{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T21:00:46Z","timestamp":1777496446153,"version":"3.51.4"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T00:00:00Z","timestamp":1699315200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T00:00:00Z","timestamp":1699315200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"National Science and Technology Council, Taiwan; National Center for Theoretical Sciences, National Taiwan University; and Chang Gung Memorial Hospital"},{"name":"the National Natural Science Foundation of China; Guangdong Basic and Applied Basic Research Foundation","award":["12001237, 2020A1515110340"],"award-info":[{"award-number":["12001237, 2020A1515110340"]}]},{"name":"NSERC of Canada","award":["RGPIN-2019-05648"],"award-info":[{"award-number":["RGPIN-2019-05648"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2024,2]]},"DOI":"10.1007\/s00332-023-09985-z","type":"journal-article","created":{"date-parts":[[2023,11,7]],"date-time":"2023-11-07T17:01:45Z","timestamp":1699376505000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Nonlocal Reaction-Diffusion Model of West Nile Virus with Vertical Transmission"],"prefix":"10.1007","volume":"34","author":[{"given":"Feng-Bin","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruiwen","family":"Wu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Qiang","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,11,7]]},"reference":[{"key":"9985_CR1","doi-asserted-by":"publisher","first-page":"1553","DOI":"10.1007\/s00285-013-0677-3","volume":"68","author":"A Abdelrazec","year":"2014","unstructured":"Abdelrazec, A., Lenhart, S., Zhu, H.: Transmission dynamics of West Nile virus in mosquitoes and corvids and non-corvids. J. Math. Biol. 68, 1553\u20131582 (2014)","journal-title":"J. Math. Biol."},{"key":"9985_CR2","doi-asserted-by":"publisher","first-page":"134","DOI":"10.4269\/ajtmh.2012.11-0473","volume":"86","author":"JF Anderson","year":"2012","unstructured":"Anderson, J.F., Main, A.J., Cheng, G., Ferrandino, F.J., Fikrig, E.: Horizontal and vertical transmission of West Nile virus genotype NY99 by Culex salinarius and genotypes NY99 and WN02 by Culex tarsalis. Am. J. Trop. Med. Hyg. 86, 134\u2013139 (2012)","journal-title":"Am. J. Trop. Med. Hyg."},{"key":"9985_CR3","doi-asserted-by":"publisher","DOI":"10.1016\/j.onehlt.2021.100330","volume":"13","author":"A Angelou","year":"2021","unstructured":"Angelou, A., Kioutsioukis, I., Stilianakis, N.I.: A climate-dependent spatial epidemiological model for the transmission risk of West Nile virus at local scale. One Health 13, 100330 (2021)","journal-title":"One Health"},{"key":"9985_CR4","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1007\/s00285-006-0015-0","volume":"53","author":"N Baca\u00ebr","year":"2006","unstructured":"Baca\u00ebr, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421\u2013436 (2006)","journal-title":"J. Math. Biol."},{"key":"9985_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106758","volume":"115","author":"Z Bai","year":"2022","unstructured":"Bai, Z., Zhao, X.-Q.: Threshold dynamics of a nonlocal and time-delayed West Nile virus model with seasonality. Commun. Nonlinear Sci. Numer. Simul. 115, 106758 (2022)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"9985_CR6","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1016\/j.bulm.2005.01.002","volume":"67","author":"C Bowman","year":"2005","unstructured":"Bowman, C., Gumel, A.B., Van den Driessche, P., Wu, J., Zhu, H.: A mathematical model for assessing control strategies against West Nile virus. Bull. Math. Biol. 67, 1107\u20131133 (2005)","journal-title":"Bull. Math. Biol."},{"key":"9985_CR7","volume-title":"Nonlinear Functional Analysis","author":"K Deimling","year":"1988","unstructured":"Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1988)"},{"key":"9985_CR8","unstructured":"Daners,D., Koch Medina, P.: Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser., vol. 279, Longman, Harlow (1992)"},{"key":"9985_CR9","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/S0025-5564(98)10003-2","volume":"150","author":"L Esteva","year":"1998","unstructured":"Esteva, L., Vargas, C.: Analysis of a dengue disease transmission model. Math. Biosci. 150, 131\u2013151 (1998)","journal-title":"Math. Biosci."},{"key":"9985_CR10","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1007\/s10884-011-9224-3","volume":"23","author":"SB Hsu","year":"2011","unstructured":"Hsu, S.B., Wang, F.B., Zhao, X.-Q.: Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone. J. Dyn. Differ. Equ. 23, 817\u2013842 (2011)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9985_CR11","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s11538-005-9018-z","volume":"68","author":"M Lewis","year":"2006","unstructured":"Lewis, M., Renc\u0141awowicz, J., van den Driessche, P.: Traveling waves and spread rates for a West Nile virus model. Bull. Math. Biol. 68, 3\u201323 (2006)","journal-title":"Bull. Math. Biol."},{"key":"9985_CR12","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1007\/s00332-019-09579-8","volume":"30","author":"F Li","year":"2020","unstructured":"Li, F., Liu, J., Zhao, X.-Q.: A West Nile virus model with vertical transmission and periodic time delays. J Nonlinear Sci. 30, 449\u2013486 (2020)","journal-title":"J Nonlinear Sci."},{"key":"9985_CR13","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.jde.2020.09.019","volume":"272","author":"F Li","year":"2021","unstructured":"Li, F., Zhao, X.-Q.: Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease. J. Differ. Equ. 272, 127\u2013163 (2021)","journal-title":"J. Differ. Equ."},{"key":"9985_CR14","doi-asserted-by":"publisher","first-page":"3603","DOI":"10.1137\/16M1108832","volume":"49","author":"X Liang","year":"2017","unstructured":"Liang, X., Zhang, L., Zhao, X.-Q.: The principal eigenvalue for degenerate periodic reaction-diffusion systems. SIAM J. Math. Anal. 49, 3603\u20133636 (2017)","journal-title":"SIAM J. Math. Anal."},{"key":"9985_CR15","doi-asserted-by":"publisher","first-page":"1247","DOI":"10.1007\/s10884-017-9601-7","volume":"31","author":"X Liang","year":"2019","unstructured":"Liang, X., Zhang, L., Zhao, X.-Q.: 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 (2019)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9985_CR16","doi-asserted-by":"publisher","first-page":"1381","DOI":"10.1007\/s00285-017-1124-7","volume":"75","author":"Z Lin","year":"2017","unstructured":"Lin, Z., Zhu, H.: Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381\u20131409 (2017)","journal-title":"J. Math. Biol."},{"key":"9985_CR17","doi-asserted-by":"publisher","first-page":"145","DOI":"10.3934\/mbe.2006.3.145","volume":"3","author":"R Liu","year":"2006","unstructured":"Liu, R., Shuai, J., Wu, J., Zhu, H.: Modeling spatial spread of west nile virus and impact of directional dispersal of birds. Math. Biosci. Eng. 3, 145\u2013160 (2006)","journal-title":"Math. Biosci. Eng."},{"key":"9985_CR18","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.ijid.2017.08.011","volume":"64","author":"Y Lustig","year":"2017","unstructured":"Lustig, Y., Kaufman, Z., Mendelson, E., Orshan, L., Anis, E., Glazer, Y., Cohen, D., Shohat, T., Bassal, R.: Spatial distribution of West Nile virus in humans and mosquitoes in Israel, 2000\u20132014. Int. J. Infect. Dis. 64, 20\u201326 (2017)","journal-title":"Int. J. Infect. Dis."},{"key":"9985_CR19","doi-asserted-by":"publisher","first-page":"311","DOI":"10.3201\/eid0903.020628","volume":"9","author":"N Komar","year":"1999","unstructured":"Komar, N., Langevin, S., Hinten, S., Nemeth, N., Edwards, E.: Experimental infection of North American birds with the New York,: strain of West Nile Virus. Emerg. Infect. Dis. 9, 311\u2013322 (1999)","journal-title":"Emerg. Infect. Dis."},{"key":"9985_CR20","first-page":"1","volume":"321","author":"R Martin","year":"1990","unstructured":"Martin, R., Smith, H.L.: Abstract functional differential equations and reaction-diffusion systems. Trans. AMS 321, 1\u201344 (1990)","journal-title":"Trans. AMS"},{"key":"9985_CR21","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1137\/S0036141003439173","volume":"37","author":"P Magal","year":"2005","unstructured":"Magal, P., Zhao, X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM. J. Math. Anal. 37, 251\u2013275 (2005)","journal-title":"SIAM. J. Math. Anal."},{"key":"9985_CR22","doi-asserted-by":"publisher","first-page":"1451","DOI":"10.1088\/0951-7715\/25\/5\/1451","volume":"25","author":"R Peng","year":"2012","unstructured":"Peng, R., Zhao, X.-Q.: A reaction-diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25, 1451\u20131471 (2012)","journal-title":"Nonlinearity"},{"key":"9985_CR23","doi-asserted-by":"crossref","unstructured":"Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations. Springer (1984)","DOI":"10.1007\/978-1-4612-5282-5"},{"key":"9985_CR24","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.prevetmed.2008.01.006","volume":"85","author":"F Rubel","year":"2008","unstructured":"Rubel, F., Brugger, K., Hantel, M., Chvala-Mannsberger, S., Bakonyi, T., WeissenB\u00f6ck, H., Nowotny, N.: Explaining Usutu virus dynamics in Austria: model development and calibration. Prev. Vet. Med. 85, 166\u2013186 (2008)","journal-title":"Prev. Vet. Med."},{"key":"9985_CR25","unstructured":"Smith, H. L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, American Mathematical Society, Providence (1995)"},{"key":"9985_CR26","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/j.nonrwa.2017.10.017","volume":"41","author":"Y-C Shyu","year":"2018","unstructured":"Shyu, Y.-C., Chien, R.-N., Wang, F.-B.: Global dynamics of a West Nile virus model in a spatially variable habitat: real world applications. Nonlinear Anal. 41, 313\u2013333 (2018)","journal-title":"Nonlinear Anal."},{"key":"9985_CR27","unstructured":"The Center for Food Security and Public Health, West Nile Virus Infection (2003) http:\/\/www.cfsph.iastate.edu\/Factsheets\/pdfs\/wes-nile-fever.pdf"},{"key":"9985_CR28","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1016\/S0895-7177(01)00098-X","volume":"34","author":"M Thomas","year":"2001","unstructured":"Thomas, M., Urena, B.: A mathematical model describing the evolution of West Nile-like encephalitis in New York City. Math. Comput. Model. 34, 771\u2013781 (2001)","journal-title":"Math. Comput. Model."},{"key":"9985_CR29","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1137\/080732870","volume":"70","author":"HR Thieme","year":"2009","unstructured":"Thieme, H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188\u2013211 (2009)","journal-title":"SIAM J. Appl. Math."},{"key":"9985_CR30","doi-asserted-by":"publisher","first-page":"1498","DOI":"10.1137\/18M1236162","volume":"18","author":"F-B Wang","year":"2019","unstructured":"Wang, F.-B., Wu, R., Zhao, X.-Q.: A West Nile virus transmission model with periodic incubation periods. SIAM J. Appl. Dyn. Syst. 18, 1498\u20131535 (2019)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"9985_CR31","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1007\/s10884-008-9111-8","volume":"20","author":"W Wang","year":"2008","unstructured":"Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699\u2013717 (2008)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9985_CR32","doi-asserted-by":"publisher","first-page":"853","DOI":"10.1137\/16M1087916","volume":"16","author":"X Wang","year":"2017","unstructured":"Wang, X., Zhao, X.-Q.: Dynamics of a time-delayed Lyme disease model with seasonality. SIAM J. Appl. Dyn. Syst. 16, 853\u2013881 (2017)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"9985_CR33","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1098\/rspb.2003.2608","volume":"271","author":"MJ Wonham","year":"2004","unstructured":"Wonham, M.J., de Camino-Beck, T., Lewis, M.A.: An epidemiological model for West Nile virus: Invasion analysis and control applications. Proc. R. Soc. Lond. B Biol. Sci. 271, 501\u2013507 (2004)","journal-title":"Proc. R. Soc. Lond. B Biol. Sci."},{"key":"9985_CR34","doi-asserted-by":"crossref","unstructured":"Wu, J.: Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Science, vol. 119. Springer, Berlin (1996)","DOI":"10.1007\/978-1-4612-4050-1"},{"key":"9985_CR35","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s00332-018-9475-9","volume":"29","author":"R Wu","year":"2019","unstructured":"Wu, R., Zhao, X.-Q.: A reaction-diffusion model of vector-borne disease with periodic delays. J. Nonlinear Sci. 29, 29\u201364 (2019)","journal-title":"J. Nonlinear Sci."},{"key":"9985_CR36","doi-asserted-by":"publisher","first-page":"3011","DOI":"10.1016\/j.jde.2014.12.032","volume":"258","author":"L Zhang","year":"2015","unstructured":"Zhang, L., Wang, Z.-C., Zhao, X.-Q.: Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period. J. Differ. Equ. 258, 3011\u20133036 (2015)","journal-title":"J. Differ. Equ."},{"key":"9985_CR37","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/s10884-015-9425-2","volume":"29","author":"X-Q Zhao","year":"2017","unstructured":"Zhao, X.-Q.: Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. 29, 67\u201382 (2017)","journal-title":"J. Dyn. Differ. Equ."},{"key":"9985_CR38","unstructured":"Zhao, X.-Q.: Dynamical Systems in Population Biology, 2nd edn. Springer, New York (2017)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09985-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-023-09985-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-023-09985-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,15]],"date-time":"2024-02-15T17:05:19Z","timestamp":1708016719000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-023-09985-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,7]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["9985"],"URL":"https:\/\/doi.org\/10.1007\/s00332-023-09985-z","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,7]]},"assertion":[{"value":"14 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 October 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 November 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"13"}}