An epidemic model with nonlocal diffusion on networks

Elisabeth Logak; Isabelle  Passat; Elisabeth Logak; Isabelle  Passat

doi:10.3934/nhm.2016014

Networks and Heterogeneous Media

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

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An epidemic model with nonlocal diffusion on networks

Elisabeth Logak ^{1
,},
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: 01 November 2015 Revised: 01 February 2016
35B35, 35B40, 35B51, 45J05, 47G20, 92C42, 92D30.

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.
- Epidemic models,
- patchy environments,
- nonlocal diffusion,
- Fisher equation,
- complex networks
Citation: Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks[J]. Networks and Heterogeneous Media, 2016, 11(4): 693-719. doi: 10.3934/nhm.2016014

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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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