  We consider a continuous-time branching random walk on a multidimensional
lattice with two types of particles and an infinite number of initial
particles. The main results are devoted to the study of the generating function
and the limiting behavior of the moments of subpopulations generated by a
single particle of each type. We assume that particle types differ from each
other not only by the laws of branching, as in multi-type branching processes,
but also by the laws of walking. For a critical branching process at each
lattice point and recurrent random walk of particles, the effect of limit
spatial clustering of particles over the lattice is studied. A model
illustrating epidemic propagation is also considered. In this model, we
consider two types of particles: infected and immunity generated. Initially,
there is an infected particle that can infect others. Here, for the local
number of particles of each type at a lattice point, we study the moments and
their limiting behavior. Additionally, the effect of intermittency of the
infected particles is studied for a supercritical branching process at each
lattice point. Simulations are presented to demonstrate the effect of limit
clustering for the epidemiological model.
