What are you planning to do int he Carribean? If you will be at tourist areas, I would definitely not worry about it and go! It sounds like fun.
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Check the WHO page for the country you are going to and you can get the country's rates for the different diseases.
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I took my unvaxed dd to visit my dh's family in Kenya for 3 months from when she was 17 - 20 months. I did tons of research before I went and I have lived in Kenya so I knew the area where we were staying and I wasn't really worried. The most likely disease she would have caught was measles, and I'm not worried about her catching that. There had also been a small polio outbreak but it was in areas where we were not going and there had been no cases reported in over 6 months by the time we arrived.
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I think if you look at the actual statistics you will find that their risk of catching anything serious is very low, and extremely low if you will be staying in tourist areas.
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I would go and take your unvaxed kids and have fun!