I’m not a mathematician, but I did my best to grind some numbers for a “friend” once who asked much the same question. Here’s what I was able to come up with (he was particularly concerned with HPV, which is why I started there):
Between 2006 and January 5, 2015, we administered 67,250,524 HPV vaccines. There were 156 claims filed through the NCVIP program for adverse events. For 10 of those, the Dept. of Health and Human Services conceded that the vaccine either caused or contributed to the injury or death. For 1, a judge ruled that the vaccine either caused or contributed to the injury or death, and while the HHS did not agree, they paid without fighting the case. For 65 cases, neither a court nor HHS found that the vaccine either caused or contributed to the injury or death, but settled anyhow because reasons*. For 83 claims, no evidence linking the vaccine to the injury or death was found, or the cause of injury/death was positively identified as due to some other factor, and the claims were dismissed.
So out of 67,250,524 doses of the vaccine administered over 9 years, 11 injuries occurred that everyone agrees were caused by or contributed to by the HPV vaccine. That’s a 0.00016% adverse reaction rate.
Let’s go ahead and add in those 65 cases that were settled (with the understanding that settled cases are not settled because there’s evidence in their favor). If out of 67,250,521 doses, 76 injuries were caused by or contributed to by the HPV vaccine, that would give the vaccine a 0.0013% adverse reaction rate.
That’s what we mean by “low risk”: Sixteen one hundred thousands of one percent to thirteen ten thousandths of one percent.
What about vaccines as a whole? Of the vaccines covered by the NCVIP program, there were 2,236,678,735 doses during this time period. 2,852 claims were filed. 1,184 were dismissed. 156 conceded by HHS, 164 awarded by court decision, and 1,351 settled.
Even if I give every benefit of the doubt, that’s 1668 claims that were compensable. Out of 2,236,678,735 doses of vaccine administered. That’s an adverse event rate for all vaccines in toto of (gah! I hate exponents…left 7, right 2…) 0.00746%
So, yes, you absolutely can find me some stories of children killed by vaccines. You can find 11 stories in the last 9 years where a dose of HPV vaccine injured or killed someone, and each one of them is a tragedy, absolutely. I can find you 67,250,510 that didn’t.
*“Settlement: The petition is resolved via a negotiated settlement between the parties. This settlement is not an admission by the United States or the Secretary of Health and Human Services that the vaccine caused the petitioner’s alleged injuries, and, in settled cases, the Court does not determine that the vaccine caused the injury. A settlement therefore cannot be characterized as a decision by HHS or by the Court that the vaccine caused an injury. Claims may be resolved by settlement for many reasons, including consideration of prior court decisions; a recognition by both parties that there is a risk of loss in proceeding to a decision by the Court making the certainty of settlement more desirable; a desire by both parties to minimize the time and expense associated with litigating a case to conclusion; and a desire by both parties to resolve a case quickly and efficiently.” Page 9 of http://www.hrsa.gov/vaccinecompensation/statisticsreport.pdf
As for side effects and adverse events that may or may have not gone through NCVIP:
Most of the childhood vaccinations share the same side effects, and they’re listed on the patient information sheet that must be given out before the vaccination is given. The most common are pain at the injection site (duh, there’s a needle going in there) and mild to moderate fever (which is actually a sign that the immune system has recognized the invaders and is working to form antibodies against it.) The prevalence of these side effects vary by the vaccine. DTP causes swelling in soreness very commonly - so commonly that I can’t find numbers on it. HepB causes pain at the injection site somewhere between 3-29% of the time, depending on which study you’re looking at.
There are definitely some risks of adverse events more worrisome than pain and fever. Luckily, they are also much rarer than pain and fever. Approximately one person per 600,000 vaccinated with HepB (HBV) has an anaphylactic reaction to it. None of these have been fatal, but they are scary as hell. Anaphylactic reaction to HepB is an absolute medical contraindication to getting another HepB shot. As we’re always balancing risks with benefits, let’s look at what happens if people aren’t vaccinated for HBV: 5% of the US population will get HBV, and approximately 2,000-5,000 persons in each U.S. birth cohort will die as a result of HBV-related liver disease. (A birth cohort is all the people born in one year.) So you have a much greater chance of being harmed by HBV than the HepB vaccine, even when its scariest most potentially fatal risk is taken into account.
MMR, in addition to causing fever in 5-15% of those who get it, can also cause a rash. The rash happens in about 5% of the people vaccinated and goes away without treatment. Central nervous system disorders, including encephalitis and encephalopathy, happen in about 1 in 1 million doses - which is actually LESS than childhood encephalitis of unknown origin without the vaccine. That strongly suggests that the vaccine isn’t causing the encephalitis, that it would have happened with or without vaccination. Febrile (fever related) seizures may also occur, which are terrifying to witness, but don’t actually cause harm, nor do they increase the risk of developing epilepsy. Thrombocytopenia (a dangerously low white blood cell count) happens in about one case per 1 million of US children vaccinated, although the number is quite a bit higher in other countries. (I don’t know if they’re using the same vaccine or not.) But again, let’s compare that to actually getting the measles or rubella: 1 in 3000 children who get the measles or rubella will develop thrombocytopenia. MMR, like HepB, as occasionally caused an anaphylactic reaction. But when I say occasionally, I mean that in the 70 million doses given since we started tracking this in 1990, only 33 cases of anaphylactic reaction have been reported, and only 11 of those happened immediately after the injection and were likely caused by it. If you’re allergic to any of the vaccine components (including gelatin and neomycin, which are not unknown to cause allergic reactions) you and your doctor should carefully weigh the risks and benefits of vaccinating.
I could go on, but I bet at this point, people are wishing I wouldn’t. But all of this information is publicly available, and it all points to the same thing: for healthy non immune compromised patients who are not allergic to the vaccine components, vaccination is vastly safer than getting the disease you’re vaccinating for, and just generally pretty darn safe overall.