{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T14:35:53Z","timestamp":1774622153381,"version":"3.50.1"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T00:00:00Z","timestamp":1679443200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T00:00:00Z","timestamp":1679443200000},"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":["12201548"],"award-info":[{"award-number":["12201548"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","award":["BK20220553"],"award-info":[{"award-number":["BK20220553"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010023","name":"Natural Science Research of Jiangsu Higher Education Institutions of China","doi-asserted-by":"publisher","award":["22KJB110006"],"award-info":[{"award-number":["22KJB110006"]}],"id":[{"id":"10.13039\/501100010023","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,6]]},"DOI":"10.1007\/s12190-023-01851-1","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T05:02:39Z","timestamp":1679461359000},"page":"2657-2673","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Global dynamics of an SIS compartment model with resource constraints"],"prefix":"10.1007","volume":"69","author":[{"given":"Huayu","family":"Liu","sequence":"first","affiliation":[]},{"given":"Chenbo","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Tao","family":"Feng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,3,22]]},"reference":[{"issue":"4","key":"1851_CR1","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1128\/mr.47.4.455-509.1983","volume":"47","author":"AM Behbehani","year":"1983","unstructured":"Behbehani, A.M.: The smallpox story: life and death of an old disease. Microbiol. Rev. 47(4), 455\u2013509 (1983)","journal-title":"Microbiol. Rev."},{"issue":"12","key":"1851_CR2","doi-asserted-by":"publisher","first-page":"2561","DOI":"10.1093\/aje\/kwy191","volume":"187","author":"P Spreeuwenberg","year":"2018","unstructured":"Spreeuwenberg, P., Kroneman, M., Paget, J.: Reassessing the global mortality burden of the 1918 influenza pandemic. Am. J. Epidemiol. 187(12), 2561\u20132567 (2018)","journal-title":"Am. J. Epidemiol."},{"issue":"4","key":"1851_CR3","first-page":"3035","volume":"22","author":"V Zoumpourlis","year":"2020","unstructured":"Zoumpourlis, V., Goulielmaki, M., Rizos, E., Baliou, S., Spandidos, D.A.: [Comment] the Covid-19 pandemic as a scientific and social challenge in the 21st century. Mol. Med. Rep. 22(4), 3035\u20133048 (2020)","journal-title":"Mol. Med. Rep."},{"issue":"4","key":"1851_CR4","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1038\/s41579-021-00639-z","volume":"20","author":"RE Baker","year":"2022","unstructured":"Baker, R.E., Mahmud, A.S., Miller, I.F., Rajeev, M., Rasambainarivo, F., Rice, B.L., Takahashi, S., Tatem, A.J., Wagner, C.E., Wang, L.-F., et al.: Infectious disease in an era of global change. Nat. Rev. Microbiol. 20(4), 193\u2013205 (2022)","journal-title":"Nat. Rev. Microbiol."},{"key":"1851_CR5","doi-asserted-by":"publisher","first-page":"549","DOI":"10.3389\/fimmu.2019.00549","volume":"10","author":"DE Bloom","year":"2019","unstructured":"Bloom, D.E., Cadarette, D.: Infectious disease threats in the twenty-first century: strengthening the global response. Front. Immunol. 10, 549 (2019)","journal-title":"Front. Immunol."},{"key":"1851_CR6","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1016\/j.jde.2022.12.045","volume":"352","author":"Z Wu","year":"2023","unstructured":"Wu, Z., Cai, Y., Wang, Z., Wang, W.: Global stability of a fractional order sis epidemic model. J. Diff. Equ. 352, 221\u2013248 (2023)","journal-title":"J. Diff. Equ."},{"issue":"6812","key":"1851_CR7","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1038\/35046270","volume":"408","author":"K Dietz","year":"2000","unstructured":"Dietz, K., Heesterbeek, J.: Bernoulli was ahead of modern epidemiology. Nature 408(6812), 513\u2013514 (2000)","journal-title":"Nature"},{"issue":"6","key":"1851_CR8","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1097\/00007611-199906000-00004","volume":"92","author":"K Rajakumar","year":"1999","unstructured":"Rajakumar, K., Weisse, M.: Centennial year of Ronald ross\u2019 epic discovery of malaria transmission: an essay and tribute. South. Med. J. 92(6), 567\u2013571 (1999)","journal-title":"South. Med. J."},{"issue":"772","key":"1851_CR9","doi-asserted-by":"publisher","first-page":"700","DOI":"10.1098\/rspa.1927.0118","volume":"115","author":"WO Kermack","year":"1927","unstructured":"Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Londn. Ser. A 115(772), 700\u2013721 (1927)","journal-title":"Proc. R. Soc. Londn. Ser. A"},{"issue":"834","key":"1851_CR10","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1098\/rspa.1932.0171","volume":"138","author":"WO Kermack","year":"1932","unstructured":"Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. ii. The problem of endemicity. Proc. R. Soc. Lond. Ser. A 138(834), 55\u201383 (1932)","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"issue":"2","key":"1851_CR11","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1137\/070700966","volume":"69","author":"Y Tang","year":"2008","unstructured":"Tang, Y., Huang, D., Ruan, S., Zhang, W.: Coexistence of limit cycles and homoclinic loops in a sirs model with a nonlinear incidence rate. SIAM J. Appl. Math. 69(2), 621\u2013639 (2008)","journal-title":"SIAM J. Appl. Math."},{"issue":"2","key":"1851_CR12","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/0025-5564(95)92756-5","volume":"125","author":"MY Li","year":"1995","unstructured":"Li, M.Y., Muldowney, J.S.: Global stability for the Seir model in epidemiology. Math. Biosci. 125(2), 155\u2013164 (1995)","journal-title":"Math. Biosci."},{"issue":"6","key":"1851_CR13","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1038\/s41592-020-0856-2","volume":"17","author":"ON Bj\u00f8rnstad","year":"2020","unstructured":"Bj\u00f8rnstad, O.N., Shea, K., Krzywinski, M., Altman, N.: The Seirs model for infectious disease dynamics. Nat. Methods 17(6), 557\u2013559 (2020)","journal-title":"Nat. Methods"},{"issue":"4","key":"1851_CR14","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/S0036144500371907","volume":"42","author":"HW Hethcote","year":"2000","unstructured":"Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599\u2013653 (2000)","journal-title":"SIAM Rev."},{"issue":"6","key":"1851_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00285-022-01745-z","volume":"84","author":"S Gao","year":"2022","unstructured":"Gao, S., Martcheva, M., Miao, H., Rong, L.: The impact of vaccination on human papillomavirus infection with disassortative geographical mixing: a two-patch modeling study. J. Math. Biol. 84(6), 1\u201334 (2022)","journal-title":"J. Math. Biol."},{"key":"1851_CR16","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1016\/j.jtbi.2013.07.004","volume":"337","author":"M Starnini","year":"2013","unstructured":"Starnini, M., Machens, A., Cattuto, C., Barrat, A., Pastor-Satorras, R.: Immunization strategies for epidemic processes in time-varying contact networks. J. Theor. Biol. 337, 89\u2013100 (2013)","journal-title":"J. Theor. Biol."},{"issue":"1","key":"1851_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41467-020-19024-0","volume":"11","author":"M Te Vrugt","year":"2020","unstructured":"Te Vrugt, M., Bickmann, J., Wittkowski, R.: Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory. Nat. Commun. 11(1), 1\u201311 (2020)","journal-title":"Nat. Commun."},{"issue":"4","key":"1851_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00285-021-01668-1","volume":"83","author":"L Bolzoni","year":"2021","unstructured":"Bolzoni, L., Della Marca, R., Groppi, M.: On the optimal control of sir model with erlang-distributed infectious period: isolation strategies. J. Math. Biol. 83(4), 1\u201321 (2021)","journal-title":"J. Math. Biol."},{"issue":"12","key":"1851_CR19","doi-asserted-by":"publisher","first-page":"7463","DOI":"10.1016\/j.jde.2015.08.024","volume":"259","author":"Y Cai","year":"2015","unstructured":"Cai, Y., Kang, Y., Banerjee, M., Wang, W.: A stochastic sirs epidemic model with infectious force under intervention strategies. J. Diff. Equ. 259(12), 7463\u20137502 (2015)","journal-title":"J. Diff. Equ."},{"key":"1851_CR20","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1016\/j.mbs.2017.09.007","volume":"294","author":"G Agaba","year":"2017","unstructured":"Agaba, G., Kyrychko, Y., Blyuss, K.: Dynamics of vaccination in a time-delayed epidemic model with awareness. Math. Biosci. 294, 92\u201399 (2017)","journal-title":"Math. Biosci."},{"key":"1851_CR21","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.ecocom.2017.03.002","volume":"31","author":"G Agaba","year":"2017","unstructured":"Agaba, G., Kyrychko, Y., Blyuss, K.: Time-delayed sis epidemic model with population awareness. Ecol. Complex. 31, 50\u201356 (2017)","journal-title":"Ecol. Complex."},{"key":"1851_CR22","volume-title":"Differential Equations and Dynamical Systems","author":"L Perko","year":"2013","unstructured":"Perko, L.: Differential Equations and Dynamical Systems, vol. 7. Springer, New York (2013)"},{"issue":"1","key":"1851_CR23","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1006\/jdeq.1993.1097","volume":"106","author":"Y Li","year":"1993","unstructured":"Li, Y., Muldowney, J.S.: On Bendixson\u2019s criterion. J. Diff. Equ. 106(1), 27\u201339 (1993)","journal-title":"J. Diff. Equ."},{"issue":"3","key":"1851_CR24","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1080\/03081087608817149","volume":"4","author":"D London","year":"1976","unstructured":"London, D.: On derivations arising in differential equations. Linear Multilinear Algebra 4(3), 179\u2013189 (1976)","journal-title":"Linear Multilinear Algebra"},{"key":"1851_CR25","doi-asserted-by":"publisher","first-page":"857","DOI":"10.1216\/rmjm\/1181073047","volume":"20","author":"JS Muldowney","year":"1990","unstructured":"Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20, 857\u2013872 (1990)","journal-title":"Rocky Mt. J. Math."},{"key":"1851_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108547","volume":"139","author":"T Feng","year":"2023","unstructured":"Feng, T., Liu, C.: A mathematical framework for collective foraging behavior of social insect colonies in multi-dynamic environments. Appl. Math. Lett. 139, 108547 (2023)","journal-title":"Appl. Math. Lett."},{"key":"1851_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2022.108910","volume":"353","author":"T Feng","year":"2022","unstructured":"Feng, T., Zhou, H., Qiu, Z., Kang, Y.: Impacts of demographic and environmental stochasticity on population dynamics with cooperative effects. Math. Biosci. 353, 108910 (2022)","journal-title":"Math. Biosci."},{"key":"1851_CR28","doi-asserted-by":"publisher","DOI":"10.2307\/j.ctv301f9v","volume-title":"Mathematics in Population Biology","author":"HR Thieme","year":"2018","unstructured":"Thieme, H.R.: Mathematics in Population Biology, vol. 12. Princeton University Press, New Jersey (2018)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01851-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01851-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01851-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T16:02:06Z","timestamp":1744214526000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01851-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,22]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["1851"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01851-1","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,3,22]]},"assertion":[{"value":"8 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 March 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"We have no conflict of interest to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}