Measles turns out to be wickedly contagious and an outbreak is currently underway in the Northwest US. It's the result of too few people being vaccinated. It turns out there's a simple way think about herd immunity.

R_{o} (r-nought) is the basic reproduction number: the average number of people an infected person will infect while he's contagious. It's a threshold. If it's less than 1.0, the disease will eventually die out. If greater than 1.0 it will spread and can become an epidemic. A lot goes into R_{o} - how contagious the disease is, how long it is contagious, social factors, variable latencies and so on. There are models for calculating it, but each disease tends to fall into a range making it easy for a non-epidemiologist to think about.

Measles had an R_{o} range between 12 and 18, smallpox 5 to 7, mumps 4 to 7, HIV/AIDs 2 to 6, the 1918 influenza 2.0 to 3.0, and Ebola between 1.2 and 2.5. We tend to be afraid of Ebola and flues like the 1918 variety as the mortality rate is very high. Even so, before vaccinations measles killed about two million people a year in a world with forty percent the current population. It is fair to say measles vaccines save about five million people a year.

The proportion of a population that needs to be vaccinated must exceed 1 - 1/Ro. To understand how it works consider a disease like smallpox with an R_{o} of 5. With no vaccination one infected person infects five others, but if four out of the five are vaccinated, only one will contract the disease. This make the effective R_{o} for the disease 1 (one person manages to infect one other). In this case if the faction of the population is greater than eighty percent, the disease won't become an epidemic.

To really protect the population from measles you need to have more than 1 - 1/18 or 0.944 of the population immunized .. 95% would be a minimum for the most infectious measles, 1 - 1/12 or about 92% for the least infectious. There are regions in the US with immunization rates under 80%. They are tempting an epidemic and are ground zero for the current epidemic.

When not enough people vaccinated (in some cases vaccines don't exist) and you lack effective treatments, quarantine is about the only option. Nigeria effectively did this in the Ebola outbreak a few years ago.

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You can talk about the spread of other things. People have done serious work on the R_{o} for ideas. Years ago two of us wrote an (ahem) April 1 math paper on how part of our corporate headquarters structure was a fairly efficient vaccine:-)

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