{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T10:59:14Z","timestamp":1785149954104,"version":"3.55.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"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":[[2026,7]]},"DOI":"10.1007\/s12190-026-02835-7","type":"journal-article","created":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T13:09:23Z","timestamp":1783948163000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Global dynamics analysis and event-triggered impulsive control for a time-delay SVIR epidemic model in complex networks with vaccination-infection gain coupling"],"prefix":"10.1007","volume":"72","author":[{"given":"ShuangYan","family":"Yang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"JinRong","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,13]]},"reference":[{"issue":"10350","key":"2835_CR1","doi-asserted-by":"publisher","first-page":"452","DOI":"10.1016\/S0140-6736(22)01214-4","volume":"400","author":"A.V. Ballering","year":"2022","unstructured":"Ballering, A.V., van Zon, S.K.R., Hartman, T.C.O., Rosmalen, J.G.M.: Persistence of somatic symptoms after COVID-19 in the Netherlands: an observational cohort study. Lancet 400(10350), 452\u2013461 (2022). https:\/\/doi.org\/10.1016\/S0140-6736(22)01214-4.","journal-title":"Lancet"},{"key":"2835_CR2","doi-asserted-by":"publisher","first-page":"101939","DOI":"10.1016\/j.tmaid.2020.101939","volume":"38","author":"M. Bielecki","year":"2020","unstructured":"Bielecki, M., Patel, D., Hinkelbein, J., Komorowski, M., Kester, J., Ebrahim, S., Rodriguez-Morales, A.J., Memish, Z.A., Schlagenhauf, P.: Reprint of: air travel and COVID-19 prevention in the pandemic and peri-pandemic period: a narrative review. Travel Med Infect Dis. 38, 101939 (2020). https:\/\/doi.org\/10.1016\/j.tmaid.2020.101939.","journal-title":"Travel Med Infect Dis."},{"issue":"6492","key":"2835_CR3","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1126\/science.abb5659","volume":"368","author":"S. Cobey","year":"2020","unstructured":"Cobey, S.: Modeling infectious disease dynamics. Science 368(6492), 713\u2013714 (2020). https:\/\/doi.org\/10.1126\/science.abb5659.","journal-title":"Science"},{"issue":"2","key":"2835_CR4","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1109\/TASE.2021.3106782","volume":"19","author":"T.Y. Luo","year":"2022","unstructured":"Luo, T.Y., Cao, Z.D., Wang, Y.J., Zeng, D., Zhang, Q.: Role of asymptomatic COVID-19 cases in viral transmission: findings from a hierarchical community contact network model. IEEE Trans. Automat. Sci. Eng. 19(2), 576\u2013585 (2022). https:\/\/doi.org\/10.1109\/TASE.2021.3106782.","journal-title":"IEEE Trans. Automat. Sci. Eng."},{"key":"2835_CR5","doi-asserted-by":"crossref","unstructured":"Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. In: Proceedings of the Royal Society of London, vol. 115, pp. 700\u2013721. Series A (1927).","DOI":"10.1098\/rspa.1927.0118"},{"issue":"4","key":"2835_CR6","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1137\/030600370","volume":"3","author":"M.E. Alexander","year":"2004","unstructured":"Alexander, M.E., Bowman, C., Moghadas, S.M., Summers, R., Gumel, A.B., Sahai, B.M.: A vaccination model for transmission dynamics of influenza. SIAM J. Appl. Dyn. Syst. 3(4), 503\u2013524 (2004). https:\/\/doi.org\/10.1137\/030600370.","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"3","key":"2835_CR7","first-page":"177","volume":"9","author":"Y.N. Zhao","year":"2015","unstructured":"Zhao, Y.N., Jiang, D.Q.: The asymptotic behavior and ergodicity of stochastically perturbed SVIR epidemic model. Int. J. Biomath. 9(3), 177\u2013190 (2015).","journal-title":"Int. J. Biomath."},{"issue":"2","key":"2835_CR8","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s12064-023-00392-2","volume":"142","author":"P. Mahato","year":"2023","unstructured":"Mahato, P., Mahato, S.K., Das, S., Karmakar, P.: Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments. Theory Biosci. 142(2), 181\u2013198 (2023). https:\/\/doi.org\/10.1007\/s12064-023-00392-2.","journal-title":"Theory Biosci."},{"key":"2835_CR9","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/j.apm.2023.03.036","volume":"120","author":"K. Wang","year":"2023","unstructured":"Wang, K., Fan, H.J., Zhu, Y.L.: Dynamics and application of a generalized SIQR epidemic model with vaccination and treatment. Appl. Math. Modell. 120, 382\u2013399 (2023). https:\/\/doi.org\/10.1016\/j.apm.2023.03.036.","journal-title":"Appl. Math. Modell."},{"key":"2835_CR10","doi-asserted-by":"publisher","first-page":"109579","DOI":"10.1016\/j.aml.2025.109579","volume":"168","author":"T. Su","year":"2025","unstructured":"Su, T., Kao, Y.G., Jiang, D.Q.: Persistence and extinction of a stochastic SVI epidemic model with standard incidence and reaction\u2013diffusion. Appl. Math. Lett. 168, 109579 (2025). https:\/\/doi.org\/10.1016\/j.aml.2025.109579.","journal-title":"Appl. Math. Lett."},{"issue":"20","key":"2835_CR11","doi-asserted-by":"publisher","first-page":"18611","DOI":"10.1007\/s11071-024-09932-x","volume":"112","author":"S. Barman","year":"2024","unstructured":"Barman, S., Jana, S., Majee, S., Das, D.K., Kar, T.K.: Complex dynamics of a fractional-order epidemic model with saturated media effect. Nonlinear Dyn. 112(20), 18611\u201318637 (2024). https:\/\/doi.org\/10.1007\/s11071-024-09932-x.","journal-title":"Nonlinear Dyn."},{"key":"2835_CR12","doi-asserted-by":"publisher","first-page":"109113","DOI":"10.1016\/j.cnsns.2025.109113","volume":"152","author":"F. Rao","year":"2026","unstructured":"Rao, F., Li, X., Xue, D.D.: Dynamics of a stochastic SAIVR epidemic model with vaccination. Commun. Nonlinear Sci. Numer. Simul. 152, 109113 (2026). https:\/\/doi.org\/10.1016\/j.cnsns.2025.109113.","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"1","key":"2835_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jtbi.2007.10.014","volume":"253","author":"X.N. Liu","year":"2008","unstructured":"Liu, X.N., Takeuchi, Y., Iwami, S.: SVIR epidemic models with vaccination strategies. J. Theor. Biol. 253(1), 1\u201311 (2008). https:\/\/doi.org\/10.1016\/j.jtbi.2007.10.014.","journal-title":"J. Theor. Biol."},{"key":"2835_CR14","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.chaos.2014.06.003","volume":"66","author":"F.Y. Wang","year":"2014","unstructured":"Wang, F.Y., Wang, X.Y., Zhang, S.W., Ding, C.: On pulse vaccine strategy in a periodic stochastic SIR epidemic model. Chaos, Solitons Fractals. 66, 127\u2013135 (2014). https:\/\/doi.org\/10.1016\/j.chaos.2014.06.003.","journal-title":"Chaos, Solitons Fractals"},{"key":"2835_CR15","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1016\/j.nahs.2017.08.004","volume":"27","author":"L.F. Nie","year":"2018","unstructured":"Nie, L.F., Shen, J.Y., Yang, C.X.: Dynamic behavior analysis of SIVS epidemic models with state-dependent pulse vaccination. Nonlinear Anal.: Hybrid Syst. 27, 258\u2013270 (2018). https:\/\/doi.org\/10.1016\/j.nahs.2017.08.004.","journal-title":"Nonlinear Anal.: Hybrid Syst."},{"issue":"2","key":"2835_CR16","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1016\/j.amc.2009.04.005","volume":"214","author":"X.Y. Song","year":"2009","unstructured":"Song, X.Y., Jiang, Y., Wei, H.M.: Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays. Appl. Math. Comput. 214(2), 381\u2013390 (2009). https:\/\/doi.org\/10.1016\/j.amc.2009.04.005.","journal-title":"Appl. Math. Comput."},{"key":"2835_CR17","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1016\/j.mbs.2015.09.005","volume":"269","author":"Z.G. Bai","year":"2015","unstructured":"Bai, Z.G.: Threshold dynamics of a time-delayed SEIRS model with pulse vaccination. Math Biosci. 269, 178\u2013185 (2015). https:\/\/doi.org\/10.1016\/j.mbs.2015.09.005.","journal-title":"Math Biosci."},{"key":"2835_CR18","doi-asserted-by":"publisher","first-page":"113206","DOI":"10.1016\/j.chaos.2023.113206","volume":"168","author":"D.D. Sun","year":"2023","unstructured":"Sun, D.D., Teng, Z.D., Wang, K., Zhang, T.: Stability and hopf bifurcation in delayed age-structured SVIR epidemic model with vaccination and incubation. Chaos, Solitons Fractals 168, 113206 (2023). https:\/\/doi.org\/10.1016\/j.chaos.2023.113206.","journal-title":"Chaos, Solitons Fractals"},{"issue":"2","key":"2835_CR19","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1007\/s11071-019-05371-1","volume":"99","author":"W. Lv","year":"2020","unstructured":"Lv, W., Ke, Q., Li, K.Z.: Dynamical analysis and control strategies of an SIVS epidemic model with imperfect vaccination on scale-free networks. Nonlinear Dyn. 99(2), 1507\u20131523 (2020). https:\/\/doi.org\/10.1007\/s11071-019-05371-1.","journal-title":"Nonlinear Dyn."},{"key":"2835_CR20","doi-asserted-by":"publisher","first-page":"108097","DOI":"10.1016\/j.cnsns.2024.108097","volume":"137","author":"G. Guan","year":"2024","unstructured":"Guan, G., Guo, Z.Y., Xiao, Y.Y.: Dynamical behaviors of a network-based SIR epidemic model with saturated incidence and pulse vaccination. Commun. Nonlinear Sci. Numer. Simul. 137, 108097 (2024). https:\/\/doi.org\/10.1016\/j.cnsns.2024.108097.","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"2835_CR21","doi-asserted-by":"publisher","first-page":"116671","DOI":"10.1016\/j.chaos.2025.116671","volume":"199","author":"X.J. Fu","year":"2025","unstructured":"Fu, X.J., Wang, J.R.: Non-periodic intermittent stochastic control for SVIQR epidemic model on complex network with saturation incidence rate. Chaos, Solitons Fractals 199, 116671 (2025). https:\/\/doi.org\/10.1016\/j.chaos.2025.116671.","journal-title":"Chaos, Solitons Fractals"},{"key":"2835_CR22","doi-asserted-by":"publisher","first-page":"110806","DOI":"10.1016\/j.chaos.2021.110806","volume":"145","author":"S.Z. Yu","year":"2021","unstructured":"Yu, S.Z., Yu, Z.Y., Jiang, H.J., Li, J.: Dynamical study and event-triggered impulsive control of rumor propagation model on heterogeneous social network incorporating delay. Chaos, Solitons Fractals 145, 110806 (2021). https:\/\/doi.org\/10.1016\/j.chaos.2021.110806.","journal-title":"Chaos, Solitons Fractals"},{"key":"2835_CR23","doi-asserted-by":"publisher","first-page":"108607","DOI":"10.1016\/j.cnsns.2025.108607","volume":"143","author":"Y. Jing","year":"2025","unstructured":"Jing, Y., Wang, Y.G., Zhai, Q.Q.: Stochastic resonance and dynamic event-triggered impulsive control of a variable-order fractional information diffusion system with hybrid noise. Commun. Nonlinear Sci. Numer. Simul. 143, 108607 (2025). https:\/\/doi.org\/10.1016\/j.cnsns.2025.108607.","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"issue":"2","key":"2835_CR24","doi-asserted-by":"publisher","first-page":"1176","DOI":"10.1109\/TAC.2024.3448128","volume":"70","author":"Q.X. Zhu","year":"2025","unstructured":"Zhu, Q.X.: Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by L\u00e9vy processes. IEEE Trans. Automat. Contr. 70(2), 1176\u20131183 (2025). https:\/\/doi.org\/10.1109\/TAC.2024.3448128.","journal-title":"IEEE Trans. Automat. Contr."},{"issue":"3","key":"2835_CR25","doi-asserted-by":"publisher","first-page":"2090","DOI":"10.1109\/TAC.2024.3486978","volume":"70","author":"X.T. Yang","year":"2025","unstructured":"Yang, X.T., Zhao, T.Z., Zhu, Q.X.: Aperiodic event-triggered controls for stochastic functional differential systems with sampled-data delayed output. IEEE Trans. Automat. Contr. 70(3), 2090\u20132097 (2025). https:\/\/doi.org\/10.1109\/TAC.2024.3486978.","journal-title":"IEEE Trans. Automat. Contr."},{"issue":"4","key":"2835_CR26","doi-asserted-by":"publisher","first-page":"2621","DOI":"10.1109\/TAC.2023.3335005","volume":"69","author":"H.F. Xu","year":"2024","unstructured":"Xu, H.F., Zhu, Q.X., Zheng, W.X.: Exponential stability of stochastic nonlinear delay systems subject to multiple periodic impulses. IEEE Trans. Automat. Contr. 69(4), 2621\u20132628 (2024). https:\/\/doi.org\/10.1109\/TAC.2023.3335005.","journal-title":"IEEE Trans. Automat. Contr."},{"issue":"4\u20135","key":"2835_CR27","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/S0895-7177(02)00177-2","volume":"36","author":"A. d\u2019Onofrio","year":"2002","unstructured":"d\u2019Onofrio, A.: Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures. Math. Comput. Modell. 36(4\u20135), 473\u2013489 (2002). https:\/\/doi.org\/10.1016\/S0895-7177(02)00177-2.","journal-title":"Math. Comput. Modell."},{"key":"2835_CR28","volume-title":"Delay Differential Equation with Application in Population Dynamics","author":"Y. Kuang","year":"1993","unstructured":"Kuang, Y.: Delay differential equation with application in population dynamics. Academic, New York (1993)."},{"issue":"35\u201336","key":"2835_CR29","doi-asserted-by":"publisher","first-page":"6037","DOI":"10.1016\/j.vaccine.2006.05.018","volume":"24","author":"S.J. Gao","year":"2006","unstructured":"Gao, S.J., Chen, L.S., Nieto, J.J., Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24(35\u201336), 6037\u20136045 (2006). https:\/\/doi.org\/10.1016\/j.vaccine.2006.05.018.","journal-title":"Vaccine"},{"issue":"2","key":"2835_CR30","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1016\/j.amc.2007.07.083","volume":"197","author":"X.Z. Meng","year":"2008","unstructured":"Meng, X.Z., Chen, L.S.: The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Appl. Math. Comput. 197(2), 582\u2013597 (2008). https:\/\/doi.org\/10.1016\/j.amc.2007.07.083.","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"2835_CR31","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/S0025-5564(00)00018-3","volume":"166","author":"H.R. Thieme","year":"2000","unstructured":"Thieme, H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math Biosci. 166(2), 173\u2013201 (2000). https:\/\/doi.org\/10.1016\/S0025-5564(00)00018-3.","journal-title":"Math Biosci."},{"issue":"3","key":"2835_CR32","doi-asserted-by":"publisher","first-page":"831","DOI":"10.1109\/TSMCB.2009.2030506","volume":"40","author":"H.G. Zhang","year":"2010","unstructured":"Zhang, H.G., Ma, T.D., Huang, G.B., Wang, Z.: Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans Syst Man Cybern B 40(3), 831\u2013844 (2010). https:\/\/doi.org\/10.1109\/TSMCB.2009.2030506.","journal-title":"IEEE Trans Syst Man Cybern B"},{"key":"2835_CR33","volume-title":"Numerical Methods for Delay Differential Equations","author":"B. Alfredo","year":"2003","unstructured":"Alfredo, B., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)."},{"issue":"3","key":"2835_CR34","doi-asserted-by":"publisher","first-page":"885","DOI":"10.1007\/s40840-017-0519-3","volume":"42","author":"Q.Q. Cui","year":"2019","unstructured":"Cui, Q.Q., Zhang, Q., Qiu, Z.P., Yang, X.: Transmission dynamics of an epidemic model with vaccination, treatment and isolation. Bull. Malays. Math. Sci. Soc. 42(3), 885\u2013896 (2019). https:\/\/doi.org\/10.1007\/s40840-017-0519-3.","journal-title":"Bull. Malays. Math. Sci. Soc."}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02835-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02835-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02835-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T10:02:11Z","timestamp":1785146531000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02835-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7]]},"references-count":34,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["2835"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02835-7","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7]]},"assertion":[{"value":"28 January 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 May 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 July 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"204"}}