  We consider integer-valued GARCH processes, where the count variable
conditioned on past values of the count and state variables follows a so-called
Skellam distribution. Using arguments for contractive Markov chains we prove
that the process has a unique stationary regime. Furthermore, we show
asymptotic regularity ($\beta$-mixing) with geometrically decaying coefficients
for the count process. These probabilistic results are complemented by a
statistical analysis, a few simulations as well as an application to recent
COVID-19 data.
