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# Polio vaccines, Nash Equilibrium and Game Theory

### Understanding the maths and the social science behind anti-vaxxer arguments.

There are some regions where Polio disease cases caused by the polio vaccine itself (yes you heard it right), is more than the cases caused by wild Polio virus. ðŸ¤¯

Let's talk about vaccine misinformation, game theory, Nash equilibrium and a lot more.

A thread ðŸ§µðŸ‘‡

Before we begin, let me just link to The Guardian as proof that the tweet above is not misinformation.

Yes, more people are getting polio from the vaccine than the virus in some African countries.

Of course, not everyone will read the article, so I will give you a tl;dr; There are 2 polio vaccines, the oral (OPV) and the injection. The oral one contains **attenuated** poliovirus. i.e. weakened poliovirus. The injectable one uses **inactive** polio virus.

It is logistically much cheaper to use OPV than IPV so most poor countries (including India - remember Amitabh Bacchchan's "2 boondh zindagi ke"?) use the OPV rather than the IPV. USA and Europe mostly use the IPV.

A risk with OPV is that 1 in 3 million cases, the attenuated virus in the OPV can mutate in the body (if the host has low immunity and/or malnourished) and evolve into an actual polio infection.

So for those rarest of rare cases where this happens, for those people, unfortunately, the truth is, they actually got polio BECAUSE THEY TOOK THE VACCINE. That isn't a false statement. But hold on ðŸš©ðŸš©ðŸš©ðŸš© this isn't a reason for all of us to stop using the polio vaccine!!

In USA in 1952, out of 165 million people, 55000+ had polio, with half of them leading to paralysis. In a healthy, first-world, well-fed, highly immune population, 1000 out of every 3 million naturally got polio. 500 of them turned paralytic.

If you vaccinate all 3 million of these people with the polio vaccine, there is a chance that out of them, 1 person will develop polio (not necessarily as severe as inducing paralysis)So 100% vaccination brings cases down from 1000 to 1. But that 1 person gets it from the vax.

This is where we will discuss a little bit about game theory. Consider you are in country of 3 million, and the other 2,999,999 people have already been vaxxed for poliovirus, what is your optimum strategy now? Get vaxxed or not? (Do reply to this before reading ahead)

Purely mathematically speaking, your chances of contracting polio are infinitely higher if you get vaxxed now. Because if everyone else is already vaxxed, they there is little likelihood that you will catch polio from anyone else. Taking the vaccine does have a 1 in 3M chance

The problem is that if everyone thinks this way, then probably no one will take the vaccine (or 50% people might not), and then we go back to the original condition of 1000 polio cases. And this is where John Nash's equilibrium and papers come in.

Quick detour, you might love to watch A Beautiful Mind and 90% accurate biography of John Nash, beautifully portrayed by Russel Crowe

https://en.wikipedia.org/wiki/A_Beautiful_Mind_(film)

Particularly, the dramatized scene of him stumbling upon his equilibrium (please watch)

Now consider a country of 3 million people where everyone knows these 2 facts.

1. If unvaxxed, 1000 in 3M get polio

2. If 100% vaxxed, 1 in 3M get polio

Everyone would want to get vaccinated to save themselves from polio and we would be at a Nash Equlibrium

The problem occurs when someone stands at the town square and starts shouting that that "why do I need to take the vaccine? Me taking the vaccine doesn't affect others right?" They say that -

1. taking the vaccine is their choice

2. if others take they are still protected

When they are questioned why they are saying this, they come up with 2 major answers

1. my body, my choice, I don't want to inject (heck in today's world these idiots would prefer the less efficient OPV over the more efficient IPV)

2. they will cite the findings (in the beginning of this thread), and say that, if 2,999,999 others are vaccinated, they are better of not getting vaccinated since that has more chance of contracting polio.

But here is where social science and mathematics tend to diverge.

You can never be the 3,000,000th person, after 2,999,999 have been vaccinated. The moment you stand on the town square and cry out about your vaccine apprehensions, another idiot joins you, then another, and soon, 1M idiots don't want to be vaccinated.

You feel awesome that you got 999,999 more people to agree to your point. But now you fucked around with Nash Equilibrium with your misinformation campaign have completely turned the math around its head.

Now because 1/3rd population is unvaccinated, you are not standing on the rock solid foundation of 2,999,999 vaxxed peopled vs 1 unvaxxed yourself. You now have a 33% unvaxxed population where not only 300 people will get polio, but it gets more room to mutate and evolve

Eventually you make it pointless even for the 2M folks who did get vaxxed to get their vaccines. They will probably now be susceptible to a new mutated form of polio, and whoops, we are back to square one, probaly with >1000 people contracting the dreaded disease.

As an administrator of this 3M sized country, what would you prefer ? 1. 1000 polio cases, 500 paralysis cases, because no one took the vaccine; or2. 100% vaccination, but 1 polio case, which happened due to the vaccine itself ?

A few closing points - do not draw 1:1 parallels with COVID vaccines. Some COVID vaccines are mRNA, some are replicating viral vector, and some are attenuated ones like OPV. You can read more here

We have gotten nowhere near the coverage with COVID as we did with polio so there HAS NOT YET BEEN ANY RESEARCH at all on whether the attenuated COVID vaccines themselves can cause COVID in rare cases. None of the most common vaccines are attenuated virus, though.

There are some side effects of some COVID vaccines. Some cause blood clots **in very very rare cases**. Wherever these numbers have been higher than permissible limits, the vaccines have been stopped. There are so far no documented cases of direct death due to COVID vaccines.

"death of a vaccinated person", is not the same as "death caused by the vaccine". A person with chronic heart disease might die with or without being vaccinated.

Anti-vaxxers have cherry-picked many such cases, do not fall for these.

## Polio vaccines, Nash Equilibrium and Game Theory

Aha! This read was simply fantastic, reminding me of the good old days when I used to take my oral polio vaccine (or was it something else? My memory is a little hazy)

Mind boggling read