Bootstrap percolation has been used to describe opinion formation in society
and other social and natural phenomena. The formal equation of the bootstrap
percolation may have more than one solution, corresponding to several stable
fixed points of the corresponding iteration process. We construct a reversible
bootstrap percolation process, which converges to these extra solutions
displaying a hysteresis typical of discontinuous phase transitions. This
process provides a reasonable model for fake news spreading and the
effectiveness of fact checking. We show that sometimes it is not sufficient to
discard all the sources of fake news in order to reverse the belief of a
population that formed under the influence of these sources
