Social media fan the flames. As in many countries, sites targeting Ukrainians carry false claims - that vaccines cause autism, for example.
These intermingle with more blatant untruths and conspiracy theories.
"Unvaccinated children's immunity is stronger than in your vaccinated ones," asserted a post on Facebook in September in the name of Svetik Lamakhova in Oleksandriya, central Ukraine, who confirmed to Reuters she had expressed that view.
Another Facebook poster, named Elisaveta Shchepova, said that doctors and officials encouraging vaccination in Ukraine "do not need our health – they need our money, grief in family, illness and death." She did not respond to requests for comment.
Online advocates of vaccination are attacked. Olena Kudryashova, a 31-year-old fitness trainer, said she came down with measles when her daughter was just over 1, just as she had decided to go ahead and give her the shots. The baby caught measles too. The mother went on to ensure her baby was immunised, which she posted on Facebook along with pictures of herself and her child.
Her post, in December 2018, was shared 14,000 times and prompted more than 4,000 comments, many of them negative. "I seriously think you were bribed," said one. Another: "We have been vaccinated since the days of the USSR, and even now 95% of our children are vaccinated mercilessly - so why have we got a measles epidemic ??? Maybe because vaccination is a profitable fiction with many unexplored side effects ???"
Jan Sciegenny, a spokesman for Facebook, said the company takes misinformation regarding vaccines on its platform very seriously and is working on ways to connect people with authoritative information on both Facebook and Instagram.
The risks of leaving children without shots may be higher than previously thought. Two scientific studies published in October found measles actually damages children's immune systems, by eliminating antibodies they built up to diseases they had before they were infected. That makes vaccination even more important.
UNICEF says that on the request of the health ministry, it now procures vaccines for Ukraine's immunization campaigns against infectious diseases including measles, diphtheria, tetanus and polio.
But the doubters have momentum. This year, from March to August, the group "Vaccination. Free choice" held demonstrations to protest the requirement that children be inoculated.
Veronica Sidorenko, its head, said she doesn't trust data cited by the government and UNICEF, and believes a powerful pharmaceutical lobby is behind "mass hysteria" about the current measles outbreak. She said the outbreak of measles itself sparked an "intensified vaccine policy" which included what she described in an email to Reuters as "psychological pressure on parents and manipulation of statistics and information."
The city of Kiev, which has 3 million residents, had just 87 cases of measles in 2017.
Between January and June this year, it recorded 5,000.