level of replication possibility of virus particles. For simplicity, in both BRW models, we assume that the vaccination process does not depend on time, in contrast, e.g., with [@BGMS-B:10], what allows to investigate spatial properties of the viral evolution.

The structure of the work is as follows. In this paper we consider only BRWs with a single branching source. In Section \[BRWI\], a formal description of a BRW with an infinite number of initial particles is recalled and the concept of a particle subpopulation is introduced. Besides, in Section \[BRWI\] the main differential and integral equations for the generating functions and the moments of the subpopulation particle number, as well as for the moments of a particle number of every subpopulation at an arbitrary lattice point, are obtained. In Section \[BRWIF\] we establish the duality of the BRW with an infinite number of initial particles and the BRW with one initial particle situated at an arbitrary lattice point. This result allows us to apply the theorems obtained earlier for a well-known model with one initial particle (see, e.g., [@YarBRW:r]), for studying the BRW with an infinite number of particles and a finite variance of jumps, see Theorem \[T3\]. Let us 