Sick Baby (and Why You Should Vaccinate)

Elena has been sick for more than a week now. She had simple cold symptoms, but last weekend she started having periodic coughing fits that went on for what seemed like a long time and were almost uncontrollable. I have a hard time describing the cough–sort of short and full of mucous, and while she’s in a coughing fit she hardly has time to breathe between the coughs. We have two effective ways to stop the coughing: give her some juice, or steam up the bathroom with hot water and take her in there until it’s over. The coughing fits only happen a few times a day, often after she’s been doing something more strenuous while playing, but they have woken her up in the middle of the night.

On Tuesday we decided to get some medical help, so Jenny called the nurse line and they had her make a doctor’s appointment. The doctor decided to treat her for whooping cough based on her symptoms and the fact that there has been a recent outbreak of whooping cough around here. The tests for the whooping cough take several days to run and are not entirely reliable, and in this case it’s best to start a course of antibiotics without waiting for results. Another name for whooping cough is pertussis, which is the P in the DTaP vaccination. Since Elena has followed the recommended immunization schedule, she’s already had three of these shots, so you might wonder why she still got sick. We discussed this briefly in my mathematical biology class so I can give an overview of the dynamics. I’ll try not to get too mathematical or launch into too much of a rant, but be forewarned, because I can’t make any guarantees.

Like almost all immunizations, the DTaP shot is only partially effective. Studies of effectiveness are always done at the population level, not on individuals, so it’s unclear what “partially effective” means. It could mean that most people are completely immunized while a few are completely unprotected, or it could mean that everyone immunized has a smaller chance of falling ill, but is still slightly susceptible. Of course, it’s probably some combination of the two: inoculated individuals are protected to varying degrees, depending on a wide set of complicated factors. This matters when combined with the idea of how contagious a disease is. Pertussis has a contagion rate of 70-100%, so if an non-immunized person is exposed, it’s almost a certain thing that they will fall ill.

The most basic model used by epidemiologists to study the spread of disease is called SIR, an acronym for “Susceptible, Infected, Recovered,” which is the progression that individuals go through. In the model, you look at the percentage of the population in each of those categories over the time of a disease outbreak. In order to do so, information about contagiousness, population susceptibility, and recovery rates is plugged into a matrix, which then can be multiplied against the population percentages. The resulting system can be analyzed to see whether the disease is likely to die out, maintain a steady level, or explode into an epidemic. All of that depends on some things called the eigenvalues of the matrix. You don’t need to know exactly what that means, but you do need to know that matrix multiplication doesn’t work like regular multiplication. Two matrices that look very similar could bring about very different behaviors, because they have qualitatively different eigenvalues. Specifically, any eigenvalues bigger than 1 mean the outbreak will grow, while eigenvalues smaller than 1 mean that it will die out. So even though there’s a much bigger difference between 1.05 and 3 than there is between .95 and 1.05, the first pair would imply similar similar behavior and the second pair gives very different behavior.

If we want to reduce the spread infectious disease (at the demographic level of policy and public health), there are only a few things that we can try to do, such as improving the effectiveness of our vaccines, but that’s likely to take a long time. There’s really just one thing that can be done on a large scale in a short time: increase the level of vaccination in the population. When you model these kinds of changes, the non-linearity of the system becomes clear. If you guess that 50% of the population is vaccinated, and see what happens when you move that up to 65%, it might be that there are no visible changes–the disease goes on as before. But then if you move it up just a little bit more, say to 70%, then you cross over the threshold and change the eigenvalues, which makes the outbreak less likely to persist. Sometimes this is called “herd immunity” because it works only on a large scale, and means that it’s actually okay for a certain number of people to remain unvaccinated, because they’re effectively protected from exposure by the immunity of the rest of the group.

So what does this all mean? First, it’s okay for vaccines to be only partially effective, because a second layer of protection kicks in when enough people get vaccinated. Second, it’s unrealistic to vaccinate every person in a population. Some people just don’t go to the doctor, some are still too young for the vaccine, some have legitimate allergies (eggs are the foundation of many vaccines, and some people are allergic). But that’s okay, because those people get the protection of the entire group. It’s even okay for a few crazy people to not want immunization for irrational reasons. What’s not okay is when those people write about it on blogs or appear on Oprah–then this influences others and it becomes a movement, which changes the dynamics and threatens everyone’s immunity.

Vaccine schedules are created by smart people for a reason, which is to optimize the health and safety for each individual. But they optimize in two dimensions–the miniscule risks presented by the vaccine itself, and the much greater risk of catching the disease if not enough people are immunized. The second dimension is the important one, but it’s the one that people ignore, because they don’t understand how it works. They don’t understand that the risks are nonlinear. By not vaccinating, they take advantage of the group, but this only works up to a point, after which outbreaks and sustained epidemics become likely.

When a non-vaccinator makes their choice, they probably don’t accurately consider the risk that their child will get sick, because serious, dangerous diseases are extremely rare in our modern world. We’ve eradicated many of the terrible diseases of yesteryear. The way we did it was–no surprises here–through vaccinations! If the vaccination campaigns hadn’t worked so well, and terrible diseases like polio and smallpox were still prevalent, I can guarantee that nobody would worry about nonexistent links between autism and vaccination. But these things can come back when large numbers of people opt out of vaccination. There was a tragic story about measles on This American Life some time ago.

I’m not worried about Elena. She’s happy and strong, the cough seems not to bother her too much and the coughing spells have been far shorter in the past few days. She may not even have whooping cough; we still haven’t received the results. But if she does, it’s a little bit frustrating to know that she might have a cough that lingers for months, when we did everything we could to avoid it. So my next step is to try to use this tiny platform of mine to sway the public discussion in the right direction, and help people understand why non-vaccinators are misguided. I hope that all of you vaccinate your children on the recommended schedule, and can encourage people to get the real facts when they are trying to understand how it works.

Updated: I changed a few small things at the beginning for clarity and accuracy, based on Jenny’s first-hand account of Elena’s doctor’s visit. She also summarized things succinctly for me:

When a person decides to not vaccinate their child, or even to follow a delayed scheduled, they are not just deciding to take on the risks for their own child of not vaccinating. They are putting every person their child comes into contact with at increased risk. This could be compared to the unfairness of people affected by second hand smoke – but at least in those cases you can see and smell the smoke.



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24 responses to “Sick Baby (and Why You Should Vaccinate)

  1. immunizecokids

    LOVE this post! Thank you so much for your honest, emotional and scientific push for vaccination. I especially like this section “What’s not okay is when those people write about it on blogs or appear on Oprah–then this influences others and it becomes a movement, which changes the dynamics and threatens everyone’s immunity.”

    Thanks for being an awesome mom and I hope that Elena is feeling better soon!

  2. immunizecokids

    Ugggh – EMBARRASSED! – Sorry, I read more of the blog and you are a awesome DAD! Thank you for your excellent work!

  3. Gregory Maxwell

    How about showing us the calculations? At least link to some place where someone has worked out some examples if you don’t have the patience to do so yourself.

    When you use scientific and mathematical terms in an opaque and wishy-washy way you debase them, making them little more than incantations.

    What makes modern science and mathematics materially different from alchemy is that it is rigorous. Math and science also have impressive symbols and incantations, but there are defined rules for working with these symbols that predict reality in a reliable and repeatable way.

    For all the details given here, instead of eigenvalues you might as well have attributed the infection to evil spirits which will be warded off by the Wakalixes contained in the vaccines…

    At the moment your argument is precisely as rigorous as the pseudo-science nonsense pushed by the anti-vax crowd. They too assert that some mystical sounding science mumbo-jumbo supports their position, so by using the same strategy you actually make their point look more reasonable. If instead you show some examples of how adding vaccination to a model of infection saves lives, and link to studies where the agreement between the models and reality are established then you’ll be providing your readers with an understanding that this isn’t just a war of opinions and buzz-words.

    Systems of differential equations are, indeed, the stuff of rocket science. But they aren’t *that* hard, considering the thousands of hours people spend constructing contrived arguments against vaccination. That is time that could be spent understanding the science and math, and I have to believe that at least a few more people would if they were exposed to it. But even if people don’t wish to learn how to use these tools, it is important to know that it is not just buzzwords and opinion and that people can independently reach the same conclusions by studying the math. Otherwise it’s all just a game of marketing, and the scare-mongers are better at marketing than rational folks.

    • Tomas

      It took me around 2 minutes to google some nice resources on these epidemic mathematics. [personal attack deleted–rothlmar]

      Maybe consider the blog post as enough of a starting point to work from instead of being pissed off that this well-written, informative, free work wasn’t spoon-feeding you exactly what you wanted.

    • rothlmar


      Thanks for your interest in seeing calculations. When I wrote this post, I had no idea it would reach such a large audience; I thought it would just be seen by my friends and family. In either case, this post was for a nontechnical advice, and follows the rule of thumb (which I first heard about when reading Stephen Hawking’s A Brief History of Time) that every equation cuts readership in half.

      I’m seriously considering a followup post with more technical details, and hopefully some illustrative graphs, but that takes a lot more work than the heuristic hand waving of this post, which I typed up on the bus while commuting to work. I welcome any contributions you might have.

      Finally, the main reference for the post is Mathematical Models in Biology by Leah Edelstein-Keshet. We used it as the main text in the class I alluded to in the post; I think it’s widely used in such contexts. It assumes some knowledge of calculus, but is still pretty readable even without the appropriate mathematical background.

  4. Kevin

    “It’s even okay for a few crazy people to not want immunization for irrational reasons.”

    What does this mean? Obviously, this is your blog and you can say what you like, but perhaps people that may disagree with you aren’t all “crazy” …

    There are three things it is hard to talk about in this world:

    I’m sure whatever choice people make, they do it in the best interest of their child. To call them crazy, seems like a simple way to dismiss something you may not be comfortable with. To blindly listen to doctors without research or to ignore them both seem like the wrong course of action.

    My wife and I reside somewhere in the middle. We accepted some vaccines for our 3 month old and have declined and delayed others. We hope to install our beliefs in nutrition and natural remedies in our daughter in order to prevents the day-to-day sicknesses prevalent in society and the youth of our country. We will have to see how it works out, but we are confident in our decisions, thus far.

    We are far from “crazy”

    • Tomas

      “It’s even okay for a few crazy people to not want immunization for irrational reasons.”

      He clearly says it’s ok for a few crazy people to not want vaccinations for **irrational** reasons. If you have a rational reason, then it’s not crazy.

      “”We are far from “crazy”””

      Do you have any rational reasons for not vaccinating your kids as much as possible? If so, then he hasn’t implied you are. If not, you might be crazy.

      • Kevin

        Doctors use scare tactics. It’s that simple. I had to sign a “Refusal to Vaccinate” form. As if I had stomped around the office throwing things about. I don’t have a rational fear that my daughter has any current risk to come into contact with Polio. Or the causes of Hepititis.
        Maybe that is because so many other Parents choose to vaccinate? “Herd Immunity”? I don’t know? Maybe because she isn’t going to Russia or exchanging fluids at 3 months old?

        Other parents choose to let their kids eat Mc Donalds and drink Red Bull – that seems “crazy” to me. Cutting those things out of the diet can prevent just as much if not more than any vaccine being injected into a newborn.

  5. Matt

    Kevin, [personal attack deleted–rothlmar]
    Your belief in pseudo science and conspiracy theories would be OK if it only affected you. It doesn’t though, you’re putting your family and community at risk.
    If a natural remedy is proven effective, IT BECOMES SCIENCE BASED MEDICINE AND YOU CAN GET IT FROM YOUR DOCTOR. If you’re willing to risk your own and your loved ones lives on unproven natural “remedies”, you are mentally deficient.
    It’s a shame that the Darwinian effects of your idiocy will be felt by your community as well. Otherwise, I’d wholeheartedly support them.

    • Kevin

      This is just attacking. When I initially said I may not refer to other parents as “crazy” when acting in the best interest of their child, in their opinion, I don’t think I was attacking anyone. More so saying, it is hard to judge another parent just because you don’t share the same belief.

      You however have called me an idiot and mentally deficient.

      I am saying that I believe a healthy lifestyle for a child is possible through a diet not based on fast food, soda, energy drinks, microwave food and candy. If you cut these things and encourage an active lifestyle as opposed to video games, there will be a lot more to gain than any doctor has to offer.

      I haven’t been sick in over 20 years. I can tell you it has nothing to do with doctors. Meanwhile I am surrounded at work by people calling in sick at least once or twice a month because either they aren’t feeling well or they have to attend to their sick child. It is sad.

      I will listen to my doctors and take their most sincere recommendations to heart, but their word is not the end all be all.

      • Brit

        I just have to say…

        Fast food, energy drinks, and microwaved foods were not in the picture 100 years ago when many of these diseases were. Much can be attributed to living a healthy lifestyle, but living this way does not prevent all sickness and disease.

  6. Jason Kerwin

    One important correction to the SIR model, if you want to apply it to a real population, is to take into account strategic effects. Namely if I know that a large share of the population is vaccinated, my incentive to vaccinate myself drops. So instead of treating the vaccination rate as exogenous you can figure it out based on a richer model. Sure, getting the values people place on protecting themselves involves some educated guesswork but so does figuring out the parameters that determine the SI and IR conversion rates. I’m not sure if this has been published anywhere; it was covered in my game theory class last term.

    In a simple model where people don’t act strategically you’d think nearly everyone would get vaccinated, since it has a large positive value when few people are getting vaccinated. Instead, many people don’t get vaccinated, effectively relying on the herd immunity for protection. One interpretation of this result is that people who give dumb reasons for not believing in vaccines don’t have the necessary incentive to understand them in the first place, let alone get one.

    • rothlmar


      Thanks for this excellent contribution. I’m not an expert by any means, but as I was writing this I realized that there must be some kind of game-theoretic principles involved in understanding the behavior of those who choose to opt out. You make a convincing argument.

      This leads to two further thoughts: first, because the non-vaccinators rely on herd immunity, it seems like a very irrational strategy to publicize the fact and encourage others to do the same. Wouldn’t it be more optimal to refuse vaccination, then not tell anyone about it?

      Second, and this is in part also a response to Gregory’s comments above, it’s clear that in order to really understand just the basics of the situation, you need to have at least a working familiarity with statistics, differential equations, linear algebra, and game theory. That’s a pretty high standard, so it makes sense that people trust authority figures instead of trying to work through the ideas on their own. I hope that they realize that there’s real difference between the bully pulpit of a doctor or public health official and that of an entertainment figure.

      • Jason Kerwin

        >Wouldn’t it be more optimal to refuse vaccination, then not tell anyone about it?

        Hah – I had the exact same criticism when the model was presented in class. The response I got, which is at least somewhat credible, is that we’re assuming that the population is very large, and people get their information from stuff like TV announcements of what share of the population has been vaccinated so far. So the non-vaccinators could still lie but their individual incentive to do so is reduced. The crucial detail is that in any given interaction you don’t know whether the person you’re talking to/making out with is vaccinated or not, or whether they’re infected. Still, a more detailed model would try to incorporate people’s decisions on whether to subject to antibody tests to see if they are vaccinated and so forth.

        >That’s a pretty high standard

        Agreed. The assumption of total rationality that is standard in the social sciences leads to silly situations like tenured economists assuming that the people in their models are better at solving optimization problems than they are (since the economists haven’t figured out the model yet but presumably the agents in the model have). This is an area of research that is just opening up in economics and I think it’s something that I’d like to work on. One promising avenue is to think about the costs of fully informed an rational thought, and then look at peoples’ alternatives such as listening to elected officials or other authorities. Unfortunately it doesn’t appear that in practice people can tell the difference between Jenny McCarthy and Regina Benjamin – again having the knowledge to differentiate between actual authorities and dumb celebrities with an uninformed viewpoint is costly.

  7. Paul

    Mark – you jumped into the deep end with this one!

    I’d like to throw out a couple more considerations:

    Many of the “healthy” people are actually carriers for contagious disease. This means that just because you are not sick, it doesn’t mean that you aren’t spreading around disease to others who are more susceptible than you. This is a complex population which includes people in an incubation stage, who may yet get sick, people who manage to fight off the disease without showing identifiable symptoms, but are still contagious, and people who have “recovered” but continue to be contagious for a bit longer. There are even a few which are simply not susceptible, but carry the disease around the way that much of the population carries long term staph populations on their skin without knowing it.

    Anecdotal evidence of never having gotten sick with something is not only invalid from a statistical and scientific perspective, it is disingenuous because it ignores the disease that an individual has managed to spread around without even being aware of it.

    Pertussis is a particularly persistent issue, though, because vaccine-derived immunity is so short lived. It requires a continuing course of vaccinations throughout childhood, when individuals are most susceptible. Without regular boosters, however, protection for adults drops off rapidly. Once an outbreak occurs, all of those unprotected adults lead to a breakdown in this group immunity, spreading it to the children they come into contact with. Also known as whooping cough (for sound of the sufferers’ breathing between coughs) or hundred day cough, it will kill those whose respiratory or immune systems are not strong enough hold up to such a lengthy onslaught.

    dTAP contains immunization against Tetanus, another booster everyone should have regularly. Hopefully, this (recent) bundling of shots will increase our population’s resistance as everyone stepping on rusty nails and getting protective boosters also picks up resistance to pertussis. It would be much better, though, if everyone were simply immunized according the optimized schedules that all of this complex math can derive for us.

    Behind every complex set of equations there are inputs and output trends that are understandable to the general population. I do believe that the huge, emotional response of the scientific and medical community to the anti-vaccination movement has been much of their own undoing. I’ve read far more tirades from people ranting that we should just trust the math or the scientists or the doctors than I have seen clear explanations of the factors that are being taken into account for the equations, research, and medical recommendations.

    Thank you for providing some of this explanation.

  8. I’m saving this page. My youngest sister (married w/three kids) doesn’t vaccinate. They (her family) are sort of preachy about it, and like to send family members goofy ‘science’. I wonder why they dismiss one logic over the other, and it always seems like, to them, the absence of evidence is proof of their being correct (regarding conspiracy). For example, the pharmaceutical’s are hiding the real story, and able to pay off the doctors… unbeknownst to the doctors, and that’s why most doctors are for vaccinations…

  9. Thank you for this. Even if some don’t agree with part of it or all of it, it was well written.

    I vaccinate both my children. And I have always wondered how the children/adults who are not vaccinated will affect the population as a whole and I’m sure that will take many years to fully see.

    As far as whooping cough, I had that myself when I was about 18 and it was terrible. We have also had an outbreak here in Idaho and I’d hate to see either of my boys go through that.

  10. DrD

    HI there,

    As a doctor and mother to an infant, I agree with your comments about vaccination. It’s easy for people to be complacent about vaccination when they haven’t seen the devastation caused by the illnesses that we have all but eradicated. Lameness, blindness, brain damage, deafness, birth defects and of course, death. With international travel so prevalent, I would never risk not vaccinating my children.

    I was also upset recently to see a ‘star’ on Oprah making strong connections between vaccination and autism, which is not backed up by science…


  11. It seems like research and opinion has parents confused these days when it comes to immunizations. It is terrible because being a new parent with a newborn is scary enough not to mention trying to figure out which choices are best for your precious baby. Great post!!!

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  13. Dawna

    Thanks for your blog. I like to view both sides of vaccinations. I am one of your “crazy” audiences. Haha. I suppose with all your research “for” vaccinations in your follow-up blog, you will remember to address the issue that vaccinations truly are not “for” every child. This includes those children who are immune-compromised. My children were immune-compromised. I vaccinated two out of three of them. I had one child ridden with autism symptoms and the next one with eczema, asthma, bloated stomach and constant diarrhea. My only child NOT vaccinated has no issues whatsoever.

    Will we ever prove vaccinations caused this to my kids? Not likely. Will we ever prove that they didn’t? Again, not likely. But for me and many other families, something happened when we vaccinated. That did not exist before-hand. These are “live” viruses we are dealing with at times. If your child was immune compromised then you should reconsider vaccinating. If they are not, then you can preach at the top of all rooftops but it’s not for everybody. We can’t throw stats at people and then say things like, “so, therefore, this vaccination is for YOUR child”!

    All I’m saying is that vaccinating is not for everyone. You make blanket statements as though it is. Not sure why I felt compelled to write on your blog. Perhaps because you made the statement implying you are getting a large audience. I see you are welcoming a “logical” debate. I hope I can contribute to that with statements made by the CDC, which clearly talks about the apprehension with vaccinating immune-compromised babies and children.

    Thanks for allowing the forum to hear another side. Many families like myself are clearly not “crazy” when the CDC themselves are using the same logic.


    • rothlmar


      I’m glad you found this old blog post and wrote a comment. This topic has been on my mind recently. In the post, I point out that some people have legitimate reasons to avoid vaccination. It sounds like you might be one of them. I can’t judge because I don’t know you and I’m not a doctor. But I hope that you discussed your concerns with your doctor and came to a reasoned conclusion together. If so, I think you would be thankful for all those who can and do get vaccinated, because their vaccination protects your unvaccinated child from disease.

      I certainly understand where you’re coming from, but let me point out the other side of the coin. Suppose a child is vaccinated, but the vaccination isn’t entirely effective. Furthermore, because other people have chosen not to vaccinate, a disease outbreak occurs and the vaccinated (but not entirely protected) child gets sick. That illness can also have long-lasting or even permanent effects.

      As parents, we all want to protect our children the best way we know how. Statistically speaking, the way to do that is to vaccinate, regardless of what you might hear on television or read on the blogs. If you choose not to vaccinate, you should have a good reason, probably one that you and your doctor agree about.

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  15. commonsensedindividual

    I think all you who don’t get your child vaccinated are guilty of neglect and idiocy. The post about being mentally deficient is the truest thing I have ever heard. stop being so selfish about spending 40 dollars on vaccines that can keep your child from dying.

    If you believe that the people who do vaccinate their child are “blindly following their doctor” you are mistaken. No parent is dumb enough to just do what someone says. We have brains and use them to keep our children safe. In defense of the doctors, they are way more educated than you are and HAVE a lot more knowledge that can save your child and keep them healthy than you do. Having a healthy diet is nice, but that doesn’t protect from Hepatitis/Whooping cough/Small pox/Tetanus. Please use some common sense: Bacteria and viruses are not picky, they infect both obese and fit people.

    Unless you want the baby coffin companies to have a boost in their income I suggest you get your child vaccinated.

    a competent individual who is not easily influenced by blog reading to the extent of deciding that i will kill my child because of the idiocy of the non-vaccinators of this page

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