An epidemic model with nonlocal diffusion on networks

Elisabeth Logak; Isabelle  Passat

doi:10.3934/nhm.2016014

Article Contents

2016, Volume 11, Issue 4: 693-719. Doi: 10.3934/nhm.2016014

This issue Previous Article Decay rates for $1-d$ heat-wave planar networks Next Article

An epidemic model with nonlocal diffusion on networks

Elisabeth Logak^1, and
Isabelle Passat^2,

1.
University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, Cergy-Pontoise, F-95000
2.
University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, F-95000 Cergy-Pontoise

Received: October 31, 2015

Revised: January 31, 2016

Published: November 2016

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Abstract

We consider a SIS system with nonlocal diffusion which is the continuous version of a discrete model for the propagation of epidemics on a metapopulation network. Under the assumption of limited transmission, we prove the global existence of a unique solution for any diffusion coefficients. We investigate the existence of an endemic equilibrium and prove its linear stability, which corresponds to the loss of stability of the disease-free equilibrium. In the case of equal diffusion coefficients, we reduce the system to a Fisher-type equation with nonlocal diffusion, which allows us to study the large time behaviour of the solutions. We show large time convergence to either the disease-free or the endemic equilibrium.

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Mathematics Subject Classification: 35B35, 35B40, 35B51, 45J05, 47G20, 92C42, 92D30.

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