This paper introduces a dynamical model of the spread of fake news countered by both inoculation and debunking. The model assumes that a fake news message is published once, then spreads as a rumor by the spreaders, that is, by deceived individuals. Inoculation is conducted for part of the population in advance of the publication. Debunking begins after the publication and is carried out continuously by mass media and by skeptics, i.e., individuals who have internalized the debunking message. Building on the empirical literature, I account for the fact that fake news messages are normally more contagious via interpersonal communication than debunking messages. Numerical experiments with the model show that the effects of inoculation are limited. The proposed explanation for the limitations is that inoculation is not contagious while debunking by mass media provides less-than-linear growth of sceptics (given a constant intensity of debunking broadcasting). At the same time, the spread of a message through interpersonal communication yields nearly exponential growth of spreaders at early stages of the process. Therefore, if a fake news message is contagious enough, its spreaders quickly outnumber the sceptics. In such cases, inoculation and debunking cannot effectively counter fake news in the short run, although debunking enables sceptics to prevail in the long run.