{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T13:08:37Z","timestamp":1771938517853,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T00:00:00Z","timestamp":1745452800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T00:00:00Z","timestamp":1745452800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"Prime Minister\u2019s Research Fellowship, Ministry of Education, Govt. of India","award":["3701830"],"award-info":[{"award-number":["3701830"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s40314-025-03208-9","type":"journal-article","created":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T07:55:31Z","timestamp":1745481331000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Analysis of a nonlinear age-structured SIS epidemic model with spatial diffusion"],"prefix":"10.1007","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-7147-3918","authenticated-orcid":false,"given":"Soumak","family":"Nag","sequence":"first","affiliation":[]},{"given":"Suman Kumar","family":"Tumuluri","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,24]]},"reference":[{"key":"3208_CR1","series-title":"Real mathematical surveys and monographs","doi-asserted-by":"crossref","DOI":"10.1090\/surv\/165","volume-title":"Nonlocal diffusion problems","author":"F Andreu-Vaillo","year":"2010","unstructured":"Andreu-Vaillo F, Maz\u00f3n JM, Rossi JD, Toledo-Melero JJ (2010) Nonlocal diffusion problems, vol 165. Real mathematical surveys and monographs. American Mathematical Society\/Sociedad Matem\u00e1tica Espa\u00f1ola, Providence\/Madrid"},{"issue":"4","key":"3208_CR2","doi-asserted-by":"publisher","first-page":"1065","DOI":"10.1137\/0522069","volume":"22","author":"SN Busenberg","year":"1991","unstructured":"Busenberg SN, Iannelli M, Thieme HR (1991) Global behavior of an age-structured epidemic model. SIAM J Math Anal 22(4):1065\u20131080","journal-title":"SIAM J Math Anal"},{"issue":"1\u20133","key":"3208_CR3","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.cnsns.2014.12.016","volume":"24","author":"M-C Casab\u00e1n","year":"2015","unstructured":"Casab\u00e1n M-C, Cort\u00e9s J-C, Romero J-V, Rosell\u00f3 M-D (2015) Probabilistic solution of random SI-type epidemiological models using the random variable transformation technique. Commun Nonlinear Sci Numer Simul 24(1\u20133):86\u201397","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"3208_CR4","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.cnsns.2015.08.009","volume":"32","author":"M-C Casab\u00e1n","year":"2016","unstructured":"Casab\u00e1n M-C, Cort\u00e9s J-C, Navarro-Quiles A, Romero J-V, Rosell\u00f3 M-D, Villanueva R-J (2016) A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Commun Nonlinear Sci Numer Simul 32:199\u2013210","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"2","key":"3208_CR5","first-page":"161","volume":"66","author":"WL Chan","year":"1989","unstructured":"Chan WL, Guo BZ (1989) On the semigroups of age-size dependent population dynamics with spatial diffusion. Manuscr Math 66(2):161\u2013181","journal-title":"Manuscr Math"},{"issue":"8","key":"3208_CR6","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1016\/j.jde.2020.04.046","volume":"269","author":"A Chekroun","year":"2020","unstructured":"Chekroun A, Kuniya T (2020) Global threshold dynamics of an infection age-structured SIR epidemic model with diffusion under the Dirichlet boundary condition. J Differ Equ 269(8):117\u2013148","journal-title":"J Differ Equ"},{"issue":"1","key":"3208_CR7","doi-asserted-by":"publisher","first-page":"366","DOI":"10.1016\/j.jmaa.2005.09.042","volume":"313","author":"M Delgado","year":"2006","unstructured":"Delgado M, Molina-Becerra M, Su\u00e1rez A (2006) A nonlinear age-dependent model with spatial diffusion. J Math Anal Appl 313(1):366\u2013380","journal-title":"J Math Anal Appl"},{"issue":"4","key":"3208_CR8","doi-asserted-by":"publisher","first-page":"929","DOI":"10.1007\/s00028-010-0077-8","volume":"10","author":"G Di Blasio","year":"2010","unstructured":"Di Blasio G (2010) Mathematical analysis for an epidemic model with spatial and age structure. J Evol Equ 10(4):929\u2013953","journal-title":"J Evol Equ"},{"issue":"1","key":"3208_CR9","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s00205-008-0203-8","volume":"195","author":"A Ducrot","year":"2010","unstructured":"Ducrot A, Magal P, Ruan S (2010) Travelling wave solutions in multigroup age-structured epidemic models. Arch Ration Mech Anal 195(1):311\u2013331","journal-title":"Arch Ration Mech Anal"},{"issue":"2","key":"3208_CR10","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1016\/j.jde.2004.10.009","volume":"218","author":"Z Feng","year":"2005","unstructured":"Feng Z, Huang W, Castillo-Chavez C (2005) Global behavior of a multi-group SIS epidemic model with age structure. J Differ Equ 218(2):292\u2013324","journal-title":"J Differ Equ"},{"issue":"1","key":"3208_CR11","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1006\/jmaa.1994.1193","volume":"184","author":"BZ Guo","year":"1994","unstructured":"Guo BZ, Chan WL (1994) On the semigroup for age dependent population dynamics with spatial diffusion. J Math Anal Appl 184(1):190\u2013199","journal-title":"J Math Anal Appl"},{"issue":"1\u20132","key":"3208_CR12","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/0025-5564(81)90075-4","volume":"54","author":"ME Gurtin","year":"1981","unstructured":"Gurtin ME, MacCamy RC (1981) Diffusion models for age-structured populations. Math Biosci 54(1\u20132):49\u201359","journal-title":"Math Biosci"},{"key":"3208_CR13","doi-asserted-by":"publisher","DOI":"10.1002\/mma.8668","author":"J Huo","year":"2022","unstructured":"Huo J, Yuan R (2022) Mathematical analysis of autonomous and non autonomous age-structured reaction\u2013diffusion\u2013advection population model. Math Meth Appl Sci. https:\/\/doi.org\/10.1002\/mma.8668","journal-title":"Math Meth Appl Sci"},{"issue":"1","key":"3208_CR14","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/BF02573475","volume":"49","author":"W Huyer","year":"1994","unstructured":"Huyer W (1994) Semigroup formulation and approximation of a linear age-dependent population problem with spatial diffusion. Semigroup Forum 49(1):99\u2013114","journal-title":"Semigroup Forum"},{"issue":"1","key":"3208_CR15","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1080\/08898488809525260","volume":"1","author":"H Inaba","year":"1988","unstructured":"Inaba H (1988) A semigroup approach to the strong ergodic theorem of the multistate stable population process. Math Popul Stud 1(1):49\u201377","journal-title":"Math Popul Stud"},{"issue":"4","key":"3208_CR16","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/BF00178326","volume":"28","author":"H Inaba","year":"1990","unstructured":"Inaba H (1990) Threshold and stability results for an age-structured epidemic model. J Math Biol 28(4):411\u2013434","journal-title":"J Math Biol"},{"issue":"2","key":"3208_CR17","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1080\/08898480.2014.891905","volume":"21","author":"H Inaba","year":"2014","unstructured":"Inaba H (2014) On a pandemic threshold theorem of the early Kermack\u2013McKendrick model with individual heterogeneity. Math Popul Stud 21(2):95\u2013111","journal-title":"Math Popul Stud"},{"key":"3208_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-10-0188-8","volume-title":"Age-structured population dynamics in demography and epidemiology","author":"H Inaba","year":"2017","unstructured":"Inaba H (2017) Age-structured population dynamics in demography and epidemiology. Springer, Singapore"},{"issue":"1","key":"3208_CR19","doi-asserted-by":"publisher","first-page":"Paper No. 5","DOI":"10.1007\/s00285-021-01634-x","volume":"83","author":"H Kang","year":"2021","unstructured":"Kang H, Ruan S (2021a) Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion. J Math Biol 83(1):Paper No. 5","journal-title":"J Math Biol"},{"key":"3208_CR20","doi-asserted-by":"publisher","first-page":"430","DOI":"10.1016\/j.jde.2021.01.004","volume":"278","author":"H Kang","year":"2021","unstructured":"Kang H, Ruan S (2021b) Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions. J Differ Equ 278:430\u2013462","journal-title":"J Differ Equ"},{"issue":"2","key":"3208_CR21","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1007\/s10884-020-09860-5","volume":"34","author":"H Kang","year":"2022","unstructured":"Kang H, Ruan S, Yu X (2022) Age-structured population dynamics with nonlocal diffusion. J Dyn Differ Equ 34(2):789\u2013823","journal-title":"J Dyn Differ Equ"},{"issue":"4","key":"3208_CR22","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s10444-004-7639-7","volume":"25","author":"M-Y Kim","year":"2006","unstructured":"Kim M-Y (2006) Global dynamics of approximate solutions to an age-structured epidemic model with diffusion. Adv Comput Math 25(4):451\u2013474","journal-title":"Adv Comput Math"},{"key":"3208_CR23","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.nonrwa.2014.10.006","volume":"23","author":"T Kuniya","year":"2015","unstructured":"Kuniya T, Oizumi R (2015) Existence result for an age-structured SIS epidemic model with spatial diffusion. Nonlinear Anal Real World Appl 23:196\u2013208","journal-title":"Nonlinear Anal Real World Appl"},{"issue":"2","key":"3208_CR24","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1007\/s13160-018-0300-5","volume":"35","author":"T Kuniya","year":"2018","unstructured":"Kuniya T, Inaba H, Yang J (2018) Global behavior of SIS epidemic models with age structure and spatial heterogeneity. Jpn J Ind Appl Math 35(2):669\u2013706","journal-title":"Jpn J Ind Appl Math"},{"issue":"3","key":"3208_CR25","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1137\/0516037","volume":"16","author":"M Langlais","year":"1985","unstructured":"Langlais M (1985) A nonlinear problem in age-dependent population diffusion. SIAM J Math Anal 16(3):510\u2013529","journal-title":"SIAM J Math Anal"},{"issue":"1","key":"3208_CR26","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/0022-0396(81)90083-8","volume":"39","author":"RC MacCamy","year":"1981","unstructured":"MacCamy RC (1981) A population model with nonlinear diffusion. J Differ Equ 39(1):52\u201372","journal-title":"J Differ Equ"},{"key":"3208_CR27","series-title":"Applied mathematical sciences","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-01506-0","volume-title":"Theory and applications of abstract semilinear Cauchy problems","author":"P Magal","year":"2018","unstructured":"Magal P, Ruan S (2018) Theory and applications of abstract semilinear Cauchy problems, vol 201. Applied mathematical sciences. Springer, Cham (With a foreword by Glenn Webb)"},{"key":"3208_CR28","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1137\/0119060","volume":"19","author":"I Marek","year":"1970","unstructured":"Marek I (1970) Frobenius theory of positive operators: comparison theorems and applications. SIAM J Appl Math 19:607\u2013628","journal-title":"SIAM J Appl Math"},{"issue":"1","key":"3208_CR29","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1137\/1023003","volume":"23","author":"AJ Pritchard","year":"1981","unstructured":"Pritchard AJ, Zabczyk J (1981) Stability and stabilizability of infinite-dimensional systems. SIAM Rev 23(1):25\u201352","journal-title":"SIAM Rev"},{"key":"3208_CR30","first-page":"53","volume":"15","author":"I Sawashima","year":"1964","unstructured":"Sawashima I (1964) On spectral properties of some positive operators. Nat Sci Rep Ochanomizu Univ 15:53\u201364","journal-title":"Nat Sci Rep Ochanomizu Univ"},{"key":"3208_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.matcom.2022.08.001","volume":"204","author":"Y Tan","year":"2023","unstructured":"Tan Y, Cai Y, Wang X, Peng Z, Wang K, Yao R, Wang W (2023) Stochastic dynamics of an sis epidemiological model with media coverage. Math Comput Simul 204:1\u201327","journal-title":"Math Comput Simul"},{"issue":"4","key":"3208_CR32","doi-asserted-by":"publisher","first-page":"1366","DOI":"10.1137\/090750044","volume":"41","author":"C Walker","year":"2009","unstructured":"Walker C (2009) Positive equilibrium solutions for age- and spatially-structured population models. SIAM J Math Anal 41(4):1366\u20131387","journal-title":"SIAM J Math Anal"},{"issue":"1","key":"3208_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10231-010-0133-6","volume":"190","author":"C Walker","year":"2011","unstructured":"Walker C (2011) Bifurcation of positive equilibria in nonlinear structured population models with varying mortality rates. Ann Mat Pura Appl (4) 190(1):1\u201319","journal-title":"Ann Mat Pura Appl (4)"},{"key":"3208_CR34","doi-asserted-by":"crossref","unstructured":"Webb GF (1980\/1981) An age-dependent epidemic model with spatial diffusion. Arch Rational Mech Anal 75(1):91\u2013102","DOI":"10.1007\/BF00284623"},{"key":"3208_CR35","series-title":"Monographs and textbooks in pure and applied mathematics","volume-title":"Theory of nonlinear age-dependent population dynamics","author":"GF Webb","year":"1985","unstructured":"Webb GF (1985) Theory of nonlinear age-dependent population dynamics, vol 89. Monographs and textbooks in pure and applied mathematics. Marcel Dekker, Inc., New York"},{"issue":"11","key":"3208_CR36","doi-asserted-by":"publisher","first-page":"2177","DOI":"10.1007\/s11425-020-1740-1","volume":"63","author":"W-B Xu","year":"2020","unstructured":"Xu W-B, Li W-T, Ruan S (2020) Spatial propagation in an epidemic model with nonlocal diffusion: the influences of initial data and dispersals. Sci China Math 63(11):2177\u20132206","journal-title":"Sci China Math"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03208-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03208-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03208-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,19]],"date-time":"2025-05-19T14:11:01Z","timestamp":1747663861000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03208-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,24]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["3208"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03208-9","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,24]]},"assertion":[{"value":"17 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 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":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"250"}}