Branching Random Walks and their Applications for Epidemic Modelling

Abstract

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.