{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T22:31:01Z","timestamp":1774477861418,"version":"3.50.1"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"S1","license":[{"start":{"date-parts":[[2025,4,30]],"date-time":"2025-04-30T00:00:00Z","timestamp":1745971200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,30]],"date-time":"2025-04-30T00:00:00Z","timestamp":1745971200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2025,9]]},"DOI":"10.1007\/s12190-025-02474-4","type":"journal-article","created":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T07:48:27Z","timestamp":1746172107000},"page":"725-760","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Traveling wave solution for a delayed reaction-diffusion two-group SIR epidemic model with a generalized nonlinear incidence function"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4030-6499","authenticated-orcid":false,"given":"Salih","family":"Djilali","sequence":"first","affiliation":[]},{"given":"Rassim","family":"Darazirar","sequence":"additional","affiliation":[]},{"given":"Ibrahim","family":"Alraddadi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,30]]},"reference":[{"issue":"1","key":"2474_CR1","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s00285-017-1191-9","volume":"77","author":"D. Bichara","year":"2018","unstructured":"Bichara, D., Iggidr, A.: Multi-patch and multi-group epidemic models: a new framework. J. Math Biol. 77(1), 107\u2013134 (2018)","journal-title":"J. Math Biol."},{"key":"2474_CR2","doi-asserted-by":"crossref","unstructured":"Cantrell, S., Cosner, C., Ruan, S.: Modeling spatial spread of communicable diseases involving animal hosts. Spatial Ecol. Chapman And Hall\/CRC 319\u2013342 (2009)","DOI":"10.1201\/9781420059861-20"},{"issue":"10","key":"2474_CR3","doi-asserted-by":"publisher","first-page":"2891","DOI":"10.1088\/0951-7715\/24\/10\/012","volume":"24","author":"A. Ducrot","year":"2011","unstructured":"Ducrot, A., Magal, P.: Travelling wave solutions for an infection-age structured epidemic model with external supplies. Nonlinearity 24(10), 2891 (2011)","journal-title":"Nonlinearity"},{"key":"2474_CR4","doi-asserted-by":"crossref","unstructured":"Freedman, H.I., Tang, M.X., Ruan, S.G.: Uniform persistence and flows near a closed positively invariant set. J. Dynam. Differ. Equations 6 (1994)","DOI":"10.1007\/BF02218848"},{"issue":"4","key":"2474_CR5","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1090\/S0033-569X-2014-01349-5","volume":"72","author":"S.-C. Fu","year":"2014","unstructured":"Fu, S.-C.: The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction. Q. Appl. Math. 72(4), 649\u2013664 (2014)","journal-title":"Q. Appl. Math."},{"issue":"1","key":"2474_CR6","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.jmaa.2015.09.069","volume":"435","author":"S.C. Fu","year":"2016","unstructured":"Fu, S.C.: Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl. 435(1), 20\u201337 (2016)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"2474_CR7","doi-asserted-by":"publisher","first-page":"99","DOI":"10.3934\/mbe.2015.12.99","volume":"12","author":"X. Feng","year":"2014","unstructured":"Feng, X., Teng, Z., Zhang, F.: Global dynamics of a general class of multi-group epidemic models with latency and relapse. Math. Biosci. And Eng. 12(1), 99\u2013115 (2014)","journal-title":"Math. Biosci. And Eng."},{"key":"2474_CR8","doi-asserted-by":"crossref","unstructured":"Guenad, B., Darazirar, R., Djilali, S., Alraddadi, I.: Traveling waves in a delayed reaction\u2013diffusion SIR epidemic model with a generalized incidence function. Nonlinear Dyn 1\u201321 (2024)","DOI":"10.1007\/s11071-024-10413-4"},{"key":"2474_CR9","first-page":"259","volume":"14","author":"H. Guo","year":"2006","unstructured":"Guo, H., Li, M.Y., Shuai, Z.: Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can. Appl. Math. Q. 14, 259\u2013284 (2006)","journal-title":"Can. Appl. Math. Q."},{"issue":"2","key":"2474_CR10","doi-asserted-by":"publisher","first-page":"417","DOI":"10.3934\/dcdsb.2016.21.417","volume":"21","author":"K.P. Hadeler","year":"2015","unstructured":"Hadeler, K.P.: Stefan problem, traveling fronts, and epidemic spread. Discrete & Contin. Dyn. Syst.-Ser. B 21(2), 417\u2013436 (2015)","journal-title":"Discrete & Contin. Dyn. Syst.-Ser. B"},{"key":"2474_CR11","doi-asserted-by":"crossref","unstructured":"Hartman, P.: Ordinary differential equations. Soc. For Ind. And Appl. Math. (2002)","DOI":"10.1137\/1.9780898719222"},{"key":"2474_CR12","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s11538-005-9037-9","volume":"68","author":"A. Korobeinikov","year":"2006","unstructured":"Korobeinikov, A.: Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68, 615\u2013626 (2006)","journal-title":"Bull. Math. Biol."},{"issue":"1","key":"2474_CR13","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1137\/18M1205947","volume":"79","author":"T. Kuniya","year":"2019","unstructured":"Kuniya, T.: Global behavior of a multi-group SIR epidemic model with age structure and an application to the chlamydia epidemic in Japan. SIAM J Appl Math 79(1), 321\u2013340 (2019)","journal-title":"SIAM J Appl Math"},{"issue":"3\u20134","key":"2474_CR14","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/0025-5564(76)90125-5","volume":"28","author":"A. Lajmanovich","year":"1976","unstructured":"Lajmanovich, A., Yorke, J.A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math Biosci 28(3\u20134), 221\u2013236 (1976)","journal-title":"Math Biosci"},{"key":"2474_CR15","unstructured":"LaSalle, J.P.: The stability of dynamical Systems. In: SIAM Regional Conference Series in Applied Mathematics. No. 25, Society for Industrial and Applied Mathematics, Philadelphia (1977)"},{"key":"2474_CR16","doi-asserted-by":"publisher","first-page":"2048","DOI":"10.1007\/s11538-009-9457-z","volume":"71","author":"J. Li","year":"2009","unstructured":"Li, J., Zou, X.: Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain. Bull. Math. Biol. 71, 2048\u20132079 (2009)","journal-title":"Bull. Math. Biol."},{"key":"2474_CR17","first-page":"723","volume":"247","author":"Y. Li","year":"2014","unstructured":"Li, Y., Li, W.T., Yang, F.Y.: Traveling waves for a nonlocal dispersal SIR model with delay and external supplies. Appl. Math. Comput. 247, 723\u2013740 (2014)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"2474_CR18","first-page":"583","volume":"14","author":"Y. Li","year":"2015","unstructured":"Li, Y., Li, W.T., Lin, G.: Traveling waves of a delayed diffusive SIR epidemic model. Commun. Pur. Appl. Anal. 14(3), 583\u2013600 (2015)","journal-title":"Commun. Pur. Appl. Anal."},{"key":"2474_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jde.2009.09.003","volume":"248","author":"M.Y. Li","year":"2010","unstructured":"Li, M.Y., Shuai, Z.: Global-stability problem for coupled systems of differential equations on networks. J. Differ. Equations 248, 1\u201320 (2010)","journal-title":"J. Differ. Equations"},{"key":"2474_CR20","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1016\/S0025-5564(99)00030-9","volume":"160","author":"M.Y. Li","year":"1999","unstructured":"Li, M.Y., Graef, J.R., Wang, L., Karsai, J.: Global dynamics of a SEIR model with varying total population size. Math Biosci 160, 191\u2013213 (1999)","journal-title":"Math Biosci"},{"key":"2474_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."},{"issue":"1","key":"2474_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/140971683","volume":"14","author":"C.-C. McCluskey","year":"2015","unstructured":"McCluskey, C.-C.: Using Lyapunov functions to construct Lyapunov functionals for delay differential equations. SIAM J Appl Dyn Syst 14(1), 1\u201324 (2015)","journal-title":"SIAM J Appl Dyn Syst"},{"key":"2474_CR23","unstructured":"Murray, J.D.: Math. Biol., Vol. 19 Of Biomath. (1989)"},{"issue":"13","key":"2474_CR24","doi-asserted-by":"publisher","first-page":"14045","DOI":"10.1002\/mma.9303","volume":"46","author":"S. Ottaviano","year":"2023","unstructured":"Ottaviano, S., Sensi, M., Sottile, S.: Global stability of multi-group SAIRS epidemic models. Math. Methods Appl. Sci. 46(13), 14045\u201314071 (2023)","journal-title":"Math. Methods Appl. Sci."},{"issue":"2","key":"2474_CR25","doi-asserted-by":"publisher","first-page":"1502","DOI":"10.1016\/j.jmaa.2019.03.029","volume":"475","author":"X.F. San","year":"2019","unstructured":"San, X.F., Wang, Z.C.: Traveling waves for a two-group epidemic model with latent period in a patchy environment. J. Math. Anal. Appl. 475(2), 1502\u20131531 (2019)","journal-title":"J. Math. Anal. Appl."},{"key":"2474_CR26","doi-asserted-by":"publisher","first-page":"2286","DOI":"10.1016\/j.camwa.2010.08.020","volume":"60","author":"R. Sun","year":"2010","unstructured":"Sun, R.: Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence. Comput. Math. Appl. 60, 2286\u20132291 (2010)","journal-title":"Comput. Math. Appl."},{"issue":"10","key":"2474_CR27","first-page":"5042","volume":"219","author":"Y. Shang","year":"2013","unstructured":"Shang, Y.: Mixed SI (R) epidemic dynamics in random graphs with general degree distributions. Appl. Math. Comput. 219(10), 5042\u20135048 (2013)","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"2474_CR28","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1007\/s11424-014-2186-x","volume":"28","author":"Y. Shang","year":"2015","unstructured":"Shang, Y.: Degree distribution dynamics for disease spreading with individual awareness. J. Syst. Sci. Complexity 28(1), 96\u2013104 (2015)","journal-title":"J. Syst. Sci. Complexity"},{"issue":"4","key":"2474_CR29","doi-asserted-by":"publisher","first-page":"1029","DOI":"10.1016\/S0252-9602(14)60067-9","volume":"34","author":"Y. Shang","year":"2014","unstructured":"Shang, Y.: Consensus formation of two-level opinion dynamics. Acta Math. Scientia 34(4), 1029\u20131040 (2014)","journal-title":"Acta Math. Scientia"},{"key":"2474_CR30","doi-asserted-by":"publisher","first-page":"107588","DOI":"10.1016\/j.cnsns.2023.107588","volume":"128","author":"T.D. Tuong","year":"2024","unstructured":"Tuong, T.D., Nguyen, D.H., Nguyen, N.N.: Stochastic multi-group epidemic SVIR models: Degenerate case. Commun. Nonlinear Sci. Numer. Simul. 128, 107588 (2024)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"2474_CR31","doi-asserted-by":"publisher","first-page":"1871","DOI":"10.1007\/s00285-018-1227-9","volume":"77","author":"L. Zhao","year":"2018","unstructured":"Zhao, L., Wang, Z.C., Ruan, S.: Traveling wave solutions in a two-group SIR epidemic model with constant recruitment. J. Math Biol. 77, 1871\u20131915 (2018)","journal-title":"J. Math Biol."},{"key":"2474_CR32","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.1088\/1361-6544\/aa59ae","volume":"30","author":"L. Zhao","year":"2017","unstructured":"Zhao, L., Wang, Z.C., Ruan, S.: Traveling wave solutions of a two-group epidemic model with latent period. Nonlinearity 30, 1287\u20131325 (2017)","journal-title":"Nonlinearity"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02474-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-025-02474-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-025-02474-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T13:14:31Z","timestamp":1757423671000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-025-02474-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,30]]},"references-count":32,"journal-issue":{"issue":"S1","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["2474"],"URL":"https:\/\/doi.org\/10.1007\/s12190-025-02474-4","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,30]]},"assertion":[{"value":"25 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 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":"Competing interests"}}]}}