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 is considered numerically and analytically. The limiting solutions describing the behaviour of the standard SIR model with a small number of initially infected individuals are derived, and expressions found for the duration of an outbreak. We study a model for a weakly mixed population distributed between the interacting centres. The centres are modelled as SIR nodes with interaction between sites determined by a diffusion-type migration process. Under the assumption of fast migration, a one-dimensional lattice of SIR nodes is studied numerically with deterministic and random 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. Explicit formulae are found for the speed of the travelling waves and compared with results of numerical simulation. Approximate formulae for the epidemic propagation speed are also derived when coupling coefficients are randomly distributed, they allow us to estimate how the average speed in random media is slowed down. © 2008 EDP Sciences.