  In this work, we derive a system of Boltzmann-type equations to describe the
spread of SARS-CoV-2 virus at the microscopic scale, that is by modeling the
human-to-human mechanisms of transmission. To this end, we consider two
populations, characterized by specific distribution functions, made up of
individuals without symptoms (population $1$) and infected people with symptoms
(population $2$). The Boltzmann operators model the interactions between
individuals within the same population and among different populations with a
probability of transition from one to the other due to contagion or, vice
versa, to recovery. In addition, the influence of innate and adaptive immune
systems is taken into account. Then, starting from the Boltzmann microscopic
description we derive a set of evolution equations for the size and mean state
of each population considered. Mathematical properties of such macroscopic
equations, as equilibria and their stability, are investigated and some
numerical simulations are performed in order to analyze the ability of our
model to reproduce the characteristic features of Covid-19.
