Multiple travelling waves for an $SI$-epidemic model

Arnaud Ducrot; Michel Langlais; Pierre Magal

doi:10.3934/nhm.2013.8.171

Article Contents

2013, Volume 8, Issue 1: 171-190. Doi: 10.3934/nhm.2013.8.171

This issue Previous Article Archimedean copula and contagion modeling in epidemiology Next Article Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems

Multiple travelling waves for an $SI$-epidemic model

1.
Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence

Received: February 29, 2012

Revised: August 31, 2012

Published: February 2013

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Abstract

In this note we analyze a spatially structured SI epidemic model with vertical transmission, a logistic effect on vital dynamics and a density dependent incidence. The dynamics of the underlying system of ordinary differential equations are first shown to exhibit an infinite number of heteroclinic orbits connecting the trivial equilibrium with an interior equilibrium. Our mathematical study of the corresponding reaction-diffusion system is concerned with travelling wave solutions. Based on a detailed study of the center-unstable manifold around the interior equilibrium, we are able to prove the existence of an infinite number of travelling wave solutions connecting the trivial equilibrium and the interior equilibrium.

Keywords:

Spatially structured SI model,
multiple travelling wave solutions.

Mathematics Subject Classification: Primary: 35A18, 35K57; Secondary: 35B35, 35B40.

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