Over at my friend Pal’s blog, in a discussion about vaccination, a commenter came up with the following in an argument against the value of vaccination:
Mathematical formula:
100% – % of population who are not/cannot be vaccinated – % of population who have been vaccinated but are not immune (1-effective rate)-% of population who have been vaccinated but immunity has waned – % of population who have become immune compromised-(any other variables an immunologist would know that I may not)
What vaccine preventable illnesses have the result of that formula above the necessary threshold to maintain herd immunity?
I don’t know if the population is still immune to Smallpox, but I would hope that that is just a science fiction question. Smallpox was eradicated, but that vaccine did have the highest number of adverse reaction (I’m sure PAL will correct me if that statement is wrong)
It’s a classic example of what I call obfuscatory mathematics: that is, it’s an attempt to use fake math in an attempt to intimidate people into believing that there’s a real argument, when in fact they’re just hiding behind the appearance of mathematics in order to avoid having to really make their argument. It’s a classic technique, frequently used by crackpots of all stripes.
It’s largely illegible, due to notation, punctuation, and general babble. That’s typical of obfuscatory math: the point isn’t to use math to be comprehensible, or to use formal reasoning; it’s to create an appearance of credibility. So let’s take that, and try to make it sort of readable.
What he wants to do is to take each group of people who, supposedly, aren’t protected by vaccines, and try to put together an argument about how it’s unlikely that vaccines can possibly create a large enough group of protected people to really provide herd immunity.
So, let’s consider the population of people. Per Chuck’s argument, we can consider the following subgroups:
-
is the percentage of the population that does not get vaccinated, for whatever reason.
-
is the percentage of people who got vaccinated; obviously equal to
.
-
is the percentage of people who were vaccinated, but who didn’t gain any immunity from their vaccination.
-
is the percentage of people who were vaccinated, but whose immunity from the vaccine has worn off.
-
is the percentage of people who were vaccinated, but who have for some reason become immune-compromised, and thus gain no immunity from the vaccine.
He’s arguing then, that the percentage of effectively vaccinated people is 
There’s a whole lot wrong with this, ranging from the trivial to the moderately interesting. We’ll start with the trivial, and move on to the more interesting.
In real modeling, we usually describe two groups of people: the portion of people who are immune to the disease (
He wants to make
Even if you did insist on doing that… the groups in the list aren’t mutually exclusive. The set of people who are immune compromised and the set of people who were vaccinated but didn’t get immunity will definitely overlap; so will the set of people who are immune compromised, and who’s immunity has waned. He doesn’t consider that – quite deliberately – because it reduces the appearance that there are all of these people out there who are unprotected by vaccines.
Getting to the more interesting stuff: In terms of modeling, the real model of a disease is actually pretty simple. There’s a factor called 
So, for the disease to survive in a population, 
For a disease like the flu, 
But this whole discussion so far is, in some sense, rather deceptive. It’s vastly oversimplifying reality. It’s looking at herd immunity as a simple yes/no: is the disease sustainable in the population according to a simple model?
When we look at a disease like whooping cough, or polio, or meningitis, we can, absolutely, get a vaccination rate which exceeds the critical threshold. When’s the last time you saw someone with polio? How many of us have ever seen a child with whooping cough? Those are incredibly rare precisely because we’ve got a high enough rate of immunity within the population (thanks to vaccines!) that when an exposure to one of those occurs in the population, it burns itself out: it’s
So to answer the original question: what vaccine preventable diseases are there where effective herd immunity is really possible? Lots. For many diseases, it’s not particularly difficult to attain herd immunity: mumps, measles, chicken-pox, smallpox, polio, whooping cough, meningitis – for all of these, we can (and in most places, do) have really solid herd immunity. But while it’s not difficult to have herd immunity against these diseases, it’s also not too hard to break it. There’ve been recent outbreaks in many places where the vaccination rate has fallen.
When it comes to things like the flu, we’re not getting complete herd immunity from vaccines. In fact, it’s quite probable that we can’t, because the 
And in some sense, this is all a distraction. “Perfect” herd immunity doesn’t mean that no one will ever catch the disease. There will be exposures; people will sometimes catch the disease when there’s a new exposure. Herd immunity means that when that happens, the disease won’t be sustainable – it will burn out.
The real question that we care about when it comes to vaccination isn’t “is it possible for people to get infected?”, because the answer to that will always be yes, as long as there’s any possible source of exposure.The real question is: “What is the risk of a susceptible individual being exposed and as a result coming to harm?” And that’s a much fuzzier question, but it’s very clear that as we reduce the population of people who can carry it – as we reduce the pool of people in
I hope to God that you are not an educator or a teacher. If you are I’m sure you must enjoy beating your students. As I categorically stated in the thread, it was a guess as I am not a medical doctor or immunologist..
ahahahahahahahahahahahaha!!!!!
now that you have been shown to be an obfuscatory idiot, you want to cry “wah wah wah you are so MEAN because I am totally not an expert and was just GUESSING and really I don’t know what I’m talking about so how dare you explain exactly how stupid I am wah wah wah”
that is freaking hilarious! Thanks for the laugh!
No Problem, I am glad I have brought some enjoyment to your life.
People formulate opinions based on the types of logic to which they’ve been exposed. This post was interesting, and pointed out a lot of flaws in the gentleman’s logic such as double counting.
I didn’t really detect anything in the original post or the subsequent replies to indicate that Chuck was trying (knowingly at least) to obfuscate the issue. He was probably just trying to understand it, and put a framework around it.
Vaccines (or any topic that involve disease, pain or death) mix ignorance and fear. Negative emotion make people want to frame the issue negatively, because they aren’t good at axiomatic reasoning and don’t approach many issues with impartial logic.
It’s constructive to post something like this because it brings out ideas he wasn’t exposed to before. It’s not constructive to point a finger and laugh at someone for it.
I’ll save you some time and post your succinct reply to me: stfu.
Interesting! What with all the anti-vaxxers around it’s good to read such a clear explanation of the stats of it.
Awesome and very clearly written post Mark!
If MarkCC were a teacher, many of his students would be absolutely thrilled because he is actually willing to take the time to explain, in detail, mathematical modeling logic. In this case, the relationship between immunization and immunity.
Chuck, it is not about beating you, it is about seeing if you are willing to try to understand.
Gauthier,
If you actually go back to the thread, you can see that I asked Dr. Lipson, not PAL, a serious question in hopes of receiving a serious answer. He never gave one. I was attempting to clarify to GOATRIDER what I thought was encompassed in the herd immunity calculation. I did not appreciate the snide response from Dr. Lipson. I fully expect snide from a blogger, which is why I specifically ask a doctor.
“-(any other variables an immunologist would know that I may not)”
“I’m sure PAL will correct me if that statement is wrong)”
“The formula is my educated guess as to what percentage of the population defines herd immunity.”
I fully expect someone with better subject matter to correct any of my mistakes. Wouldn’t you?
Tara and Daedalus were gracious in their response. Dr. Lipson was not.
Gaythia, I didn’t go back and verify your ID, My apologies.
You may also notice that I will be the first one to apologize if I discovered that I made a mistake. Human beings are known to do that as part of the learning process
Good argument, but a notational nitpick:
“v is the percentage of people who got vaccinated; obviously equal to 1-u”
Either you have “percentage” where you meant “fraction”, or you have “1-u” where you meant “100-u”.
Oh man, you just hit on one of my #1 pet peeves. I apologize in advance for the rudeness of my tone here — I get dispropotionately enrage by this. It’s not you, it’s me, so to speak.
Anyway: No, even if he meant percentage, he did not mean “100-u”. 100% = 1!!!!!!!! By the same token, 100 = 10000%.
In school, often a formula that was intended to produce a percentage would be expressed as “blahblahblah*100%”. NO NO NO NO NO! Saying “multiply by 100 percent” is the same fucking thing as saying “Now multiply by one, duh-herp!!!!” It’s totally unnecessary.
7/8 = 87.5% END OF SENTENCE. There is no need to say any more about it. There is no need to “multiply by 100%” or any such other pointless no-ops.
If u is the percentage of unvaccinated, then 100-u is equal to v + 9900%. which is the same as v + 99. Which is NOT the same as v + 99%! Argh argh argh I wish this didn’t piss me off so much.
seriously though, it’s not you, it’s me. You approach is how virtually all educators teach it, at least in my experience and in the US. But they and you are all wrong. 100 != 100%. 100% = 1. It can be no other way.
While I certainly agree that 100% = 1, I don’t see anything wrong with a formula ending “*100%”. It expresses the combination of “Please give the answer using percentage notation” and “Here’s how you convert to percentage notation, in case you’ve forgotten” in 5 characters, which is pretty concise.
I multiply by things that are equal to 1 all the time, such as 100%, (2.2 lb/kg), (3600 s/hour), or (1000 g/kg).
Maybe they could emphasize more that 100%=1. (I don’t know if they even say that in school; it’s been long enough since I learned about percentages that I don’t remember. If not, they should.)
As far as measuring units are concerned, I tend to think of the prefix as belonging to the number, not the unit (with, of course, one exception). So I think of my own height, for example, not as “172 centimetres” but “172 centi-” metres. Where “centi” happens to be short for “times ten to the power of minus two”. Likewise, an appliance’s power consumption “2 kilo-” Watts.
Casio calculators even have a button to normalise the exponent to a multiple of 3, so as to correspond with a named prefix. (I confess to finding “589e-9” more readable than “5.89e-7”.)
Fair enough. I retain it as an irrational pet peeve myself, but that is a fair argument. However:
“Forgotten”?!?!? Oy.
I suppose this is okay for fifth-graders, but seriously, how many people over the age of 12 can’t remember how to convert to a percentage?
(Don’t answer that question. I don’t actually want to know.)
This is true, but you don’t put them in a formula (unless it is a conversion formula, in which case it is then the only thing in the formula, right?)
I mean… have you ever seen Newton’s Second Law expressed as:
No you haven’t, because that would be stupid. If you want the answer in Joules, then you convert it to Joules.
Sorry, it’s just me. It is a pet peeve.
Force is measured in Newtons, not Joules. Anyway, the answer comes out of Newton’s Second Law formula already in Newtons, as long as you supply the mass in kilograms and the acceleration in metres per second squared.
That’s the entire point of the base units: if you put everything into the formula in unprefixed base units (all lengths in metres, all times in seconds, all masses in kilograms, all forces in Newtons, all magnetic fluxes in Teslas and so forth), the answer will always come out in unprefixed base units.
The formula for the reactance of a capacitor does not expect microfarads and megahertz, but Farads and Hertz respectively. Similarly, the formula for magnetic flux does not expect square millimetres and return an answer in microwebers; it expects square metres and returns Webers.
No matter how inappropriate the unprefixed base unit may appear to be for a particular calculation, the prefix is just an easier-to-say form of the exponent.
Well that’s an embarrassing slip-up I made on units. I don’t know what I was thinking. Oy…
The only thing I can offer in my defense is I’ve been trying to renovate my kitchen, all by myself, in time to host Thanksgiving dinner, while also trying to keep my full-time job.
You know, I remember starting out googling for the conversion between newtons and foot-pounds… and then I saw that foot-pounds were equivalent to Joules and not newtons.. how I decided to ignore my initial recollection that the output of Newton’s Second Law was, uh, newtons.. and instead decided to go with my other initial recollection that foot-pounds was somehow a unit of force… even though I know full well that pounds is the unit of force. Oh my. I just don’t know what else to say. My brain ain’t working right at the moment.
Mark,
Taking a step back and a deep breath, this is a very well written piece.
“The entire division into sub-categories is solely for the purposes of deception.”
The laundry list was an off the top of my head everything I guessed may be relevant in the calculation.
“people who are immune compromised, and who’s immunity has waned.”
Immune compromised is a function of physiological disorders or medical treatment. Immunity has waned is a function of the vaccine. That is how I envisioned it. And I intended them to be unique, non-overlapping populations. I also envision that these are time series calculation that may not have a mathematical function.
I did not properly define all of the variable that are necessary in the calculation. I will be the first to admit that. There is a great deal of fuzzy math and subjective measurement when it comes to herd immunity and vaccine effectiveness. I am just trying to feel my way around the subject matter like everybody else who isn’t an immunologist.
Chuck:
Give me a break. It’s remarkably obvious that you went into the discussion over at Pal’s with a very specific goal. And then in the ensuing discussion, you slapped together a pile of bullshit pseudo-math in order to make it look as if there was some credibility to your argument.
You did not properly define any of the variables in your “calculation”. You clearly did not attempt to do so. You clearly did not even attempt to understand the real math of immunology/epidemiology.
Before I saw your rubbish, I didn’t either. It took me under five minutes to pick up some minimal basics. Obviously, I’m a math guy with some background in mathematical modeling. But the basic math of epidemiology is really simple, and there are numerous very clear introductions to it on the net.
You pulled a typical anti-vaxxer trick: you just piled bullshit together, in order to create the impression that there’s something mysterious and deceptive about the efficacy of vaccines.
And you’re still playing that same game.
“I also envision that there are time series calculations that may not have a mathematical function”. Do you know what a function is? If you did, you’d realize how utterly stupid that statement is. There is no such thing as a time series that doesn’t have a mathematical function: a time series is a mapping from time-points to values. That is, by definition, a function.
And finally, “there is a great deal of fuzzy math and subjective measurement”… That’s a very strong claim. And, as usual, you make no attempt to actual back it up.
You’re a dishonest dumbass who’s trying to pretend to know things that he doesn’t, throwing around terminology in a desperate attempt to cover up your own cluelessness, in order to support an unsupportable bullshit antivax argument.
“I’m a math guy with some background in mathematical modeling.”
A PHD is a little more than “some” and yes I agree that I “slapped together a pile of bullshit pseudo-math”. Actually in retrospect it was probably more of a flaming pile of BS pseudo-math. It was a steep learning curve, but I do have better models to rely upon now. I wish the very best to you. I will not comment any more,
Oh I’m sorry, but this is abuse. You want room 12A, just along the corridor.
My PhD is not in math or in mathematical modeling. It’s in computer science with a specialization in programming languages.
In terms of modeling, I’ve got no formal background as a result of my PhD.
You’re just tying to make excuses for your own laziness. Basic modeling of this sort doesn’t take a phd to understand. To pick up the basics, it takes no background beyond high school level algebra. If you actually had any interest in the truth, you would have spent the twenty minutes or so that it would take to read a wikipedia article.
But you didn’t. You’ve been arguing about vaccines for how long? And yet, you’ve never even tried to understand this. Why, do you suppose, that is?
“A disease can survive in a population when R ≥ 1 – that is, if the probability of exposure times the percentage of the population that’s susceptible to the disease is greater than one.”
My brain isn’t working right now. If the max probability of exposure is 1 (guaranteed exposure) and the max percentage of the population susceptible to the disease (1-q) is 1 (the entire population), then how does R ever get to be greater than one?
Can Ro rise above 1 because an infected patient can expose more than one other? Or is there something else I’m missing?
Sorry if this is a dumb question. This was a very nicely explained post.
R is, basically, the average number of people who any given infected individual will infect.
If R = 1, then on average each infected person will infect one other person before they recover – so the population of infected people is stable when R=1.
If R=2, then each infected individual will, on average, infect two people. For a novel flu virus, R=2 isn’t unusual – it’s what leads to epidemics.
Mark, thank you so much for writing this response. The clarity with which you write and your honest presentation of the mathematics is something we rarely see. I know that you’re a hard-core computer guy but you have a terrific gift in communicating public health facts.
As for, “When’s the last time you saw someone with polio?” – my wife used to work with an older woman who was a polio survivor and walked with a brace and walker. Our daughter, who she adored, proudly got her vaccine because she knew it would protect her and others from getting the same thing as Ms. [redacted]
When’s the last time you saw someone with polio? How many of us have ever seen a child with whooping cough?
I haven’t actually seen a child with whooping cough, mostly because my friend didn’t invite me over to potentially expose anyone in my family that might not have gotten immunity from the vaccination. Her vaccinated 5-year-old was taken to the ER around this time last year for whooping cough, because the vaccine hadn’t given her sufficient immunity to combat it in the epidemic environment of central Texas. (Her identical twin sister was fine, so whether or not the vaccine provides immunity is not entirely genetic.)
Nice math argument, I didn’t need it, but I’ll be happy to pass along the link when I think it’s needed.
I started to read this blog post but I stopped when I realized it was senseless to do so. My reason is simple:
100% of this post is true – % of the post which is purposely untrue – % of the post which is inadvertently entirely untrue – (∑ % of each statement that is partially untrue * % of the post each statement occupies) – (∑ % of each response that is untrue or misleading * % of relevance given to this response) = some largely incalculable yet inarguably statistically insignificant percentage.
I laughed hard. Thank you.
Take a look at http://www.lagriffedulion.com/ . I’m no mathematician, just a psychologist, but it seems to me that much of the arguments of that blog is based on maths that is made to be more difficult than needs be.
I do not have a constructive comment, but might I point out there’s a couple of instances where you mistake its/it’s and whose/who’s.
Also, this didn’t make much sense to me:
” In fact, it’s quite probable that we can’t, because the for seasonal flu is high enough, and the population-wide effectiveness of the flu vaccine is low enough that even with 100% vaccination, [R_0 x S > 1] and [S]. If you cut in half, that you cut rate in half as well!”
Where that [S] is, maybe half a formula missing?
Now, if you want an example of a good treatment of infectious disease spread, I suggest Munz, Hudea, and Imad et al.’s “When Zombies Attack!: Mathematical Modeling of an Outbreak of Zombie Infection” Inf. Dis. Mod. Research Programs, Nova Science Pubslishers, 2009. Link.
Unfortunately they don’t discuss the need for vaccination in detail. 🙂
Thank you. I ran across this kind obfuscatory argument yesterday here:
http://shotofprevention.com/2010/11/05/shot-of-great-media-coverage-promotes-good-health/
I gave it shot, and noted he was using a kind of word salad. Oh, deer. I see he has returned. Unfortunately this is not Sunday, and I need to get things done.
(a different Chris)
“When’s the last time you saw someone with polio? How many of us have ever seen a child with whooping cough?”
Julia, in QLD Australia we are now experiencing a whooping cough epidemic. The reason? In some parts of the state vaccination levels have dropped to the point of allowing the disease to resurge, I believe California is going through the same thing.
Pertussis vaccinations in particular have to be managed as boosters are required at certain points and the immune resonse varies more than with other vaccines.
Another real world example of what happens if vaccination levels drop below a certain threshold.
Kathleen M. Carley, et al., 2006, Modeling Community Containment for Pandemic Influenza: A Letter Report, Washington, DC: National Academy Press.
Is there a prisoner’s dilemma in vaccination?
http://atdotde.blogspot.com/2010/10/is-there-prisoners-dilemma-in.html
Recently, I have been thinking about vaccination strategies as I was confronted with opinions which I consider at least very risk inviting. Not to spill oil in the fire I will thus anonymize illnesses and use made up probabilities. But let me assure you that for the illness I have in mind the probabilities are such that the story is similar.
Let’s consider illness X. For simplicity assume that if you meet somebody with that illness you’ll have it yourself a bit later with 100% probability. If you have X then in 1 in 2000 cases you will develop complication C which is lethal. But C itself is not contagious.
Luckily, there exists a vaccination against X that is 100% effective, i.e. if vaccinated you are immune to X. But unfortunately, the veccination [sic] itself causes the deadly C in 1 in a million cases.
So, the question is: Should you get vaccinated?
Unfortunately, the answer is not clear: It depends on the probability that if not vaccinated you will run into somebody spreading X. If X is essentially eradicated there is no point in taking the vaccination risk but if X is common it is much safer to vaccinate….
Jonathan, the 0.5-1% number is only for one strain, variola minor. The chance of death from the other prominent strain is closer to 30%. Also keep in mind that those rates were in populations naturally exposed to the disease every generation; their resistance would be higher than ours. So it is likely that even variola minor would be much deadlier to 21st century populations than historically expected…to say nothing of the deadliness of major…
I’ve given this question some thought. The answer is, “It depends”. So let’s divide diseases into three categories:
1) Diseases where you stand a reasonable chance of getting it (e.g. flu, varicella up until pretty recently, etc.)
2) Diseases where you stand a very small but non-trivial chance of getting it (e.g. pertussis)
3) Diseases where you are virtually certain not to get it (e.g. polio, at least in the US)
For #1 there is most definitely not a Prisoner’s Dilemma. Vaccine side-effects are so rare that if you stand a non-trivial chance of getting the illness, even if all the illness is likely to do is make life a little unpleasant for you for a week or two, your cost-benefit still swings strongly towards vaccination.
For #3, there probably is a Prisoner’s Dilemma. The best answer I have come up with for that one is, “Do it for your grandchildren.” I don’t have to get a smallpox vaccine — a particularly nasty, side-effect laden vaccine, by the way — because my parents and grandparents did instead. I owe it to them. So if you won’t do it for your fellow humans, do it for your grandchildren. (BTW, I made this argument once at a meeting of anti-vaxers that I had infiltrated. They looked at me like I had said that the National Socialists had a good day care program. hahahaha…)
For #2, it’s highly uncertain. We’re talking about such low numbers on both sides, that small perturbations can upset your cost/benefit analysis. With eradication still far off (and in some cases, not really technically feasible) the “do it for your grandkids” argument is much harder to support.
Which is one reason why I support making it a pain-in-the-ass to forgo vaccines. No public schools for you (unless of course you have a medical reason not to get vaccinated). Now your cost-benefit is pretty clear, ain’t it? No more Prisoner’s Dilemma. (I oppose truly mandatory vaccination on ideological grounds, but I have no problem with making it damn-near-mandatory)
For #3, there probably is a Prisoner’s Dilemma. The best answer I have come up with for that one is, “Do it for your grandchildren.”
If someone really, honestly thinks they are in a prisoner’s dilemma, then you might point out to them that the existence of prisoner’s dilemmas is typically used as an argument for compulsory government programs. I.e. the solution to it is to have some outside authority forcing people to cooperate. So if that’s what they think vaccination is, they should support governmentally mandated vaccination programs.
You might also point out that the best strategy to win iterative PD games is not ‘always-defect.’ That strategy consistently loses. The best strategy is to cooperate the first time, then play tit-for-tat. So again, if they really think they’re in a prisoner’s dilemma, they should go along with vaccination at first, and only defect in response to defections by others.
The lesson of the prisoner’s dilemma is that the most rational one-shot strategy can lead to a sub-optimal outcome for all players, which all players would really like to avoid. If they don’t get that they, too should want to avoid the result of mass defections, they don’t understand the prisoner’s dilemma.
I guess it’s not a true PD, because there are far more than two participants. It’s more a Tragedy of the Commons thing. You can’t really employ a tit-for-tat strategy when there are 300 million simultaneous tats…
Still, your argument makes a lot of sense, and as I alluded to, that’s why I support making vaccination almost-but-not-quite-mandatory. (As I said, I oppose truly mandatory vaccination on narrow ideological grounds — but I support making the opt-out carry a heavy enough price to strongly discourage it)
I can live with that.
I could even almost live with forcing patients to go Private, if they refused their publicly-funded vaccinations for no good reason.
Mark,
I am curious about the model you describe. From what I gather is its a SIR model without any recovery (an SI model?). I imagine you could model discretely it using the following difference equations: (S[t] susceptible at time t, I[t] infected at time t)
S[t+1] = -R0*S[t]*I[t]
I[t+1] = I[t] + R0*S[t]*I[t].
But for this model wouldn’t there always be exponential growth as long as R0 > 0? Would this be very realistic?
I’ve been playing around with these SIR models on paper, and it seems the recovery rate (which is a parameter dependent of the disease and independent on the population) plays a vital role in the model.
I suspect that without including the recovery rate in your model you’ll ALWAYS get exponential blowup provided R0 is greater than 0, which it always is. Moreover, using a normal SIRS model you’ll find that the number of immune required for herd immunity is linearly dependent on the recovery rate, so I guess it is really an essential consideration when talking about herd immunity.
I know very little beyond Wikipedia, forgive me if I’m wrong.
YOU ARE A FRICKIN MORON! I have been studying vaccines for aprox 13 years now and the more I read the more I realize how insane the practice actually is. It is directly drivin by a corrupt medical and governmental system. I sugest you start to look at the actual diseases and vaccination problems, then corelate the driving force (finances) behind the push, before you make any other dumb ass remarks!
Dano: thank you for your unemotional post, so free of name-calling. I looked into the finances as you suggested and was appalled at what I found – corrupt doctors being paid by lawyers to fabricate studies about vaccine ‘ dangers’; thank you for bringing this corruption to my attention.