I've learned to cope with needles (donating blood has helped a lot), but as I have grown more comfortable with immunizations, an increasing share of the population has become less comfortable with them. In states that allow it, about 2.5% of children exemptions from immunization requirements for personal beliefs.
From an economic standpoint, skipping immunizations is a rational decision based on perceived risks of disease and vaccines. Because innoculation has been so successful for diseases like measles, parents percieve their children to be at low risk of contracting the disease. At the same time, misinformation about ties between vaccines and autism, as well as actual but very small risks of adverse reactions to vaccines, lead parents to perceive vaccines as risky. With these perceived risks, it makes sense for parents to choose to forgo the risk of vaccinating their children, while depending on herd immunity to protect them from the risks of the disease. Essentially, these parents are free-riding off of those who choose to vaccinate their children.
At the same time, they are imposing negative externalities on the rest of the population. Most vaccines are less than 100% effective, and some people cannot be vaccinated for medical reasons. Those who choose not to vaccinate are more likely to get sick and pass diseases on to vulnerable segments of the population.
So what does economics predict the outcome could or should be? If nothing is done to compel more immunizations, it is likely that the situation will reach an equilibrium as fewer people immunize and diseases like measles become more common. As the diseases become more common, their perceived (and actual) risk will increase, providing more people to immunize their children. Unfortunately, this equilibrium will not be efficient, since parents will not account for the herd immunity benefits of vaccinating, nor the negative externalities of failing to vaccinate. More people will get sick or even die than would if people were required to vaccinate.
One solution to this, besides requiring vaccination, could be to require people choosing not to vaccinate to pay a fee. That fee could be used to reimburse people who have adverse reactions to vaccinations (since people who don't vaccinate are free-riding off those who do vaccinate and face the small risk of the vaccination). The fee could also be used to pay for medical care for those who get sick from diseases, since these people are the ones who bear the cost of the negative externality from not vaccinating.