attention will be paid to the properties of two BRW models with a single branching source on $\mathbb{Z}^d$, one of these models with a finite and another with an infinite number of initial particles. Some results about connection between these models are presented.

We will show that there exists a kind of duality between the BRW with a finite number of initial particles and the BRW with an infinite number of initial particles which is associated with the possibility of their twofold description, see Sections \[BRWI\] and \[BRWIF\], and which is useful from biological point of view. The transition from the model with a finite number of initial particles to another model, with an infinite number of particles, gives an opportunity to describe the behavior of some infected populations in a more natural way. For example, in connection with this it is worth to mention the phenomenon called virus persistence [@RG:17; @BAP:96] according to which some viruses can get into a latent state, weakly interfering in cell processes and become active only under specific conditions. Such a state, called latency, is typical for herpes viruses, in particular the varicella or Epstein-Barr virus [@BAP:96], which results in infectious mononucleosis, see, e.g., 