The Speed of Epidemic Waves in a One-Dimensional Lattice of SIR Models

Abstract

A one-dimensional lattice of SIR (susceptible/infected/removed) epidemic centres isconsidered numerically and analytically. The limiting solutions describing the behaviour of thestandard SIR model with a small number of initially infected individuals are derived, and expressionsfound for the duration of an outbreak. We study a model for a weakly mixed populationdistributed between the interacting centres. The centres are modelled as SIR nodes with interactionbetween sites determined by a diffusion-type migration process. Under the assumption of fastmigration, a one-dimensional lattice of SIR nodes is studied numerically with deterministic andrandom coupling, and travelling wave-like solutions are found in both cases. For weak coupling,the main part of the travelling wave is well approximated by the limiting SIR solution. Explicitformulae are found for the speed of the travelling waves and compared with results of numericalsimulation. Approximate formulae for the epidemic propagation speed are also derived whencoupling coefficients are randomly distributed, they allow us to estimate how the average speed inrandom media is slowed down.