一类具有非线性发生率与时滞的非局部扩散SIR模型的临界波的存在性
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摘要
研究了一类具有时滞的非局部扩散SIR传染病模型的行波解.首先,利用反证法证明了I是有界的,并根据I的有界性研究了波速c>c~*时行波解(波速大于最小波速的行波)的存在性.其次,利用c>c~*的行波的存在性结果证明了临界波(波速等于最小波速的行波)的存在性.最后,讨论了R_0对临界波存在性的影响.
The existence of traveling wave solutions for nonlocal dispersal SIR epidemic models with delay was studied. Firstly, the boundedness of I was proved by contradiction. Then according to the boundedness of I, the existence of traveling waves with c>c~* was established. Secondly, through further analysis of traveling waves with super-critical speeds, the existence of traveling waves with the critical speed was derived. Finally, the influence of basic reproduction number R_0 on the existence of c>c~* was discussed.
The existence of traveling wave solutions for nonlocal dispersal SIR epidemic models with delay was studied. Firstly, the boundedness of I was proved by contradiction. Then according to the boundedness of I, the existence of traveling waves with c>c~* was established. Secondly, through further analysis of traveling waves with super-critical speeds, the existence of traveling waves with the critical speed was derived. Finally, the influence of basic reproduction number R_0 on the existence of c>c~* was discussed.
引文
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[3] WANG H Y.Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems[J].Journal of Nonlinear Science,2011,21(5):747-783.
[4] WANG H Y,WANG X S.Traveling wave phenomena in a Kermack-Mckendrick SIR model[J].Journal of Dynamics & Differential Equations,2016,28(1):143-166.
[5] WANG J B,LI W T,YANG F Y.Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission[J].Communications in Nonlinear Science & Numerical Simulations,2015,27(1/3):136-152.
[6] FU S C,GUO J S,WU C C.Traveling wave solutions for a discrete diffusive epidemic model[J].Journal of Nonlinear and Convex Analysis,2016,17(9):1739-1751.
[7] ZHANG T R,WANG W D,WANG K F.Minimal wave speed for a class of non-cooperative diffusion-reaction system[J].Journal of Differential Equations,2016,260(3):2763-2791.
[8] WANG X S,WANG H Y,WU J H.Traveling waves of diffusive predator-prey systems:disease outbreak propagation[J].Discrete & Continuous Dynamical Systems,2017,32(9):3303-3324.
[9] LI Y,LI W T,YANG F Y.Traveling waves for a nonlocal dispersal SIR model with delay and external supplies[J].Applied Mathematics and Computation,2014,247:723-740.
[10] LI Y,LI W T,LIN G.Traveling waves of a delayed diffusive SIR epidemic model[J].Communications on Pure & Applied Analysis,2015,14(3):1001-1022.
[11] WANG Z C,WU J H.Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission[J].Proceedings Mathematical Physical & Engineering Sciences,2010,466(2113):237-261.
[12] LI W T,YANG F Y,MA C,et al.Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold[J].Discrete and Continuous Dynamical Systems(Series B),2014,19(2):467-484.
[13] BAI Z G,WU S L.Traveling waves in a delayed SIR epidemic model with nonlinear incidence[J].Applied Mathematics and Computation,2015,263:221-232.
[14] 邹霞,吴事良.一类具有非线性发生率与时滞的非局部扩散SIR模型的行波解[J].数学物理学报,2018,38(3):496-513.(ZOU Xia,WU Shiliang.Traveling waves in a nonlocal dispersal SIR epidemic model with delay and nonlinear incidence[J].Acta Mathematica Scientia,2018,38(3):496-513.(in Chinese))
[15] ZHANG S P,YANG Y R,ZHOU Y H.Traveling waves in a delayed SIR model with nonlocal dispersal and nonlinear incidence[J].Journal of Mathematical Physics,2018,59(1):011513.DOI:10.1063/1.5021761.
[16] CHEN Y Y,GUO J S,HAMEL F.Traveling waves for a lattice dynamical system arising in a diffusive endemic model[J].Nonlinearity,2016,30(6).DOI:10.1088/1361-6544/aa6b0a.
[17] YANG F Y,LI W T.Traveling waves in a nonlocal dispersal SIR model with critical wave speed[J].Journal of Mathematical Analysis & Applications,2017,458(2):1131-1146.
[18] WU C C.Existence of traveling waves with the critical speed for a discrete diffusive epidemic model[J].Journal of Differential Equations,2017,262(1):272-282.
[19] ZHANG G B,LI W T,WANG Z C.Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity[J].Journal of Differential Equations,2012,252(9):5096-5124.
[20] CAPASSO V,SERIO G.A generalization of the Kermack-McKendrick deterministic epidemic model[J].Mathematical Biosciences,1978,42(1/2):43-61.