{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T18:18:13Z","timestamp":1781374693737,"version":"3.54.1"},"reference-count":62,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T00:00:00Z","timestamp":1736985600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T00:00:00Z","timestamp":1736985600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071068"],"award-info":[{"award-number":["12071068"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12126206"],"award-info":[{"award-number":["12126206"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007847","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["20210101482JC"],"award-info":[{"award-number":["20210101482JC"]}],"id":[{"id":"10.13039\/100007847","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007847","name":"Natural Science Foundation of Jilin Province","doi-asserted-by":"publisher","award":["20210101482JC"],"award-info":[{"award-number":["20210101482JC"]}],"id":[{"id":"10.13039\/100007847","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1007\/s00285-024-02181-x","type":"journal-article","created":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T15:55:18Z","timestamp":1737042918000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Reaction\u2013advection\u2013diffusion model of highly pathogenic avian influenza with behavior of migratory wild birds"],"prefix":"10.1007","volume":"90","author":[{"given":"Liu","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4596-7093","authenticated-orcid":false,"given":"Meng","family":"Fan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,1,16]]},"reference":[{"key":"2181_CR1","unstructured":"American Chemical Society. Highly pathogenic bird flu virus can survive months on steel or glass at cooler temperatures. https:\/\/www.sciencedaily.com\/releases\/2010\/10\/101013124334.htm, 2024-04-17"},{"issue":"1","key":"2181_CR2","doi-asserted-by":"crossref","first-page":"33161","DOI":"10.1038\/srep33161","volume":"6","author":"J Belkhiria","year":"2016","unstructured":"Belkhiria J, Alkhamis MA, Mart\u00ednez-L\u00f3pez B (2016) Application of species distribution modeling for avian influenza surveillance in the United States considering the North America migratory flyways. Sci Rep 6(1):33161","journal-title":"Sci Rep"},{"key":"2181_CR3","unstructured":"Blue-winged Teal Ducks. Smithsonian\u2019s National Zoo and Conservation Biology Institute. https:\/\/nationalzoo.si.edu\/animals\/blue-winged-teal, 2024-04-17"},{"issue":"52","key":"2181_CR4","doi-asserted-by":"crossref","first-page":"1627","DOI":"10.1098\/rsif.2010.0126","volume":"7","author":"L Bourouiba","year":"2010","unstructured":"Bourouiba L, Wu J, Newman S, Takekawa J, Natdorj T, Batbayar N, Wikelski M (2010) Spatial dynamics of bar-headed geese migration in the context of H5N1. J R Soc Interface 7(52):1627\u20131639","journal-title":"J R Soc Interface"},{"issue":"2","key":"2181_CR5","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1137\/100803110","volume":"71","author":"L Bourouiba","year":"2011","unstructured":"Bourouiba L, Gourley SA, Liu R, Wu J (2011) The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza. SIAM J Appl Math 71(2):487\u2013516","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"2181_CR6","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1000346","volume":"5","author":"R Breban","year":"2009","unstructured":"Breban R, Drake JM, Stallknecht DE, Rohani P (2009) The role of environmental transmission in recurrent avian influenza epidemics. PLoS Comput Biol 5(4):e1000346","journal-title":"PLoS Comput Biol"},{"issue":"1","key":"2181_CR7","doi-asserted-by":"crossref","first-page":"11729","DOI":"10.1038\/s41598-022-13447-z","volume":"12","author":"V Caliendo","year":"2022","unstructured":"Caliendo V, Lewis NS, Pohlmann A, Baillie SR, Banyard AC, Beer M, Berhane Y (2022) Transatlantic spread of highly pathogenic avian influenza H5N1 by wild birds from Europe to North America in 2021. Sci Rep 12(1):11729","journal-title":"Sci Rep"},{"key":"2181_CR8","doi-asserted-by":"crossref","DOI":"10.1002\/0470871296","volume-title":"Spatial ecology via reaction\u2013diffusion equations","author":"RS Cantrell","year":"2004","unstructured":"Cantrell RS, Cosner C (2004) Spatial ecology via reaction\u2013diffusion equations. Wiley, New York"},{"key":"2181_CR9","unstructured":"Duck Unlimted. Northern shovelers and blue-winged and cinnamon teal are among the most closely related of North America\u2019s ducks. https:\/\/www.ducks.org\/conservation\/waterfowl-research-science\/understanding-waterfowl-the-blue-winged-ducks, 2024-04-17"},{"key":"2181_CR10","doi-asserted-by":"crossref","unstructured":"European Food Safety Authority, European Centre for Disease Prevention and Control and European Union Reference Laboratory for Avian Influenza, Adlhoch C, Fusaro A, Gonzales JL, Kuiken T, Baldinelli F (2021) Avian influenza overview December 2020\u2013February 2021. EFSA J 19(3):e06497","DOI":"10.2903\/j.efsa.2021.6497"},{"key":"2181_CR11","doi-asserted-by":"crossref","unstructured":"Fourni\u00e9 G, Walker P, Porphyre T, M\u00e9tras R, Pfeiffer D (2012) Mathematical models of infectious diseases in livestock: concepts and application to the spread of highly pathogenic avian influenza virus strain type H5N1. Health and Animal Agriculture in Developing Countries, pp 183\u2013205","DOI":"10.1007\/978-1-4419-7077-0_11"},{"issue":"4","key":"2181_CR12","doi-asserted-by":"crossref","first-page":"581","DOI":"10.3201\/eid1904.120903","volume":"19","author":"TL Fuller","year":"2013","unstructured":"Fuller TL, Gilbert M, Martin V, Cappelle J, Hosseini P, Njabo KY, Smith TB (2013) Predicting hotspots for influenza virus reassortment. Emerg Infect Dis 19(4):581","journal-title":"Emerg Infect Dis"},{"issue":"10","key":"2181_CR13","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0026118","volume":"6","author":"SJ Galsworthy","year":"2011","unstructured":"Galsworthy SJ, Ten Bosch QA, Hoye BJ, Heesterbeek JA, Klaassen M, Klinkenberg D (2011) Effects of infection-induced migration delays on the epidemiology of avian influenza in wild mallard populations. PLoS ONE 6(10):e26118","journal-title":"PLoS ONE"},{"issue":"3","key":"2181_CR14","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.sste.2012.01.002","volume":"3","author":"M Gilbert","year":"2012","unstructured":"Gilbert M, Pfeiffer DU (2012) Risk factor modelling of the spatio-temporal patterns of highly pathogenic avian influenza (HPAIV) H5N1: a review. Spat Spatio Temporal 3(3):173\u2013183","journal-title":"Spat Spatio Temporal"},{"issue":"2","key":"2181_CR15","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1137\/090767261","volume":"9","author":"SA Gourley","year":"2010","unstructured":"Gourley SA, Liu R, Wu J (2010) Spatiotemporal distributions of migratory birds: Patchy models with delay. SIAM J Appl Dyn Syst 9(2):589\u2013610","journal-title":"SIAM J Appl Dyn Syst"},{"key":"2181_CR16","unstructured":"Government of Canada. Poultry Import and export reports. https:\/\/agriculture.canada.ca\/en\/sector\/animal-industry\/poultry-egg-market-information\/imports-exports, 2024-04-17"},{"key":"2181_CR17","unstructured":"Hess P (1991) Periodic\u2013parabolic boundary value problems and positivity"},{"issue":"1","key":"2181_CR18","doi-asserted-by":"crossref","first-page":"2592","DOI":"10.1038\/s41598-020-59077-1","volume":"10","author":"JM Humphreys","year":"2020","unstructured":"Humphreys JM, Ramey AM, Douglas DC, Mullinax JM, Soos C, Link P, Prosser DJ (2020) Waterfowl occurrence and residence time as indicators of H5 and H7 avian influenza in North American Poultry. Sci Rep 10(1):2592","journal-title":"Sci Rep"},{"issue":"10","key":"2181_CR19","doi-asserted-by":"crossref","first-page":"2040","DOI":"10.1111\/1365-2664.13963","volume":"58","author":"JM Humphreys","year":"2021","unstructured":"Humphreys JM, Douglas DC, Ramey AM, Mullinax JM, Soos C, Link P, Prosser DJ (2021) The spatial\u2013temporal relationship of blue-winged teal to domestic poultry: movement state modelling of a highly mobile avian influenza host. J Appl Ecol 58(10):2040\u20132052","journal-title":"J Appl Ecol"},{"issue":"2","key":"2181_CR20","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.jtbi.2009.05.031","volume":"260","author":"E Jung","year":"2009","unstructured":"Jung E, Iwami S, Takeuchi Y, Jo TC (2009) Optimal control strategy for prevention of avian influenza pandemic. J Theor Biol 260(2):220\u2013229","journal-title":"J Theor Biol"},{"issue":"1","key":"2181_CR21","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1186\/s13567-023-01219-0","volume":"54","author":"S Lambert","year":"2023","unstructured":"Lambert S, Bauzile B, Mugnier A, Durand B, Vergne T, Paul MC (2023) A systematic review of mechanistic models used to study avian influenza virus transmission and control. Vet Res 54(1):96","journal-title":"Vet Res"},{"issue":"10","key":"2181_CR22","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0007289","volume":"4","author":"C Lebarbenchon","year":"2009","unstructured":"Lebarbenchon C, Albespy F, Brochet AL, Grandhomme V, Renaud F, Fritz H, Gauthier-Clerc M (2009) Spread of avian influenza viruses by common teal (Anas crecca) in Europe. PLoS ONE 4(10):e7289","journal-title":"PLoS ONE"},{"key":"2181_CR23","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1007\/s11425-013-4652-7","volume":"57","author":"C Lei","year":"2014","unstructured":"Lei C, Kim K, Lin Z (2014) The spreading frontiers of avian-human influenza described by the free boundary. Sci China Math 57:971\u2013990","journal-title":"Sci China Math"},{"issue":"5","key":"2181_CR24","doi-asserted-by":"crossref","first-page":"3603","DOI":"10.1137\/16M1108832","volume":"49","author":"X Liang","year":"2017","unstructured":"Liang X, Zhang L, Zhao XQ (2017) The principal eigenvalue for degenerate periodic reaction\u2013diffusion systems. SIAM J Math Anal 49(5):3603\u20133636","journal-title":"SIAM J Math Anal"},{"key":"2181_CR25","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1007\/s10884-017-9601-7","volume":"31","author":"X Liang","year":"2019","unstructured":"Liang X, Zhang L, Zhao XQ (2019) Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J Dyn Differ Equ 31:1247\u20131278","journal-title":"J Dyn Differ Equ"},{"issue":"6","key":"2181_CR26","doi-asserted-by":"crossref","first-page":"2963","DOI":"10.1111\/1365-2664.13154","volume":"55","author":"S Lisovski","year":"2018","unstructured":"Lisovski S, van Dijk JG, Klinkenberg D, Nolet BA, Fouchier RA, Klaassen M (2018) The roles of migratory and resident birds in local avian influenza infection dynamics. J Appl Ecol 55(6):2963\u20132975","journal-title":"J Appl Ecol"},{"issue":"4","key":"2181_CR27","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s00285-010-0346-8","volume":"62","author":"Y Lou","year":"2011","unstructured":"Lou Y, Zhao XQ (2011) A reaction\u2013diffusion malaria model with incubation period in the vector population. J Math Biol 62(4):543\u2013568","journal-title":"J Math Biol"},{"issue":"1","key":"2181_CR28","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1137\/S0036141003439173","volume":"37","author":"P Magal","year":"2005","unstructured":"Magal P, Zhao XQ (2005) Global attractors and steady states for uniformly persistent dynamical systems. SIAM J Math Anal 37(1):251\u2013275","journal-title":"SIAM J Math Anal"},{"issue":"1","key":"2181_CR29","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.jtbi.2008.04.011","volume":"254","author":"S Marino","year":"2008","unstructured":"Marino S, Hogue IB, Ray CJ, Kirschner DE (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254(1):178\u2013196","journal-title":"J Theor Biol"},{"issue":"1","key":"2181_CR30","first-page":"1","volume":"321","author":"RH Martin","year":"1990","unstructured":"Martin RH, Smith HL (1990) Abstract functional\u2013differential equations and reaction\u2013diffusion systems. T. Am. Math. Soc. 321(1):1\u201344","journal-title":"T. Am. Math. Soc."},{"key":"2181_CR31","unstructured":"National Wildlife Health Center. Distribution of Highly Pathogenic Avian Influenza in North America, 2021\/2022. US Geological Survey. https:\/\/www.usgs.gov\/centers\/nwhc\/science\/distribution-highly-pathogenic-avian-influenza-north-america-20212022#overview, 2024-04-17"},{"key":"2181_CR32","unstructured":"North America Population (2023) World Population Review. https:\/\/worldpopulationreview.com\/continents\/north-america-population, 2024-04-17"},{"issue":"5772","key":"2181_CR33","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1126\/science.1122438","volume":"312","author":"B Olsen","year":"2006","unstructured":"Olsen B, Munster VJ, Wallensten A, Waldenstr\u00f6m J, Osterhaus AD, Fouchier RA (2006) Global patterns of influenza A virus in wild birds. Science 312(5772):384\u2013388","journal-title":"Science"},{"issue":"1","key":"2181_CR34","doi-asserted-by":"crossref","first-page":"2175","DOI":"10.1038\/srep02175","volume":"3","author":"PS Pandit","year":"2013","unstructured":"Pandit PS, Bunn DA, Pande SA, Aly SS (2013) Modeling highly pathogenic avian influenza transmission in wild birds and poultry in West Bengal, India. Sci Rep 3(1):2175","journal-title":"Sci Rep"},{"issue":"3","key":"2181_CR35","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0017622","volume":"6","author":"DJ Prosser","year":"2011","unstructured":"Prosser DJ, Cui P, Takekawa JY, Tang M, Hou Y, Collins BM, Newman SH (2011) Wild bird migration across the Qinghai-Tibetan plateau: a transmission route for highly pathogenic H5N1. PLoS ONE 6(3):e17622","journal-title":"PLoS ONE"},{"key":"2181_CR36","volume-title":"Maximum principles in differential equations","author":"MH Protter","year":"2012","unstructured":"Protter MH, Weinberger HF (2012) Maximum principles in differential equations. Springer, Berlin"},{"key":"2181_CR37","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-18222-9","volume-title":"Superlinear parabolic problems","author":"P Quittner","year":"2019","unstructured":"Quittner P, Souplet P (2019) Superlinear parabolic problems. Springer, Berlin"},{"issue":"2","key":"2181_CR38","doi-asserted-by":"crossref","DOI":"10.1002\/jwmg.22171","volume":"86","author":"AM Ramey","year":"2022","unstructured":"Ramey AM, Hill NJ, DeLiberto TJ, Gibbs SE, Camille Hopkins M, Lang AS, Wan XF (2022) Highly pathogenic avian influenza is an emerging disease threat to wild birds in North America. J Wildl Manag 86(2):e22171","journal-title":"J Wildl Manag"},{"issue":"5","key":"2181_CR39","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1016\/j.meegid.2009.04.009","volume":"9","author":"B Roche","year":"2009","unstructured":"Roche B, Lebarbenchon C, Gauthier-Clerc M, Chang CM, Thomas F, Renaud F, Gu\u00e9gan JF (2009) Water-borne transmission drives avian influenza dynamics in wild birds: the case of the 2005\u20132006 epidemics in the Camargue area. Infect Genet Evol 9(5):800\u2013805","journal-title":"Infect Genet Evol"},{"issue":"12","key":"2181_CR40","first-page":"5135","volume":"13","author":"B Sharma","year":"2020","unstructured":"Sharma B, Khurana P, Kumar D (2020) Spread of avian influenza-a mathematical approach. Int J Eng Sci 13(12):5135\u20135141","journal-title":"Int J Eng Sci"},{"key":"2181_CR41","unstructured":"Statista. Total number of all chickens in the U.S. from 2000 to 2020. https:\/\/www.statista.com\/statistics\/196028\/total-number-of-all-chickens-in-the-us-since-2000\/, 2024-04-17"},{"issue":"2","key":"2181_CR42","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1080\/00401706.1987.10488205","volume":"29","author":"M Stein","year":"1987","unstructured":"Stein M (1987) Large sample properties of simulations using Latin hypercube sampling. Technometrics 29(2):143\u2013151","journal-title":"Technometrics"},{"issue":"s1","key":"2181_CR43","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1637\/8914-043009-Reg.1","volume":"54","author":"JY Takekawa","year":"2010","unstructured":"Takekawa JY, Newman SH, Xiao X, Prosser DJ, Spragens KA, Palm EC, Ji W (2010) Migration of waterfowl in the East Asian flyway and spatial relationship to HPAI H5N1 outbreaks. Avian Dis 54(s1):466\u2013476","journal-title":"Avian Dis"},{"key":"2181_CR44","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/BF00173267","volume":"30","author":"HR Thieme","year":"1992","unstructured":"Thieme HR (1992) Convergence results and a Poincar\u00e9\u2013Bendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30:755\u2013763","journal-title":"J Math Biol"},{"key":"2181_CR45","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1137\/080732870","volume":"70","author":"HR Thieme","year":"2009","unstructured":"Thieme HR (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70:188\u2013211","journal-title":"SIAM J Appl Math"},{"issue":"1","key":"2181_CR46","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1073\/pnas.1405216112","volume":"112","author":"H Tian","year":"2015","unstructured":"Tian H, Zhou S, Dong L, Van Boeckel TP, Cui Y, Newman SH, Xu B (2015) Avian influenza H5N1 viral and bird migration networks in Asia. P Natl Acad Sci 112(1):172\u2013177","journal-title":"P Natl Acad Sci"},{"issue":"2","key":"2181_CR47","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.jinf.2007.11.001","volume":"56","author":"S Tsiodras","year":"2008","unstructured":"Tsiodras S, Kelesidis T, Kelesidis I, Bauchinger U, Falagas ME (2008) Human infections associated with wild birds. J Infect 56(2):83\u201398","journal-title":"J Infect"},{"issue":"04","key":"2181_CR48","doi-asserted-by":"crossref","first-page":"1340004","DOI":"10.1142\/S0218339013400044","volume":"21","author":"N Tuncer","year":"2013","unstructured":"Tuncer N, Martcheva M (2013) Modeling seasonality in avian influenza H5N1. J Biol Syst 21(04):1340004","journal-title":"J Biol Syst"},{"issue":"2","key":"2181_CR49","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1137\/140966642","volume":"75","author":"NK Vaidya","year":"2015","unstructured":"Vaidya NK, Wahl LM (2015) Avian influenza dynamics under periodic environmental conditions. SIAM J Appl Math 75(2):443\u2013467","journal-title":"SIAM J Appl Math"},{"issue":"8","key":"2181_CR50","doi-asserted-by":"crossref","first-page":"2829","DOI":"10.3934\/dcdsb.2012.17.2829","volume":"17","author":"NK Vaidya","year":"2012","unstructured":"Vaidya NK, Wang FB, Zou X (2012) Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment. Discrete Contin Dyn B 17(8):2829\u20132848","journal-title":"Discrete Contin Dyn B"},{"issue":"3","key":"2181_CR51","first-page":"275","volume":"19","author":"XS Wang","year":"2012","unstructured":"Wang XS, Wu J (2012) Approximating periodic patterns and dynamic threshold for patchy model of migratory birds with delay. Can Appl Math Q 19(3):275\u2013292","journal-title":"Can Appl Math Q"},{"issue":"1","key":"2181_CR52","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1137\/090775890","volume":"71","author":"WD Wang","year":"2011","unstructured":"Wang WD, Zhao XQ (2011) A nonlocal and time-delayed reaction\u2013diffusion model of dengue transmission. SIAM J Appl Math 71(1):147\u2013168","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"2181_CR53","doi-asserted-by":"crossref","first-page":"1652","DOI":"10.1137\/120872942","volume":"11","author":"WD Wang","year":"2012","unstructured":"Wang WD, Zhao XQ (2012) Basic reproduction numbers for reaction\u2013diffusion epidemic models. SIAM J Appl Dyn Syst 11(4):1652\u20131673","journal-title":"SIAM J Appl Dyn Syst"},{"issue":"2","key":"2181_CR54","volume":"7","author":"RH Wang","year":"2012","unstructured":"Wang RH, Jin Z, Liu QX, van de Koppel J, Alonso D (2012) A simple stochastic model with environmental transmission explains multi-year periodicity in outbreaks of avian flu. PLoS ONE 7(2):e28873","journal-title":"PLoS ONE"},{"key":"2181_CR55","unstructured":"World Health Organization. Avian Influenza Weekly Update Number 883. https:\/\/www.who.int\/docs\/default-source\/wpro-documents\/emergency\/surveillance\/avian-influenza\/ai_20230217.pdf, 2024-04-17"},{"key":"2181_CR56","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jtbi.2014.02.038","volume":"352","author":"Y Xiao","year":"2014","unstructured":"Xiao Y, Sun X, Tang S, Wu J (2014) Transmission potential of the novel avian influenza A (H7N9) infection in mainland China. J Theor Biol 352:1\u20135","journal-title":"J Theor Biol"},{"issue":"1","key":"2181_CR57","doi-asserted-by":"crossref","first-page":"4846","DOI":"10.1038\/srep04846","volume":"4","author":"J Zhang","year":"2014","unstructured":"Zhang J, Jin Z, Sun GQ, Sun XD, Wang YM, Huang B (2014) Determination of original infection source of H7N9 avian influenza by dynamical model. Sci Rep 4(1):4846","journal-title":"Sci Rep"},{"key":"2181_CR58","doi-asserted-by":"crossref","first-page":"3011","DOI":"10.1016\/j.jde.2014.12.032","volume":"258","author":"L Zhang","year":"2015","unstructured":"Zhang L, Wang C, Zhao X (2015) Threshold dynamics of a time periodic reaction\u2013diffusion epidemic model with latent period. J Differ Equ 258:3011\u20133036","journal-title":"J Differ Equ"},{"issue":"1","key":"2181_CR59","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10884-015-9425-2","volume":"29","author":"XQ Zhao","year":"2017","unstructured":"Zhao XQ (2017) Basic reproduction ratios for periodic compartmental models with time delay. J Dyn Differ Equ 29(1):67\u201382","journal-title":"J Dyn Differ Equ"},{"key":"2181_CR60","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-56433-3","volume-title":"Dynamical systems in population biology","author":"XQ Zhao","year":"2017","unstructured":"Zhao XQ (2017) Dynamical systems in population biology, 2nd edn. Springer, Cham","edition":"2"},{"issue":"7","key":"2181_CR61","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s11538-023-01166-z","volume":"85","author":"X Zhao","year":"2023","unstructured":"Zhao X, Liu LD, Wang H, Fan M (2023) Ecological effects of predator harvesting and environmental noises on oceanic coral reefs. Bull Math Biol 85(7):59","journal-title":"Bull Math Biol"},{"key":"2181_CR62","doi-asserted-by":"crossref","DOI":"10.1016\/j.nonrwa.2022.103567","volume":"67","author":"T Zheng","year":"2022","unstructured":"Zheng T, Nie L, Zhu H, Luo Y, Teng Z (2022) Role of seasonality and spatial heterogeneous in the transmission dynamics of avian influenza. Nonlinear Anal Real 67:103567","journal-title":"Nonlinear Anal Real"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-024-02181-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-024-02181-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-024-02181-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,8]],"date-time":"2025-02-08T19:46:45Z","timestamp":1739044005000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-024-02181-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,16]]},"references-count":62,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["2181"],"URL":"https:\/\/doi.org\/10.1007\/s00285-024-02181-x","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,16]]},"assertion":[{"value":"21 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 December 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Both authors declare that there is no conflict of interest in this study.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"18"}}