point out that the assumption about the finiteness of the variance of jumps is used only in Theorem \[T3\]. At last, in Section \[BRWV\] we generalize both BRW models for possible applications taking into account the vaccination process.

BRW with infinitely many initial particles {#BRWI}
==========================================

We consider a continuous-time branching random walk on $\mathbb{Z}^d$, $d\ge 1$, with an infinite number of initial particles. The process of birth and death of particles occurs only at a single lattice point called the branching source. Without loss of generality, we can assume that the source is at the origin. Let just one particle be at every point $x\in \mathbb{Z}^d$ at instant $t=0$. Informal description of the model is rather simple. Being outside of the source every particle can walk on $\mathbb{Z}^{d}$ until reaching the source. At the source it spends an exponentially distributed time and then either jumps to a point $x'\neq 0$ or can give a random number of offsprings (we assume that the particle itself is included in this number; in this case, when the number of offsprings equals zero we say that the particle dies). The newborn particles behave independently and stochastically in the same way as a parent 