{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T04:29:04Z","timestamp":1788496144240,"version":"build-2803163510"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"1","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. Math. Biol."],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s00285-026-02430-1","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T04:10:03Z","timestamp":1783051803000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Spatiotemporal SEIQR Epidemic Modeling with Optimal Control for Vaccination, Treatment, and Social Measures"],"prefix":"10.1007","volume":"93","author":[{"given":"Achraf","family":"Zinihi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2561-8854","authenticated-orcid":false,"given":"Matthias","family":"Ehrhardt","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Moulay Rchid","family":"Sidi Ammi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,3]]},"reference":[{"issue":"4","key":"2430_CR1","first-page":"228","volume":"2","author":"S Ahmad","year":"2022","unstructured":"Ahmad S, Qiu D, Rahman M (2022) Dynamics of a fractional-order COVID-19 model under the nonsingular kernel of Caputo-Fabrizio operator. Mathematical Modelling and Numerical Simulation with Applications 2(4):228\u2013243","journal-title":"Mathematical Modelling and Numerical Simulation with Applications"},{"issue":"12","key":"2430_CR2","doi-asserted-by":"publisher","first-page":"12529","DOI":"10.1002\/mma.9196","volume":"46","author":"F Auricchio","year":"2023","unstructured":"Auricchio F, Colli P, Gilardi G, Reali A, Rocca E (2023) Well-posedness for a diffusion\u2013reaction compartmental model simulating the spread of COVID-19. Mathematical Methods in the Applied Sciences 46(12):12529\u201312548","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"2430_CR3","doi-asserted-by":"crossref","unstructured":"Balderrama R, Peressutti J, Pinasco JP, Vazquez F, S\u00e1nchez de\u00a0la Vega C (2022) Optimal control for a SIR epidemic model with limited quarantine. Scientific Reports, 12(1)","DOI":"10.1038\/s41598-022-16619-z"},{"key":"2430_CR4","doi-asserted-by":"crossref","unstructured":"Ball F, Critcher L (2026) Multitype SIR epidemics among a population partitioned into households with proportionate global mixing. J Math Biol 92(5):","DOI":"10.1007\/s00285-026-02359-5"},{"key":"2430_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.matcom.2022.04.001","volume":"200","author":"SR Bandekar","year":"2022","unstructured":"Bandekar SR, Ghosh M (2022) A co-infection model on TB-COVID-19 with optimal control and sensitivity analysis. Math Comput Simul 200:1\u201331","journal-title":"Math Comput Simul"},{"key":"2430_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-0760-0","volume-title":"Mathematical methods in optimization of differential systems","author":"V Barbu","year":"1994","unstructured":"Barbu V (1994) Mathematical methods in optimization of differential systems. Springer, The Netherlands"},{"issue":"3","key":"2430_CR7","doi-asserted-by":"publisher","first-page":"3201","DOI":"10.1007\/s40808-021-01279-1","volume":"8","author":"AS Bhadauria","year":"2021","unstructured":"Bhadauria AS, Devi S, Gupta N (2021) Modelling and analysis of a SEIQR model on COVID-19 pandemic with delay. Modeling Earth Systems and Environment 8(3):3201\u20133214","journal-title":"Modeling Earth Systems and Environment"},{"issue":"7","key":"2430_CR8","doi-asserted-by":"publisher","first-page":"4239","DOI":"10.1002\/mma.6190","volume":"43","author":"XC Duan","year":"2020","unstructured":"Duan XC, Jung IH, Li XZ, Martcheva M (2020) Dynamics and optimal control of an age-structured SIRVS epidemic model. Mathematical Methods in the Applied Sciences 43(7):4239\u20134256","journal-title":"Mathematical Methods in the Applied Sciences"},{"issue":"4","key":"2430_CR9","first-page":"199","volume":"34","author":"JW Green","year":"1961","unstructured":"Green JW, Valentine FA (1961) On the Arzel\u00e0-Ascoli theorem. Math Mag 34(4):199\u2013202","journal-title":"Math Mag"},{"key":"2430_CR10","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.swevo.2015.09.007","volume":"27","author":"G Huang","year":"2016","unstructured":"Huang G (2016) Artificial infectious disease optimization: A SEIQR epidemic dynamic model-based function optimization algorithm. Swarm Evol Comput 27:31\u201367","journal-title":"Swarm Evol Comput"},{"issue":"772","key":"2430_CR11","first-page":"700","volume":"115","author":"WO Kermack","year":"1927","unstructured":"Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A 115(772):700\u2013721","journal-title":"Proceedings of the Royal Society A"},{"issue":"5","key":"2430_CR12","doi-asserted-by":"publisher","first-page":"1739","DOI":"10.18576\/amis\/100513","volume":"10","author":"NH Khanh","year":"2016","unstructured":"Khanh NH (2016) Dynamics of a worm propagation model with quarantine in wireless sensor networks. Applied Mathematics & Information Sciences 10(5):1739\u20131746","journal-title":"Applied Mathematics & Information Sciences"},{"key":"2430_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2024.109265","volume":"375","author":"WR KhudaBukhsh","year":"2024","unstructured":"KhudaBukhsh WR, Rempa\u0142a GA (2024) How to correctly fit an SIR model to data from an SEIR model? Math Biosci 375:109265","journal-title":"Math Biosci"},{"issue":"4","key":"2430_CR14","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/s10444-004-7639-7","volume":"25","author":"MY Kim","year":"2006","unstructured":"Kim MY (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"},{"issue":"1","key":"2430_CR15","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s11538-008-9352-z","volume":"71","author":"A Korobeinikov","year":"2008","unstructured":"Korobeinikov A (2008) Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages. Bull Math Biol 71(1):75\u201383","journal-title":"Bull Math Biol"},{"key":"2430_CR16","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1016\/j.matcom.2023.05.024","volume":"213","author":"A Kumar","year":"2023","unstructured":"Kumar A, Gupta A, Dubey US, Dubey B (2023) Stability and bifurcation analysis of an infectious disease model with different optimal control strategies. Math Comput Simul 213:78\u2013114","journal-title":"Math Comput Simul"},{"issue":"2","key":"2430_CR17","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/BF00163027","volume":"32","author":"YA Kuznetsov","year":"1994","unstructured":"Kuznetsov YA, Piccardi C (1994) Bifurcation analysis of periodic SEIR and SIR epidemic models. J Math Biol 32(2):109\u2013121","journal-title":"J Math Biol"},{"key":"2430_CR18","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1051\/mmnp\/2022027","volume":"17","author":"AP Lemos-Pai\u00e3o","year":"2022","unstructured":"Lemos-Pai\u00e3o AP, Maurer H, Silva CJ, Torres DFM (2022) A SIQRB delayed model for cholera and optimal control treatment. Mathematical Modelling of Natural Phenomena 17:25","journal-title":"Mathematical Modelling of Natural Phenomena"},{"key":"2430_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2023.107615","volume":"128","author":"Q Liu","year":"2024","unstructured":"Liu Q, Xiang H, Zhou M (2024) Dynamic behaviors and optimal control of a new delayed epidemic model. Commun Nonlinear Sci Numer Simul 128:107615","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1","key":"2430_CR20","first-page":"1","volume":"321","author":"RH Martin","year":"1990","unstructured":"Martin RH, Smith HL (1990) Abstract functional-differential equations and reaction-diffusion systems. Trans Am Math Soc 321(1):1\u201344","journal-title":"Trans Am Math Soc"},{"key":"2430_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.mbs.2020.108368","volume":"325","author":"J Mohammed-Awel","year":"2020","unstructured":"Mohammed-Awel J, Iboi EA, Gumel AB (2020) Insecticide resistance and malaria control: A genetics-epidemiology modeling approach. Math Biosci 325:108368","journal-title":"Math Biosci"},{"issue":"1\u20132","key":"2430_CR22","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.mcm.2010.12.051","volume":"54","author":"N \u00d6zalp","year":"2011","unstructured":"\u00d6zalp N, Demirci E (2011) A fractional order SEIR model with vertical transmission. Math Comput Model 54(1\u20132):1\u20136","journal-title":"Math Comput Model"},{"key":"2430_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2021.100216","volume":"5","author":"S Paul","year":"2022","unstructured":"Paul S, Mahata A, Mukherjee S, Roy B (2022) Dynamics of SIQR epidemic model with fractional order derivative. Partial Differential Equations in Applied Mathematics 5:100216","journal-title":"Partial Differential Equations in Applied Mathematics"},{"key":"2430_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5561-1","volume-title":"Semigroups of linear operators and applications to partial differential equations","author":"A Pazy","year":"1983","unstructured":"Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New York"},{"issue":"1","key":"2430_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2024.128192","volume":"535","author":"H Qiu","year":"2024","unstructured":"Qiu H, Hou R (2024) Dynamics and optimal control of an SEIAQR epidemic model with media coverage. J Math Anal Appl 535(1):128192","journal-title":"J Math Anal Appl"},{"key":"2430_CR26","doi-asserted-by":"crossref","unstructured":"Salwahan S, Abbas S, Tridane A (2025) Optimal control of a periodically switched epidemic model. International Journal of Dynamics and Control 13(2):","DOI":"10.1007\/s40435-025-01600-1"},{"key":"2430_CR27","doi-asserted-by":"crossref","unstructured":"Schurz H, Tosun K (2026) Numerical analysis of BIMs for stochastic SIR and SIS models with variable contact diffusion rates. J Math Biol 92(6):","DOI":"10.1007\/s00285-026-02392-4"},{"issue":"1","key":"2430_CR28","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01762360","volume":"146","author":"J Simon","year":"1986","unstructured":"Simon J (1986) Compact sets in the space $$L^p(O, T; B)$$. Annali di Matematica 146(1):65\u201396","journal-title":"Annali di Matematica"},{"key":"2430_CR29","doi-asserted-by":"crossref","unstructured":"Srivastava A, N (2024) Optimal control of a fractional order SEIQR epidemic model with non-monotonic incidence and quarantine class. Comput Biol Med 178:108682","DOI":"10.1016\/j.compbiomed.2024.108682"},{"issue":"1","key":"2430_CR30","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s00332-018-9475-9","volume":"29","author":"R Wu","year":"2018","unstructured":"Wu R, Zhao XQ (2018) A reaction-diffusion model of vector-borne disease with periodic delays. Journal of Nonlinear Science 29(1):29\u201364","journal-title":"Journal of Nonlinear Science"},{"key":"2430_CR31","doi-asserted-by":"publisher","first-page":"446","DOI":"10.1016\/j.nonrwa.2018.09.023","volume":"46","author":"M Zhou","year":"2019","unstructured":"Zhou M, Xiang H, Li Z (2019) Optimal control strategies for a reaction-diffusion epidemic system. Nonlinear Anal Real World Appl 46:446\u2013464","journal-title":"Nonlinear Anal Real World Appl"},{"key":"2430_CR32","first-page":"2024","volume":"1","author":"P Zhou","year":"2024","unstructured":"Zhou P, Wang J, Teng Z, Zheng Y, Wang K (2024) Optimal control analysis in a reaction-diffusion SIRC model with cross-immune class. Advances in Continuous and Discrete Models 1:2024","journal-title":"Advances in Continuous and Discrete Models"},{"key":"2430_CR33","unstructured":"Zinihi A, Sidi Ammi MR (2025) Global stability of a parabolic system involving a nonlinear diffusion operator with an example in epidemiology. Journal of Nonlinear Modeling and Analysis 7(2):453\u2013462"},{"key":"2430_CR34","doi-asserted-by":"crossref","unstructured":"Zinihi A, Sidi Ammi MR, Ehrhardt M (2025) Optimal control of a diffusive epidemiological model involving the Caputo-Fabrizio fractional time-derivative. Partial Differential Equations in Applied Mathematics 14:101188","DOI":"10.1016\/j.padiff.2025.101188"},{"key":"2430_CR35","doi-asserted-by":"crossref","unstructured":"Zinihi A, Sidi Ammi MR, Torres DFM (2025) Fractional differential equations of a reaction-diffusion SIR model involving the Caputo-fractional time-derivative and a nonlinear diffusion operator. Evolution Equations and Control Theory 14(5):944\u2013967","DOI":"10.3934\/eect.2025018"},{"key":"2430_CR36","doi-asserted-by":"crossref","unstructured":"Zinihi A, Sidi Ammi MR, Bachir A (2025) Multi-city modeling of epidemics using a topology-based SIR model: Neural network-enhanced SAIRD model. Journal of Computational Science 92:102721","DOI":"10.1016\/j.jocs.2025.102721"},{"key":"2430_CR37","doi-asserted-by":"crossref","unstructured":"Zinihi A, Ehrhardt M, Sidi Ammi MR (2026) A nonstandard finite difference scheme for an SEIQR epidemiological PDE model. Appl Math Comput 520:129953","DOI":"10.1016\/j.amc.2026.129953"},{"key":"2430_CR38","doi-asserted-by":"crossref","unstructured":"Zinihi A, Sidi\u00a0Ammi MR, Ehrhardt M (2025) Mathematical modeling and Hyers-Ulam stability for a nonlinear epidemiological model with $$\\Phi _p$$ operator and Mittag-Leffler kernel. Advances in Applied Mathematics and Mechanics, 17","DOI":"10.4208\/aamm.OA-2024-0134"},{"key":"2430_CR39","doi-asserted-by":"crossref","unstructured":"Zinihi A, Sidi\u00a0Ammi MR, Ehrhardt M, Bachir A (2025) Dynamical analysis of a cocaine-heroin epidemiological model with spatial distributions. Advances in Continuous and Discrete Models, 2025(1)","DOI":"10.1186\/s13662-025-03924-w"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02430-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-026-02430-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02430-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,31]],"date-time":"2026-07-31T04:04:47Z","timestamp":1785470687000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-026-02430-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["2430"],"URL":"https:\/\/doi.org\/10.1007\/s00285-026-02430-1","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7]]},"assertion":[{"value":"28 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 June 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 June 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that there are no problems or conflicts of interest between them that may affect the study in this paper.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"10"}}