{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,24]],"date-time":"2026-05-24T19:03:34Z","timestamp":1779649414179,"version":"3.53.1"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:00:00Z","timestamp":1777420800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T00:00:00Z","timestamp":1777420800000},"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":["42450105"],"award-info":[{"award-number":["42450105"]}],"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":["12571541"],"award-info":[{"award-number":["12571541"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100020595","name":"National Science and Technology Council","doi-asserted-by":"publisher","award":["NSTC 112-2115-M-008-005-MY2"],"award-info":[{"award-number":["NSTC 112-2115-M-008-005-MY2"]}],"id":[{"id":"10.13039\/100020595","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":[[2026,5]]},"DOI":"10.1007\/s00285-026-02405-2","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:04:42Z","timestamp":1777453482000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Threshold dynamics of a HBV infection model with degenerate diffusion, DNA-containing capsids and time-delays in heterogeneous environment"],"prefix":"10.1007","volume":"92","author":[{"given":"Yu","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2443-0817","authenticated-orcid":false,"given":"Lan","family":"Zou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cheng-Hsiung","family":"Hsu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,29]]},"reference":[{"key":"2405_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3934\/dcds.2008.21.1","volume":"21","author":"LJS Allen","year":"2008","unstructured":"Allen LJS, Bolker BM, Lou Y, Nevai AL (2008) Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Contin Dyn Syst 21:1\u201320","journal-title":"Discrete Contin Dyn Syst"},{"key":"2405_CR2","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 (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53:421\u2013436","journal-title":"J Math Biol"},{"key":"2405_CR3","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s00285-017-1193-7","volume":"77","author":"Z Bai","year":"2018","unstructured":"Bai Z, Peng R, Zhao X-Q (2018) A reaction-diffusion malaria model with seasonality and incubation period. J Math Biol 77:201\u2013228","journal-title":"J Math Biol"},{"key":"2405_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-00547-7","volume-title":"Nonlinear Functional Analysis","author":"K Deimling","year":"1985","unstructured":"Deimling K (1985) Nonlinear Functional Analysis. Springer-Verlag, Berlin, Heidelberg"},{"issue":"6","key":"2405_CR5","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1016\/S2468-1253(24)00089-X","volume":"9","author":"PJ Easterbrook","year":"2024","unstructured":"Easterbrook PJ, Luhmann N, Bajis S et al (2024) WHO 2024 hepatitis B guidelines: An opportunity to transform care. Lancet Gastroenterol Hepatol 9(6):493\u2013495","journal-title":"Lancet Gastroenterol Hepatol"},{"key":"2405_CR6","first-page":"282","volume":"316","author":"Y Geng","year":"2018","unstructured":"Geng Y, Xu J, Hou J (2018) Discretization and dynamic consistency of a delayed and diffusive viral infection model. Appl Math Comput 316:282\u2013295","journal-title":"Appl Math Comput"},{"key":"2405_CR7","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1080\/17513750701769873","volume":"2","author":"SA Gourley","year":"2008","unstructured":"Gourley SA, Kuang Y, Nagy JD (2008) Dynamics of a delay differential equation model of hepatitis B virus infection. J Biol Dyn 2:140\u2013153","journal-title":"J Biol Dyn"},{"key":"2405_CR8","first-page":"4223","volume":"23","author":"T Guo","year":"2018","unstructured":"Guo T, Liu H, Xu C, Yan F (2018) Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate, Discrete Contin. Dyn. Syst. Ser B 23:4223\u20134242","journal-title":"Ser B"},{"key":"2405_CR9","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1007\/s10884-011-9224-3","volume":"23","author":"S-B Hsu","year":"2011","unstructured":"Hsu S-B, Wang F-B, Zhao X-Q (2011) Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone. J Dyn Diff Equat 23:817\u2013842","journal-title":"J Dyn Diff Equat"},{"key":"2405_CR10","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1016\/j.jde.2023.12.015","volume":"384","author":"Y Hu","year":"2024","unstructured":"Hu Y, Wang J (2024) Dynamics of an EIS spatially heterogeneous rabies model. J Differential Equations 384:326\u2013351","journal-title":"J Differential Equations"},{"key":"2405_CR11","doi-asserted-by":"publisher","first-page":"860","DOI":"10.1137\/21M1397234","volume":"54","author":"M Huang","year":"2022","unstructured":"Huang M, Wu S, Zhao X-Q (2022) Propagation dynamics for time-periodic and partially degenerate reaction-diffusion systems. SIAM J Math Anal 54:860\u20131897","journal-title":"SIAM J Math Anal"},{"key":"2405_CR12","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1016\/j.jde.2023.06.024","volume":"371","author":"M Huang","year":"2023","unstructured":"Huang M, Wu S, Zhao X-Q (2023) The principal eigenvalue for partially degenerate and periodic reaction-diffusion systems with time delay. J Differential Equations 371:396\u2013449","journal-title":"J Differential Equations"},{"issue":"1","key":"2405_CR13","first-page":"3","volume":"3","author":"MG Krein","year":"1948","unstructured":"Krein MG, Rutman MA (1948) Linear operator leaving invariant a cone in Banach spaces. Usp Mat Nauk 3(1):3\u201395 (23)","journal-title":"Usp Mat Nauk"},{"key":"2405_CR14","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 (2021) Global dynamics of a nonlocal periodic reaction-diffusion model of bluetongue disease. J Differential Equations 272:127\u2013163","journal-title":"J Differential Equations"},{"key":"2405_CR15","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/j.jviromet.2019.01.014","volume":"266","author":"M Li","year":"2019","unstructured":"Li M, Zu J (2019) The review of differential equation models of HBV infection dynamics. J Virol Methods 266:103\u2013113","journal-title":"J Virol Methods"},{"key":"2405_CR16","doi-asserted-by":"publisher","first-page":"3073","DOI":"10.1007\/s10884-023-10267-1","volume":"36","author":"Z Li","year":"2024","unstructured":"Li Z, Zhao X-Q (2024) Global dynamics of a nonlocal periodic reaction-diffusion model of Chikungunya disease. J Dyn Diff Equat 36:3073\u20133107","journal-title":"J Dyn Diff Equat"},{"key":"2405_CR17","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 (2019) Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J Dyn Diff Equat 31:1247\u20131278","journal-title":"J Dyn Diff Equat"},{"key":"2405_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2025.113284","volume":"437","author":"S Liu","year":"2025","unstructured":"Liu S, Zhao S, Bai Z, Lou Y, Sun G, Li L (2025) Global dynamics of a degenerate reaction-diffusion model for Brucellosis transmission. J Differential Equations 437:113284","journal-title":"J Differential Equations"},{"key":"2405_CR19","doi-asserted-by":"publisher","first-page":"3743","DOI":"10.1007\/s10884-023-10254-6","volume":"36","author":"Y Lou","year":"2024","unstructured":"Lou Y, Wang F-B (2024) A reaction-diffusion model with spatially inhomogeneous delays. J Dyn Diff Equat 36:3743\u20133758","journal-title":"J Dyn Diff Equat"},{"key":"2405_CR20","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1007\/s00285-010-0346-8","volume":"62","author":"Y Lou","year":"2011","unstructured":"Lou Y, Zhao X-Q (2011) A reaction-diffusion malaria model with incubation period in the vector population. J Math Biol 62:543\u2013568","journal-title":"J Math Biol"},{"key":"2405_CR21","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s00332-016-9344-3","volume":"27","author":"Y Lou","year":"2017","unstructured":"Lou Y, Zhao X-Q (2017) A theoretical approach to understanding population dynamics with seasonal developmental durations. J Nonlinear Sci 27:573\u2013603","journal-title":"J Nonlinear Sci"},{"key":"2405_CR22","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1137\/18M1182243","volume":"79","author":"P Magal","year":"2019","unstructured":"Magal P, Webb GF, Wu Y (2019) On the basic reproduction number of reaction-diffusion epidemic models. SIAM J Appl Math 79:284\u2013304","journal-title":"SIAM J Appl Math"},{"key":"2405_CR23","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1016\/j.cnsns.2014.08.036","volume":"22","author":"K Manna","year":"2015","unstructured":"Manna K, Chakrabarty SP (2015) Chronic hepatitis B infection and HBV DNA-containing capsids: Modeling and analysis. Commun Nonlinear Sci Numer Simulat 22:383\u2013395","journal-title":"Commun Nonlinear Sci Numer Simulat"},{"key":"2405_CR24","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s40314-015-0242-3","volume":"36","author":"K Manna","year":"2017","unstructured":"Manna K, Chakrabarty SP (2017) Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids. Comput Appl Math 36:525\u2013536","journal-title":"Comput Appl Math"},{"key":"2405_CR25","doi-asserted-by":"publisher","first-page":"1750062","DOI":"10.1142\/S1793524517500620","volume":"10","author":"K Manna","year":"2017","unstructured":"Manna K (2017) Dynamics of a diffusion-driven HBV infection model with capsids and time delay. Int J Biomath 10:1750062","journal-title":"Int J Biomath"},{"key":"2405_CR26","doi-asserted-by":"publisher","first-page":"918","DOI":"10.1080\/10236198.2015.1056524","volume":"21","author":"K Manna","year":"2015","unstructured":"Manna K, Chakrabarty SP (2015) Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids. J Differ Equ Appl 21:918\u2013933","journal-title":"J Differ Equ Appl"},{"key":"2405_CR27","first-page":"1","volume":"321","author":"R Martin Jr","year":"1990","unstructured":"Martin R Jr, Smith HL (1990) Abstract functional differential equations and reaction-diffusion systems. Trans Amer Math Soc 321:1\u201344","journal-title":"Trans Amer Math Soc"},{"key":"2405_CR28","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1002\/hep.21364","volume":"44","author":"JM Murray","year":"2006","unstructured":"Murray JM, Purcell RH, Wieland SF (2006) The half-life of Hepatitis B virions. Hepatology 44:1117\u20131121","journal-title":"Hepatology"},{"key":"2405_CR29","doi-asserted-by":"crossref","unstructured":"Nussbaum RD (1981) Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, in: Fixed Point Theory, in: Lecture Notes in Math., vol. 886, Springer-Verlag, pp 309-331","DOI":"10.1007\/BFb0092191"},{"key":"2405_CR30","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s00526-023-02436-3","volume":"62","author":"L Pang","year":"2023","unstructured":"Pang L, Wu S, Ruan S (2023) Long time behavior for a periodic Lotka-Volterra reaction-diffusion system with strong competition. Calc Var Partial Differ Equ 62:99","journal-title":"Calc Var Partial Differ Equ"},{"key":"2405_CR31","volume-title":"Nonlinear Parabolic and Elliptic Equations","author":"CV Pao","year":"1992","unstructured":"Pao CV (1992) Nonlinear Parabolic and Elliptic Equations. Plenum, New York"},{"key":"2405_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2022.126477","volume":"516","author":"H Shu","year":"2022","unstructured":"Shu H, Ma Z, Wang H (2022) Diffusive host-pathogen model revisited: nonlocal infections, incubation period and spatial heterogeneity. J Math Anal Appl 516:126477","journal-title":"J Math Anal Appl"},{"key":"2405_CR33","volume-title":"Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems","author":"HL Smith","year":"1995","unstructured":"Smith HL (1995) Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence"},{"key":"2405_CR34","doi-asserted-by":"publisher","first-page":"5084","DOI":"10.1016\/j.jde.2019.05.022","volume":"267","author":"P Song","year":"2019","unstructured":"Song P, Lou Y, Xiao Y (2019) A spatial SEIRS reaction-diffusion model in heterogeneous environment. J Differential Equations 267:5084\u20135114","journal-title":"J Differential Equations"},{"key":"2405_CR35","doi-asserted-by":"publisher","first-page":"755","DOI":"10.1007\/BF00173267","volume":"30","author":"HR Thieme","year":"1992","unstructured":"Thieme HR (1992) Convergence results and Poincar$$\\acute{\\rm e}$$-Bendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30:755\u2013763","journal-title":"J Math Biol"},{"key":"2405_CR36","doi-asserted-by":"publisher","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"},{"key":"2405_CR37","doi-asserted-by":"publisher","first-page":"2312","DOI":"10.1080\/00036811.2014.955797","volume":"93","author":"F-B Wang","year":"2014","unstructured":"Wang F-B, Huang Y, Zou X (2014) Global dynamics of a PDE in-host viral model. Appl Anal 93:2312\u20132329","journal-title":"Appl Anal"},{"key":"2405_CR38","doi-asserted-by":"crossref","unstructured":"Wang F-B, Wu R, Zhao X-Q (2024a) A nonlocal reaction-diffusion model of West Nile virus with vertical transmission. J Nonlinear Sci 34:13","DOI":"10.1007\/s00332-023-09985-z"},{"key":"2405_CR39","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.mbs.2007.05.004","volume":"210","author":"K Wang","year":"2007","unstructured":"Wang K, Wang W (2007) Propagation of HBV with spatial dependence. Math Biosci 210:78\u201395","journal-title":"Math Biosci"},{"key":"2405_CR40","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.jtbi.2007.11.007","volume":"253","author":"K Wang","year":"2008","unstructured":"Wang K, Wang W, Song S (2008) Dynamics of an HBV model with diffusion and delay. J Theor Biol 253:36\u201344","journal-title":"J Theor Biol"},{"key":"2405_CR41","doi-asserted-by":"crossref","unstructured":"Wang M (2021) Nonlinear Second Order Parabolic Equations, CRC Press","DOI":"10.1201\/9781003150169"},{"key":"2405_CR42","doi-asserted-by":"crossref","unstructured":"Wang W, Wang X, Wang H (2024b) Spatial dynamics of a generalized cholera model with nonlocal time delay in a heterogeneous environment. J Differential Equations 405:103\u2013150","DOI":"10.1016\/j.jde.2024.05.049"},{"key":"2405_CR43","doi-asserted-by":"publisher","first-page":"1652","DOI":"10.1137\/120872942","volume":"11","author":"W Wang","year":"2012","unstructured":"Wang W, Zhao X-Q (2012) Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J Appl Dyn Syst 11:1652\u20131673","journal-title":"SIAM J Appl Dyn Syst"},{"key":"2405_CR44","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1137\/15M1046277","volume":"77","author":"W Wang","year":"2017","unstructured":"Wang W, Zhao X-Q (2017) A periodic vector-bias malaria model with incubation period. SIAM J Appl Math 77:181\u2013201","journal-title":"SIAM J Appl Math"},{"key":"2405_CR45","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1016\/j.jde.2023.08.007","volume":"374","author":"S Wu","year":"2023","unstructured":"Wu S, Zhao H, Hsu C-H (2023) Spatial dynamics for a time-periodic epidemic model in discrete media. J Differential Equations 374:699\u2013736","journal-title":"J Differential Equations"},{"key":"2405_CR46","doi-asserted-by":"publisher","first-page":"4989","DOI":"10.1016\/j.jde.2017.12.027","volume":"264","author":"Y Wu","year":"2018","unstructured":"Wu Y, Zou X (2018) Dynamics and profiles of a diffusive host-pathogen system with distinct dispersal rates. J Differential Equations 264:4989\u20135024","journal-title":"J Differential Equations"},{"key":"2405_CR47","doi-asserted-by":"publisher","first-page":"499","DOI":"10.1016\/j.jtbi.2009.01.001","volume":"257","author":"R Xu","year":"2009","unstructured":"Xu R, Ma Z (2009) An HBV model with diffusion and time delay. J Theor Biol 257:499\u2013509","journal-title":"J Theor Biol"},{"key":"2405_CR48","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1111\/sapm.12643","volume":"152","author":"Y Yang","year":"2024","unstructured":"Yang Y, Chen J, Zou L (2024) Dynamical analysis of a degenerate and time delayed virus infection model with spatial heterogeneity. Stud Appl Math 152:279\u2013306","journal-title":"Stud Appl Math"},{"key":"2405_CR49","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s00332-024-10017-7","volume":"34","author":"Y Yang","year":"2024","unstructured":"Yang Y, Hsu C-H, Zou L, Zhou J (2024) Threshold dynamics of a degenerate diffusive HBV infection model with DNA-containing capsids in heterogeneous environment. J Nonlinear Sci 34:41","journal-title":"J Nonlinear Sci"},{"key":"2405_CR50","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1016\/j.jde.2016.03.014","volume":"261","author":"X Yu","year":"2016","unstructured":"Yu X, Zhao X-Q (2016) A nonlocal spatial model for Lyme disease. J Differential Equations 261:340\u2013372","journal-title":"J Differential Equations"},{"key":"2405_CR51","doi-asserted-by":"publisher","first-page":"278","DOI":"10.1016\/j.jde.2022.12.012","volume":"348","author":"Y Zha","year":"2023","unstructured":"Zha Y, Jiang W (2023) Global dynamics and asymptotic profiles for a degenerate Dengue fever model in heterogeneous environment. J Differential Equations 348:278\u2013319","journal-title":"J Differential Equations"},{"key":"2405_CR52","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1137\/23M154546X","volume":"23","author":"H Zhang","year":"2024","unstructured":"Zhang H, Wang K (2024) Dynamics on hepatitis B virus infection in vivo with interval delay, SIAM. J Appl Dyn Syst 23:205\u2013235","journal-title":"J Appl Dyn Syst"},{"key":"2405_CR53","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2024.109234","volume":"158","author":"H Zhang","year":"2024","unstructured":"Zhang H, Dai C, Wang K (2024) Stochastic dynamics on HBV infection in vivo with interval delay. Appl Math Lett 158:109234","journal-title":"Appl Math Lett"},{"key":"2405_CR54","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1016\/j.nonrwa.2013.06.005","volume":"15","author":"Y Zhang","year":"2014","unstructured":"Zhang Y, Xu Z (2014) Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response. Nonlinear Anal RWA 15:118\u2013139","journal-title":"Nonlinear Anal RWA"},{"key":"2405_CR55","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-56433-3","volume-title":"Dynamical Systems in Population Biology","author":"X-Q Zhao","year":"2017","unstructured":"Zhao X-Q (2017) Dynamical Systems in Population Biology, 2nd edn. Springer, New York","edition":"2"},{"key":"2405_CR56","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1137\/23M154546X","volume":"23","author":"H Zhong","year":"2024","unstructured":"Zhong H, Wang K (2024) Dynamics on hepatitis B virus infection in vivo with interval delay. SIAM J Appl Dyn Syst 23:205\u2013235","journal-title":"SIAM J Appl Dyn Syst"},{"key":"2405_CR57","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s00332-024-10048-0","volume":"34","author":"R Zhou","year":"2024","unstructured":"Zhou R, Wu S, Bao X (2024) Propagation dynamics for a degenerate delayed system with nonlocal dispersal in periodic habitats. J Nonlinear Sci 34:69","journal-title":"J Nonlinear Sci"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02405-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-026-02405-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02405-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,24]],"date-time":"2026-05-24T19:01:53Z","timestamp":1779649313000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-026-02405-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,29]]},"references-count":57,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["2405"],"URL":"https:\/\/doi.org\/10.1007\/s00285-026-02405-2","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,29]]},"assertion":[{"value":"9 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 October 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional research committee (TU ethics committee, reference 154\/17) and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Statement"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interest"}}],"article-number":"80"}}