# American Institute of Mathematical Sciences

• Previous Article
Threshold dynamics of a time periodic and two–group epidemic model with distributed delay
• MBE Home
• This Issue
• Next Article
Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network
October  2017, 14(5&6): 1515-1533. doi: 10.3934/mbe.2017079

## 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

* Corresponding author: Shuni Song

Received  August 29, 2016 Accepted  December 30, 2016 Published  May 2017

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.

Citation: Xue Zhang, Shuni Song, Jianhong Wu. Onset and termination of oscillation of disease spread through contaminated environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1515-1533. doi: 10.3934/mbe.2017079
##### References:

show all references

##### References:
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
The critical value of time delay $\tau_{2}$ with respect to varying $\alpha\in(0, 0.8)$
 [1] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 [2] Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095 [3] Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085 [4] Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102 [5] Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 [6] Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 [7] Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 [8] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [9] Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407 [10] Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63 [11] Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 [12] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033 [13] Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65 [14] Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048 [15] M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079 [16] Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784 [17] Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029 [18] Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255 [19] Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319 [20] Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

2018 Impact Factor: 1.313