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December  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:
 [1] L. Bourouiba, S. Gourley, R. Liu, J. Takekawa and J. Wu, Avian Influenza Spread and Transmission Dynamics. In Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases, John Wiley and Sons Inc., 2014. [2] N. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, London, 1986. [3] S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differ. Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031. [4] S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differ. Equaitons, 260 (2016), 218-240.  doi: 10.1016/j.jde.2015.08.038. [5] K. Deng and Y. Wu, Global stability for a nonlocal reaction-diffusion population model, Nonlinear Anal. Real World Appl., 25 (2015), 127-136.  doi: 10.1016/j.nonrwa.2015.03.006. [6] S. Gourley, R. Lui and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay, SIAM Journal on Applied Dynamical Systems, 9 (2010), 589-610.  doi: 10.1137/090767261. [7] S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differ. Equations, 259 (2015), 1409-1448.  doi: 10.1016/j.jde.2015.03.006. [8] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [9] Y. Hsieh, J. Wu, J. Fang, Y. Yang and J. Lou, Quantification of bird-to-bird and bird-to-human infections during 2013 novel H7N9 avian influenza outbreak in China, PLoS One, 9 (2014), e111834. doi: 10.1371/journal.pone.0111834. [10] R. Hu and Y. Yuan, Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay, J. Differ. Equations, 250 (2011), 2779-2806.  doi: 10.1016/j.jde.2011.01.011. [11] R. Liu, V. Duvvuri and J. Wu, Spread patternformation of H5N1-avian influenza and its implications for control strategies, Math. Model. Nat. Phenom., 3 (2008), 161-179.  doi: 10.1051/mmnp:2008048. [12] Z. P. Ma, Stability and Hopf bifurcation for a three-component reaction-diffeusion population model with delay effect, Appl. Math. Model., 37 (2013), 5984-6007.  doi: 10.1016/j.apm.2012.12.012. [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [14] J. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I travelling wavefronts on unbounded domains, Proceedings of the Royal Society: London A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789. [15] Y. Su, J. Wei and J. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dyn. Differ. Equ., 24 (2012), 897-925.  doi: 10.1007/s10884-012-9268-z. [16] X. Wang and J. Wu, Periodic systems of delay differential equations and avian influenza dynamics, J. Math. Sci., 201 (2014), 693-704. [17] Z. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946.  doi: 10.1090/S0002-9939-2012-11246-8. [18] Z. C. Wang, W. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312. [19] J. Wu, Theory and Applications of Partial Functional Differential Equations, Spinger-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [20] T. Yi and X. Zou, Global dynamics of a delay differential equaiton with spatial non-locality in an unbounded domain, J. Differ. Equations, 251 (2011), 2598-2611.  doi: 10.1016/j.jde.2011.04.027. [21] G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005. [22] W. Zuo and Y. Song, Stability and bifurcation analysis of a reaction-diffusion equaiton with spatio-temporal delay, J. Math. Anal. Appl., 430 (2015), 243-261.  doi: 10.1016/j.jmaa.2015.04.089.

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##### References:
 [1] L. Bourouiba, S. Gourley, R. Liu, J. Takekawa and J. Wu, Avian Influenza Spread and Transmission Dynamics. In Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases, John Wiley and Sons Inc., 2014. [2] N. Britton, Reaction-diffusion Equations and Their Applications to Biology, Academic Press, London, 1986. [3] S. Chen and J. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differ. Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031. [4] S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differ. Equaitons, 260 (2016), 218-240.  doi: 10.1016/j.jde.2015.08.038. [5] K. Deng and Y. Wu, Global stability for a nonlocal reaction-diffusion population model, Nonlinear Anal. Real World Appl., 25 (2015), 127-136.  doi: 10.1016/j.nonrwa.2015.03.006. [6] S. Gourley, R. Lui and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay, SIAM Journal on Applied Dynamical Systems, 9 (2010), 589-610.  doi: 10.1137/090767261. [7] S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differ. Equations, 259 (2015), 1409-1448.  doi: 10.1016/j.jde.2015.03.006. [8] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [9] Y. Hsieh, J. Wu, J. Fang, Y. Yang and J. Lou, Quantification of bird-to-bird and bird-to-human infections during 2013 novel H7N9 avian influenza outbreak in China, PLoS One, 9 (2014), e111834. doi: 10.1371/journal.pone.0111834. [10] R. Hu and Y. Yuan, Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay, J. Differ. Equations, 250 (2011), 2779-2806.  doi: 10.1016/j.jde.2011.01.011. [11] R. Liu, V. Duvvuri and J. Wu, Spread patternformation of H5N1-avian influenza and its implications for control strategies, Math. Model. Nat. Phenom., 3 (2008), 161-179.  doi: 10.1051/mmnp:2008048. [12] Z. P. Ma, Stability and Hopf bifurcation for a three-component reaction-diffeusion population model with delay effect, Appl. Math. Model., 37 (2013), 5984-6007.  doi: 10.1016/j.apm.2012.12.012. [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [14] J. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I travelling wavefronts on unbounded domains, Proceedings of the Royal Society: London A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789. [15] Y. Su, J. Wei and J. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dyn. Differ. Equ., 24 (2012), 897-925.  doi: 10.1007/s10884-012-9268-z. [16] X. Wang and J. Wu, Periodic systems of delay differential equations and avian influenza dynamics, J. Math. Sci., 201 (2014), 693-704. [17] Z. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946.  doi: 10.1090/S0002-9939-2012-11246-8. [18] Z. C. Wang, W. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312. [19] J. Wu, Theory and Applications of Partial Functional Differential Equations, Spinger-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [20] T. Yi and X. Zou, Global dynamics of a delay differential equaiton with spatial non-locality in an unbounded domain, J. Differ. Equations, 251 (2011), 2598-2611.  doi: 10.1016/j.jde.2011.04.027. [21] G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005. [22] W. Zuo and Y. Song, Stability and bifurcation analysis of a reaction-diffusion equaiton with spatio-temporal delay, J. Math. Anal. Appl., 430 (2015), 243-261.  doi: 10.1016/j.jmaa.2015.04.089.
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 $\alpha\in(0, 0.8)$
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