that plays in the hands of anti-vaxxers.
  • I did follow the suggestions in the comments to pose this question in the medical and the statistics communities, but it seems to get there even less traction.

Update 3
Recent suspensions of Astrazeneca vaccine in 9 European countries demonstrate the importance of this issue.

Update 4 Pfizer vaccine has been fully approved - that is from now on it is considered a fully tested vaccine (till now it had been used under the emergency authorization).

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    $\begingroup$ What is your question? $\endgroup$
    – David
    Dec 21 '20 at 16:58
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    $\begingroup$ I’m voting to close this question because this is a statement of opinion, not a question. $\endgroup$
    – David
    Dec 21 '20 at 16:59
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    $\begingroup$ @David The question is: how does one prove the long-term efficacy and absence of side effects in such a short time. $\endgroup$ Dec 21 '20 at 17:00
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    $\begingroup$ Your question is then one of medical policy and public health, not a specific question of biology and is therefore off-topic here. It may be on topic on SE Medical Sciences, but you would be better to read one of the many detailed accounts available via reputable news sources. $\endgroup$
    – David
    Dec 21 '20 at 19:11
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    $\begingroup$ Everyone, lets please be civil to one another. $\endgroup$
    – Bryan Krause
    Dec 21 '20 at 21:11
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You are correct that we have been creating and approving vaccines for a very long time, so the procedures for looking at efficacy and side effects of vaccines in general long-term are quite well developed. They occur via so-called "Phase IV" trials, also known as post-marketing surveillance.

However, one complicating factor with at least the Pfizer/BioNTech and Moderna vaccines is that they're the first mRNA vaccines ever approved. While I hope the companies have been following some of the volunteers from previous clinical studies of other mRNA vaccines, we're in somewhat uncharted territory right now. The vaccines appear to be safe and effective, at least on the scale of tens of thousands of Phase III test subjects, but we just don't know exactly how they'll behave long-term. They were approved in the US as "Emergency Use" products, meaning the upside of having them available outweighs any known or potential downsides at this point.

So, the technical answer to your question "how does one prove the long-term efficacy and absence of side effects in such a short time?" is that they can't prove it. Based on the available data to date, there's no reason to suspect that some previously-unobserved deleterious effect will become common, but we can't be 100% certain until the products have been in widespread use for years and possibly decades. However, given the seriousness of COVID-19 and its currently unchecked spread across much of the world, the benefits of having "highly, strongly, very most likely safe" vaccines (that are also very effective) are huge.

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I will try here to sketch an answer starting from where @MattDMo left off.

Question: How do we evaluate possible long-term side effects of a vaccine? (With specific application to the recently authorized vaccines for SARS-CoV-2).

Phase IV clinical trials As @MattDMo has pointed out, the definitive answer can be given only in the Phase IV of the clinical trials, which is the follow-up of the vaccine/medication after its commercialization, which typically lasts throughout the lifetime of the medication. Obviously, such data are not available for the vaccines against SARS-CoV-2 (or any new vaccines). Thus, we have to rely on the information obtained in the earlier stages of the vaccine development. This leaves us with the following options:

Phase III of the clinical trials Vaccine development process typically lasts 5-10 years, and the phase III of the clinical trials typically lasts about 6 months. Thus, typically one has the data for at least 6 months. This is however not the case for the vaccines against SARS-CoV-2, which were given emergency approvals after only 8 weeks of Phase III trials.

Animal models Several years of vaccine development also allow for additional information: e.g., the vaccine could have been extensively tested on relevant animal models for extended periods of time. How one quantitatively estimates human risks from those for the animal model deserves a separate question, but the established animal model likely means decades of data avalable for comparing it with human. Again, this is not the case of SARS-CoV-2, which has been known for a bit more than a year, and for which no reliable animal model has been tested yet.

See this, this, and this articles for the general review of the use of animal models for vaccine development, and this one for the specifics of the animal models for COVID-19.

Similar viruses In our case there are at least two very similar viruses that have been studied for years: SARS-CoV (first outbreak in 2003) and MERS-CoV (first outbreak in 2012). Although no vaccine has been available for either virus, much research effort has been put into this direction, and the data (at least on animal models) should exist. There might be even data for mRNA vaccines, which have been under development for some time already.

Nature of vaccines Finally, the very nature of vaccination may explain why vaccines are approved faster than other types of medications, which can be for years in clinical trials. Adaptive immune system develops response to a new pathogen within about a week time. Thus, any adverse side effects are likely to appear during a week after vaccination, whereas the later developments have to do with the functioning of the immune system (other medications can have cumulative effects, if taken over a long period of time, or stay in the body for longer periods - such as medical implants). How one models quantitatively the long-term immune response is another question that deserves to be asked separately.

On this respect the new SARS-CoV-2 vaccines are taking a conservative approach, with the vaccines being counter-inducated to those with possibly different immune system responses: allergics, pregnant women, children.

Disclaimer This answer could benefit from more reliable references, which I hope to pinpoint later. Also, it doesn't contain quantitative methods of risk estimation that I hoped for - rather it opens way to several more precise questions.

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