Branching Random Walks with Two Types of Particles on Multidimensional Lattices

TL;DR

This paper analyzes a multidimensional branching random walk with two particle types, studying their generating functions, moments, and clustering behavior, with applications to epidemic spread and immunity dynamics.

Contribution

It introduces a model with two particle types differing in branching and walking laws, and examines their limiting spatial clustering and epidemic propagation effects.

Findings

Particles tend to cluster spatially over time.

Infection and immunity dynamics influence particle distribution.

Simulations confirm clustering effects in epidemiological models.

Abstract

We consider a continuous-time branching random walk on a multidimensional lattice with two types of particles and an infinite number of initial particles. The main results are devoted to the study of the generating function and the limiting behavior of the moments of subpopulations generated by a single particle of each type. We assume that particle types differ from each other not only by the laws of branching, as in multi-type branching processes, but also by the laws of walking. For a critical branching process at each lattice point and recurrent random walk of particles, the effect of limit spatial clustering of particles over the lattice is studied. A model illustrating epidemic propagation is also considered. In this model, we consider two types of particles: infected and immunity generated. Initially, there is an infected particle that can infect others. Here, for the local number of particles of each type at a lattice point, we study the moments and their limiting behavior. Additionally, the effect of intermittency of the infected particles is studied for a supercritical branching process at each lattice point. Simulations are presented to demonstrate the effect of limit clustering for the epidemiological model.