# American Institute of Mathematical Sciences

March  2013, 8(1): 171-190. doi: 10.3934/nhm.2013.8.171

## Multiple travelling waves for an $SI$-epidemic model

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

Received  March 2012 Revised  September 2012 Published  April 2013

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.
Citation: Arnaud Ducrot, Michel Langlais, Pierre Magal. Multiple travelling waves for an $SI$-epidemic model. Networks & Heterogeneous Media, 2013, 8 (1) : 171-190. doi: 10.3934/nhm.2013.8.171
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##### References:
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