Onset and termination of oscillation of disease spread through contaminated environment

Xue Zhang; Shuni Song; Jianhong Wu; Xue Zhang; Shuni Song; Jianhong Wu

doi:10.3934/mbe.2017079

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1515-1533. doi: 10.3934/mbe.2017079

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Onset and termination of oscillation of disease spread through contaminated environment

1.
College of Science, Northeastern University, Shenyang, Liaoning 110819, China
2.
Center for Disease Modelling, York Institute for Health Research, York University, Toronto, Ontario, M3J 1P3, Canada

Received: 29 August 2016 Accepted: 30 December 2016 Published: 01 October 2017
MSC : Primary: 34K18, 35B35, 35B32; Secondary: 35K57, 92D40

We consider a reaction diffusion equation with a delayed nonlocal nonlinearity and subject to Dirichlet boundary condition. The model equation is motivated by infection dynamics of disease spread (avian influenza, for example) through environment contamination, and the nonlinearity takes into account of distribution of limited resources for rapid and slow interventions to clean contaminated environment. We determine conditions under which an equilibrium with positive value in the interior of the domain (disease equilibrium) emerges and determine conditions under which Hope bifurcation occurs. For a fixed pair of rapid and slow response delay, we show that nonlinear oscillations can be avoided by distributing resources for both fast or slow interventions.
- Reaction-diffusion equation,
- nonlocal delay,
- Dirichlet boundary condition,
- stability,
- Hopf bifurcation
Citation: Xue Zhang, Shuni Song, Jianhong Wu. Onset and termination of oscillation of disease spread through contaminated environment[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1515-1533. doi: 10.3934/mbe.2017079

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Abstract

We consider a reaction diffusion equation with a delayed nonlocal nonlinearity and subject to Dirichlet boundary condition. The model equation is motivated by infection dynamics of disease spread (avian influenza, for example) through environment contamination, and the nonlinearity takes into account of distribution of limited resources for rapid and slow interventions to clean contaminated environment. We determine conditions under which an equilibrium with positive value in the interior of the domain (disease equilibrium) emerges and determine conditions under which Hope bifurcation occurs. For a fixed pair of rapid and slow response delay, we show that nonlinear oscillations can be avoided by distributing resources for both fast or slow interventions.

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