Mathematical Biosciences and Engineering (MBE)

Onset and termination of oscillation of disease spread through contaminated environment

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

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Xue Zhang - College of Science, Northeastern University, Shenyang, Liaoning 110819, China (email)
Shuni Song - College of Science, Northeastern University, Shenyang, Liaoning 110819, China (email)
Jianhong Wu - Center for Disease Modelling, York Institute for Health Research, York University, Toronto, Ontario, M3J 1P3, Canada (email)

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.

Keywords:  Reaction-diffusion equation, nonlocal delay, Dirichlet boundary condition, stability, Hopf bifurcation.
Mathematics Subject Classification:  Primary: 34K18, 35B35, 35B32; Secondary: 35K57, 92D40.

Received: August 2016;      Accepted: December 2016;      Available Online: May 2017.