In the present work, we consider a compartmental model to mimic the temporal diffusion of fake news. The three compartments in which the population is divided are represented by spreaders, ignorants and skeptics. The spreaders are individuals that know the fake news and transmit it among the population. The ignorants do not know the rumor and when contacted by a spreader they become spreaders too. The skeptics are individuals able to discover the evidence of the received fake news and thus they persuade the spreader to not transmit it anymore. Firstly, the model is studied from a deterministic point of view. In particular, we determine the asymptotic behavior of the compartments and the maximum number of spreaders and we evaluate its transient evolution through numerical evaluation. Then, a stochastic counterpart is presented. More specifically, we define a two-dimensional time-homogeneous birth-death process specifying its state space and the Kolmogorov equations for the transition probabilities. We use the probability generating function approach that leads to the partial differential equation for such function that unfortunately cannot be solved analitically. However, we obtain the system of differential equations solved by the means of the categories, coinciding with the deterministic case. Finally, we present an algorithm to simulate the paths of the compartments to perform a qualitative study of the process and of the absorbing time.