{"title":"Propagation direction of traveling waves for a class of nonlocal dispersal bistable epidemic models","authors":"Yu-Xia Hao, Guo-Bao Zhang","doi":"10.1016/j.aml.2025.109458","DOIUrl":null,"url":null,"abstract":"This work is devoted to studying the propagation direction of the following nonlocal dispersal epidemic model <ce:display><ce:formula><ce:label>(0.1)</ce:label><mml:math altimg=\"si1.svg\" display=\"block\"><mml:mfenced close=\"\" open=\"{\"><mml:mrow><mml:mtable align=\"axis\" columnlines=\"none none none none none none none none\" columnspacing=\"0.27em\" equalcolumns=\"false\" equalrows=\"false\"><mml:mtr><mml:mtd columnalign=\"right\"><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign=\"left\"><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mrow><mml:mo linebreak=\"badbreak\">∫</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:msub><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>α</mml:mi><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"></mml:mspace><mml:mspace width=\"0.2777em\"></mml:mspace><mml:mspace width=\"0.2777em\"></mml:mspace><mml:mspace width=\"2em\"></mml:mspace><mml:mspace width=\"1em\"></mml:mspace><mml:mspace width=\"0.16667em\"></mml:mspace></mml:mtd><mml:mtd columnalign=\"right\"></mml:mtd><mml:mtd columnalign=\"left\"><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"></mml:mspace><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign=\"left\"><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mrow><mml:mo linebreak=\"badbreak\">∫</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:msub><mml:mi>J</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mi>β</mml:mi><mml:mi>v</mml:mi><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"0.16667em\"></mml:mspace><mml:mspace width=\"1em\"></mml:mspace><mml:mspace width=\"1em\"></mml:mspace><mml:mspace width=\"1em\"></mml:mspace></mml:mtd><mml:mtd columnalign=\"right\"></mml:mtd><mml:mtd columnalign=\"left\"><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"></mml:mspace><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></ce:formula></ce:display>where <mml:math altimg=\"si4.svg\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"></mml:mspace><mml:msub><mml:mrow><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"></mml:mspace><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"></mml:mspace><mml:mi>β</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>. By discussing the case <mml:math altimg=\"si5.svg\" display=\"inline\"><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and using the monotone dependence of the wave speed of traveling wave solutions on parameters, we state the sufficient conditions for the speed <mml:math altimg=\"si6.svg\" display=\"inline\"><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mrow><mml:mi>c</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> under some calculations and analysis. Compared to the known works for classical diffusive epidemic models, we have to overcome difficulties due to the appearance of nonlocal dispersal operators in the current paper.","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"24 1","pages":""},"PeriodicalIF":2.9000,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.aml.2025.109458","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This work is devoted to studying the propagation direction of the following nonlocal dispersal epidemic model (0.1)∂u∂t=d1∫RJ(y−x)u(y,t)dy−u−u+αv,x∈R,t>0,∂v∂t=d2∫RJ(y−x)v(y,t)dy−v−βv+g(u),x∈R,t>0,where d1,d2,α,β>0. By discussing the case c=0 and using the monotone dependence of the wave speed of traveling wave solutions on parameters, we state the sufficient conditions for the speed c>0 and c<0 under some calculations and analysis. Compared to the known works for classical diffusive epidemic models, we have to overcome difficulties due to the appearance of nonlocal dispersal operators in the current paper.
期刊介绍：
The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.
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