  Branching processes are widely used to model the viral epidemic evolution.
For more adequate investigation of viral epidemic modelling, we suggest to
apply branching processes with transport of particles usually called branching
random walks (BRWs). This allows to investigate not only the number of
particles (infected individuals), but also their spatial spread. We consider
two models of continuous-time BRWs on a multidimensional lattice in which the
transport of infected individuals is described by a symmetric random walk on a
multidimensional lattice whereas the processes of birth and death of infected
individuals are represented by a continuous-time Bienayme-Galton-Watson
processes at the lattice points (branching sources). A special attention is
paid to the properties of branching random walks with one branching source on
the lattice and finitely or infinitely many initial particles. We show that
there exists a kind of duality between the branching random walk with a finite
number of initial particles and the branching random walk with an infinite
number of initial particles, which is associated with the possibility of their
twofold description. The fact of duality is useful from the biological point of
view. Each of the models can be considered taking into account the vaccination
process. We suppose the vaccination to be a proportion of immune individuals in
the population, who are resistant to disease. For simplicity, in all our BRW
models, we assume that the vaccination process does not depend on time, what
allows to investigate spatial properties of viral evolution.
