## Why You Should Vaccinate

I came across this accidentally from some random blog, and thought it was a very good explanation of why we need vaccinations.

[Originally posted at: Paren(t)hesis]

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.

[read the rest of this post at: Paren(t)hesis]

I hope that you can take the time to read the rest of this post as the author makes a complex (and VERY important) topic simple to understand.