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

Abstract

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.

A Leslie-Gower Holling-type II ecoepidemic model

“…This branch of science developed [41], with the aim to control the spread of diseases by means of suitable vaccination policies [14,32,34]. A notable result of this kind has been the worldwide discontinuation of the vaccination against smallpox, ordered by the WHO in 1980, a decision for which mathematical models have played a most relevant role.…”

Epidemic fronts in complex networks with metapopulation structure

“…works having prescribed degree distributions -a generalization of the paradigmatic, spatial SIR model in one dimension, where the assumption of well mixed populations is relaxed to include complexity in agent interactions [18,19]. A schematic is shown in Fig.1-(b).…”

“…Fig. 2 shows the transport coefficients, (18), (19), and (20), as functions of T /T c and f , with partial scaling collapse (17) for the corresponding class of network configurations. The expected reaction-diffusion scaling can be observed near the critical point, and far away from the critical region, when T T c , q * ,…”

Predicting an epidemic based on syndromic surveillance

“…An alternative is to derive analytic expressions. From asymptotic formulas for the epidemic curve of well-mixed population [18], we can write in the limit s 0 ≈ 1:…”

“…where G(ρ) is a known function of ρ = α/β (tabulated in [18]) with the maximum G ≈ 0.4 at 1 ≤ ρ ≤ 2 and the saturation limit G ≈ 0.25 as ρ → ∞. From analysis of (1) - (3) it can be shown that as a first approximation the mixing inhomogeneity can be incorporated in (19), (20) by a simple substitute β → β 1/ν .…”