  We consider a two-dimensional time-inhomogeneous birth-death process to model
the time-evolution of fake news in a population. The two components of the
process represent, respectively, (i) the number of individuals (say spreaders)
who know the rumor and intend to spread it, and (ii) the number of individuals
(say inactives) who have forgotten the rumor previously received. We employ the
probability generating function-based approach to obtain the moments and the
covariance of the two-dimensional process. We also analyze a new adimensional
index to study the correlation between the two components. Some special cases
are considered in which both the expected numbers of spreaders and inactives
are equal to suitable sigmoidal curves, which are often adopted in modelling
growth phenomena. Finally, we provide an application based on real data related
to the diffusion of fake news, in which the optimal choice of the sigmoidal
curve that fit the datasets is based on the minimization of the mean square
error and of the relative absolute error.\
