Abstract
This paper is concerned with a nonlocal epidemic model arising from the spread of an epidemic by oral-fecal transmission. Comparing with the previous works, here we extend the model in Capasso and Maddalena, Nonlinear Phenom Math Sci. 41:207–217 (1982) by including a spatial convolution term and a discrete delay term corresponding to the dispersal of bacteria in the environment and the latent period of the virus, respectively. Besides existence and asymptotic behavior, the main part of the paper is devoted to the stability of the traveling wavefronts under some monostable assumptions. By using a comparison theorem and the weighted energy method with a suitably selected weight function, we show that all the non-critical traveling waves are exponentially stable. Finally, we apply our results to a specific epidemic model and discuss the effect of time delay on the stability of wavefront.
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References
Brown KJ, Carr J. Deterministic epidemic waves of critical velocity. Math Proc Camb Philos Soc 1977;81:431–3.
Capasso V. Asymptotic stability for an integro-differential reaction-diffusion system. J Math Anal Appl 1984;103:575–88.
Capasso V, Maddalena L. A nonlinear diffusion system modelling the spread of oro-faecal diseases. Nonlinear Phenom Math Sci 1982;41:207–17.
Capasso V, Kunisch K. A reaction-diffusion system arising in modelling man-environment disease. Q Appl Math 1988;46:431–50.
Capasso V, Maddalena L. Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases. J Math Biol 1981;13:173–84.
Capasso V, Wilson RE. Analysis of a reaction-diffusion system modeling man-environment-man epidemics. Siam J Appl Math 1997;57:327–46.
Carr J, Chmaj A. Uniquensee of travelling waves for nonlocal monostable equations. Proc Amer Math Soc 2004;132:2433–39.
Hsu C-H, Yang T-S. Existence uniqueness monotonicity and asymptotic behaviour of travelling waves for epidemic models. Nonlinearity 2013;26:121–39.
Huang R, Mei M, Wang Y. Planar traveling waves for nonlocal dispersion equations with monostable nonlinearity. Discrete Contin Dyn Syst 2012;32:3621–49.
Huang R, Mei M, Zhang K, Zhang Q. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete Contin Dyn Syst 2016;36:1331–53.
Lee CT et al. Non-local concepts in models in biology. J Theor Biol 2001;210: 201–19.
Li W -T, Ruan S, Wang ZC. On the diffusive Nicholson’s blowflies equation with nonlocal delays. J Nonlinear Sci 2007;17:505–25.
Murray J. Mathematical biology. Berlin: Springer; 1993.
Mei M. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Proceedings of the 7th AIMS International Conference. Texas: Discrete Contin Dyn Syst., Supplement; 2009. p. 526–35.
Mei M, Ou C, Zhao X -Q. Global stability of monostable traveling waves for nonlocal time-delayed reaction-diffusion equations. SIAM J Math Anal 2010;42:2762–90.
Mei M, Lin C -K, Lin C -T, So J W -H. Traveling wavefronts for time-delayed reaction-diffusion equation. I. Local nonlinearity. J Differ Equ 2009;247: 495–510.
Mei M, Lin C -K, Lin C -T, So J W -H. Traveling wavefronts for time-delayed reaction-diffusion equation. II. Nonlocal nonlinearity. J Differ Equ 2009; 247:511–29.
Pan S. Invasion speed of a predator-prey system. Appl Math Lett 2017;74:46–51.
Pan S, Li W -T, Lin G. Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay. Nonlinear Anal 2010;72:3150–58.
Thieme HR, Zhao X -Q. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J Differ Equ 2003;195:430–70.
Wang ZC, Li W -T, Ruan S. Traveling fronts in monostable equations with nonlocal delayed effects. J Dyn Diff Equat 2008;20:573–607.
Wu S -L, Hsu C -H. Existence of entire solution for delayed monostable epidemic models. Trans Amer Math Soc 2015;368:6033–62.
Wu S -L, Hsu C -H, Xiao Y. Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems. J Differ Equ 2015;258: 1058–105.
Wu S -L, Liu S -Y. Existence and uniqueness of traveling waves for non-monotone integral equations with application. J Math Anal Appl 2010;365:729–41.
Wu S -L, Li W -T, Liu S -Y. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete Contin Dyn Syst Ser B 2012;17:347–66.
Xu D, Zhao X -Q. Bistable waves in an epidemic model. J Dyn Diff Equat 2004; 16:679–707.
Xu D, Zhao X -Q. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discret Contin Dynam syst Ser B 2005;5:1043–56.
Yang Y -R, Li W -T, Wu S -L. Exponential stability of traveling fronts in a diffusion epidemic system with delay. Nonlinear Anal 2011;12:1223–34.
Zhang L, Li W -T, Wu S -L. Multi-type entire solutions in a nonlocal dispersal epidemic model. J Dyn Diff Equat 2016;28:189–224.
Zhao X -Q, Wang W. Fisher waves in an epidemic model. Discret Contin Dyn Syst Ser B 2004;4:1117–28.
Zou X. Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type. J Comput Appl Math 2002;146:309–21.
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We are very grateful to the referees for their helpful comments and suggestions which led to an improvement of our original manuscript.
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Partially supported by the NSF of China (11671315), the NSF of Shaanxi Province of China (2017JM1003) and the Science and Technology Activities Funding of Shaanxi Province of China
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Guo, Z., Wu, SL. Stability of Traveling Wavefronts for a Nonlocal Dispersal System with Delay. J Dyn Control Syst 25, 175–195 (2019). https://doi.org/10.1007/s10883-018-9405-z
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DOI: https://doi.org/10.1007/s10883-018-9405-z