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Onset and termination of oscillation of disease spread through contaminated environment

  • * Corresponding author: Shuni Song

    * Corresponding author: Shuni Song 
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  • 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.

    Mathematics Subject Classification: Primary: 34K18, 35B35, 35B32; Secondary: 35K57, 92D40.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Solutions of model (1) approach to a positive steady state with $\tau_{2}=0.6$ and a periodically oscillatory orbit with $\tau_{2}=1.2$, respectively

    Figure 2.  The critical value of time delay $\tau_{2}$ with respect to varying $\alpha\in(0, 0.8)$

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