{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,2]],"date-time":"2026-11-02T20:23:11Z","timestamp":1793650991219,"version":"4.5.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T00:00:00Z","timestamp":1771891200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T00:00:00Z","timestamp":1774569600000},"content-version":"vor","delay-in-days":31,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100020912","name":"King Faisal University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100020912","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-026-40853-4","type":"journal-article","created":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T14:33:30Z","timestamp":1771943610000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Enhancing rabies epidemic modeling with neural networks and fractional calculus"],"prefix":"10.1038","volume":"16","author":[{"given":"Ramsha","family":"Shafqat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"family":"Imran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ashraf","family":"Al-Quran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdelhamid Mohammed","family":"Djaouti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,24]]},"reference":[{"key":"40853_CR1","unstructured":"Kyaw, Phyo Sandar. A Study on the Awareness of Rabies Disease on Community Health (Phyo Sandar Kyaw, 2025). PhD diss., MERAL Portal."},{"key":"40853_CR2","unstructured":"Fooks, Anthony R., Alan C. Jackson, eds. Rabies: scientific basis of the disease and its management. Academic Press, (2020)."},{"issue":"3\u20134","key":"40853_CR3","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/0025-5564(75)90089-9","volume":"24","author":"R Bartoszy\u0144ski","year":"1975","unstructured":"Bartoszy\u0144ski, R. On the risk of rabies. Math. Biosci. 24(3\u20134), 355\u2013377. https:\/\/doi.org\/10.1016\/0025-5564(75)90089-9 (1975).","journal-title":"Math. Biosci."},{"key":"40853_CR4","doi-asserted-by":"publisher","DOI":"10.1515\/9781400841035","volume-title":"Modeling infectious diseases in humans and animals","author":"MJ Keeling","year":"2008","unstructured":"Keeling, M. J. & Rohani, P. Modeling infectious diseases in humans and animals (Princeton University Press, 2008)."},{"issue":"9","key":"40853_CR5","doi-asserted-by":"publisher","first-page":"1337","DOI":"10.3201\/eid1109.050172","volume":"11","author":"J Bingham","year":"2005","unstructured":"Bingham, J. Canine rabies ecology in southern Africa. Emerg. Infect. Dis. 11(9), 1337. https:\/\/doi.org\/10.3201\/eid1109.050172 (2005).","journal-title":"Emerg. Infect. Dis."},{"issue":"3","key":"40853_CR6","doi-asserted-by":"publisher","first-page":"681","DOI":"10.2307\/1382740","volume":"76","author":"JW Krebs","year":"1995","unstructured":"Krebs, J. W., Wilson, M. L. & Childs, J. E. Rabies\u2013epidemiology, prevention, and future research. J. Mammal. 76(3), 681\u2013694. https:\/\/doi.org\/10.2307\/1382740 (1995).","journal-title":"J. Mammal."},{"issue":"3","key":"40853_CR7","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/S0022-5193(05)80679-4","volume":"156","author":"JD Murray","year":"1992","unstructured":"Murray, J. D. & Seward, W. L. On the spatial spread of rabies among foxes with immunity. J. Theor. Biol. 156(3), 327\u2013348. https:\/\/doi.org\/10.1016\/S0022-5193(05)80679-4 (1992).","journal-title":"J. Theor. Biol."},{"issue":"10","key":"40853_CR8","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1111\/j.1439-0450.2004.00801.x","volume":"51","author":"P Bugnon","year":"2004","unstructured":"Bugnon, P., Breitenmoser, U., Peterhans, E. & Zanoni, R. Efficacy of oral vaccination in the final stage of fox rabies elimination in Switzerland. Journal of Veterinary Medicine, Series B 51(10), 433\u2013437. https:\/\/doi.org\/10.1111\/j.1439-0450.2004.00801.x (2004).","journal-title":"Journal of Veterinary Medicine, Series B"},{"issue":"1","key":"40853_CR9","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.jtbi.2005.08.026","volume":"240","author":"A Harnos","year":"2006","unstructured":"Harnos, A., Reiczigel, J., Solymosi, N. & Vattay, G. Analysis of the effect of immunization in rabies time series. J. Theor. Biol. 240(1), 72\u201377. https:\/\/doi.org\/10.1016\/j.jtbi.2005.08.026 (2006).","journal-title":"J. Theor. Biol."},{"issue":"18","key":"40853_CR10","doi-asserted-by":"publisher","first-page":"7717","DOI":"10.1073\/pnas.0609122104","volume":"104","author":"K Hampson","year":"2007","unstructured":"Hampson, K. et al. Synchronous cycles of domestic dog rabies in sub-Saharan Africa and the impact of control efforts. Proceedings of the National Academy of Sciences 104(18), 7717\u20137722. https:\/\/doi.org\/10.1073\/pnas.0609122104 (2007).","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"1","key":"40853_CR11","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1890\/1540-9295","volume":"3","author":"KF Smith","year":"2005","unstructured":"Smith, K. F. et al. Ecological theory to enhance infectious disease control and public health policy.. Front. Ecol. Environ. 3(1), 29\u201337. https:\/\/doi.org\/10.1890\/1540-9295 (2005).","journal-title":"Front. Ecol. Environ."},{"issue":"2\u20133","key":"40853_CR12","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/S0304-3800(98)00198-7","volume":"117","author":"H-H Thulke","year":"1999","unstructured":"Thulke, H.-H. et al. From pattern to practice: A scaling-down strategy for spatially explicit modelling illustrated by the spread and control of rabies. Ecol. Modell. 117(2\u20133), 179\u2013202. https:\/\/doi.org\/10.1016\/S0304-3800(98)00198-7 (1999).","journal-title":"Ecol. Modell."},{"key":"40853_CR13","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.chaos.2018.06.031","volume":"114","author":"N Hamdan","year":"2018","unstructured":"Hamdan, N. & Kilicman, A. A fractional order SIR epidemic model for dengue transmission. Chaos Solitons Fractals 114, 55\u201362. https:\/\/doi.org\/10.1016\/j.chaos.2018.06.031 (2018).","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"40853_CR14","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-019-1981-z","volume":"2019","author":"N Hamdan","year":"2019","unstructured":"Hamdan, N. & Kilicman, A. Analysis of the fractional order dengue transmission model: A case study in Malaysia. Adv. Differ. Equations 2019(1), 31. https:\/\/doi.org\/10.1186\/s13662-019-1981-z (2019).","journal-title":"Adv. Differ. Equations"},{"key":"40853_CR15","unstructured":"Atangana, A., Araz, Seda \u0130\u011fret Araz. New numerical scheme with Newton polynomial: theory, methods, and applications. Academic Press, (2021)."},{"key":"40853_CR16","doi-asserted-by":"crossref","unstructured":"Atangana, Abdon., Dumitru Baleanu. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408 (2016).","DOI":"10.2298\/TSCI160111018A"},{"key":"40853_CR17","doi-asserted-by":"publisher","first-page":"789","DOI":"10.1016\/j.rinp.2017.01.025","volume":"7","author":"NA Sheikh","year":"2017","unstructured":"Sheikh, N. A. et al. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys. 7, 789\u2013800 (2017).","journal-title":"Results Phys."},{"issue":"3","key":"40853_CR18","doi-asserted-by":"publisher","first-page":"953","DOI":"10.32604\/cmes.2020.010236","volume":"124","author":"D Baleanu","year":"2020","unstructured":"Baleanu, D., Ghanbari, B., Asad, J. H., Jajarmi, A. & Pirouz, H. M. Planar system-masses in an equilateral triangle: Numerical study within fractional calculus. Comput. Model. Eng. Sci. 124(3), 953\u2013968. https:\/\/doi.org\/10.32604\/cmes.2020.010236 (2020).","journal-title":"Comput. Model. Eng. Sci."},{"key":"40853_CR19","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.aej.2023.01.027","volume":"69","author":"S Rashid","year":"2023","unstructured":"Rashid, S. et al. Stochastic dynamical analysis of the co-infection of the fractional pneumonia and typhoid fever disease model with cost-effective techniques and crossover effects. Alex. Eng. J. 69, 35\u201355. https:\/\/doi.org\/10.1016\/j.aej.2023.01.027 (2023).","journal-title":"Alex. Eng. J."},{"issue":"11","key":"40853_CR20","doi-asserted-by":"publisher","first-page":"9175","DOI":"10.1016\/j.aej.2022.02.054","volume":"61","author":"D Baleanu","year":"2022","unstructured":"Baleanu, D., Ghassabzade, F. A., Nieto, J. J. & Jajarmi, A. On a new and generalized fractional model for a real cholera outbreak. Alex. Eng. J. 61(11), 9175\u20139186. https:\/\/doi.org\/10.1016\/j.aej.2022.02.054 (2022).","journal-title":"Alex. Eng. J."},{"key":"40853_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.104950","volume":"31","author":"A Jajarmi","year":"2021","unstructured":"Jajarmi, A., Dumitru Baleanu, K., Zarghami Vahid, H., Pirouz, M. & Asad, J. H. A new and general fractional Lagrangian approach: A capacitor microphone case study. Results Phys. 31, 104950. https:\/\/doi.org\/10.1016\/j.rinp.2021.104950 (2021).","journal-title":"Results Phys."},{"key":"40853_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2019.109450","volume":"130","author":"DK Das","year":"2020","unstructured":"Das, D. K., Khajanchi, S. & Kar, T. K. Transmission dynamics of tuberculosis with multiple re-infections. Chaos Solitons Fractals 130, 109450. https:\/\/doi.org\/10.1016\/j.chaos.2019.109450 (2020).","journal-title":"Chaos Solitons Fractals"},{"key":"40853_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2022.127206","volume":"430","author":"S Bera","year":"2022","unstructured":"Bera, S., Khajanchi, S. & Roy, T. K. Dynamics of an HTLV-I infection model with delayed CTLs immune response. Appl. Math. Comput. 430, 127206. https:\/\/doi.org\/10.1016\/j.amc.2022.127206 (2022).","journal-title":"Appl. Math. Comput."},{"issue":"8","key":"40853_CR24","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/s13360-022-03156-x","volume":"137","author":"J Mondal","year":"2022","unstructured":"Mondal, J., Khajanchi, S. & Samui, P. Impact of media awareness in mitigating the spread of an infectious disease with application to optimal control. Eur. Phys. J. Plus 137(8), 983. https:\/\/doi.org\/10.1140\/epjp\/s13360-022-03156-x (2022).","journal-title":"Eur. Phys. J. Plus"},{"issue":"1","key":"40853_CR25","doi-asserted-by":"publisher","first-page":"32023","DOI":"10.1038\/s41598-025-15195-2","volume":"15","author":"A Al-Quran","year":"2025","unstructured":"Al-Quran, A., Shafqat, R., Alsaadi, A. & Djaouti, A. M. Poliomyelitis dynamics with fractional order derivatives and deep neural networks. Sci. Rep. 15(1), 32023 (2025).","journal-title":"Sci. Rep."},{"issue":"10","key":"40853_CR26","doi-asserted-by":"publisher","first-page":"23261","DOI":"10.3934\/math.20251032","volume":"10","author":"MZ Meetei","year":"2025","unstructured":"Meetei, M. Z., Shafqat, R., Msmali, A. H. & Hamali, W. Deep neural network applications in mathematical epidemiology: Case of rabies virus. AIMS Math. 10(10), 23261\u201323291 (2025).","journal-title":"AIMS Math."},{"issue":"1","key":"40853_CR27","doi-asserted-by":"publisher","first-page":"21853","DOI":"10.1038\/s41598-024-70788-7","volume":"14","author":"A Turab","year":"2024","unstructured":"Turab, A. et al. Predictive modeling of hepatitis B viral dynamics: A caputo derivative-based approach using artificial neural networks. Sci. Rep. 14(1), 21853 (2024).","journal-title":"Sci. Rep."},{"key":"40853_CR28","unstructured":"Wiraningsih, Eti Dwi, W. L. Aryati, Syamsuddin Toaha., Suzanne Lenhart. Optimal control for SEIR rabies model between dogs and human with vaccination effect in dogs. In Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010)\u00a0\u00a06, 161\u20131175\u00a0(Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia, 2010)."},{"key":"40853_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.104652","volume":"29","author":"A Khan","year":"2021","unstructured":"Khan, A., Zarin, R., Ahmed, I., Yusuf, A. & Humphries, U. W. Numerical and theoretical analysis of Rabies model under the harmonic mean type incidence rate. Results Phys. 29, 104652. https:\/\/doi.org\/10.1016\/j.rinp.2021.104652 (2021).","journal-title":"Results Phys."},{"issue":"12","key":"40853_CR30","doi-asserted-by":"publisher","first-page":"33495","DOI":"10.3934\/math.20241599","volume":"9","author":"R Shafqat","year":"2024","unstructured":"Shafqat, R. & Alsaadi, A. Artificial neural networks for stability analysis and simulation of delayed rabies spread models. AIMS Math. 9(12), 33495\u201333531. https:\/\/doi.org\/10.3934\/math.20241599 (2024).","journal-title":"AIMS Math."},{"issue":"1","key":"40853_CR31","doi-asserted-by":"publisher","DOI":"10.1155\/2017\/2451237","volume":"2017","author":"JKK Asamoah","year":"2017","unstructured":"Asamoah, J. K. K., Oduro, F. T., Bonyah, E. & Seidu, B. Modelling of rabies transmission dynamics using optimal control analysis. J. Appl. Math. 2017(1), 2451237. https:\/\/doi.org\/10.1155\/2017\/2451237 (2017).","journal-title":"J. Appl. Math."},{"key":"40853_CR32","first-page":"34","volume":"83","author":"Paritosh Bhattacharya","year":"2015","unstructured":"Bhattacharya, Paritosh, Paul, Susmita & Biswas, Prantik. Mathematical modeling of treatment SIR model with respect to variable contact rate. International Proceedings of Economics Development and Research 83, 34\u201341 (2015).","journal-title":"International Proceedings of Economics Development and Research"},{"issue":"4","key":"40853_CR33","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. https:\/\/doi.org\/10.1137\/S0036144500371907 (2000).","journal-title":"SIAM Rev."},{"issue":"1","key":"40853_CR34","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1111\/j.1467-842X.1988.tb00464.x","volume":"30","author":"K Dietz","year":"1988","unstructured":"Dietz, K. The first epidemic model: A historical note on PD En\u2019ko. Aust. J. Stat. 30(1), 56\u201365. https:\/\/doi.org\/10.1111\/j.1467-842X.1988.tb00464.x (1988).","journal-title":"Aust. J. Stat."},{"issue":"2","key":"40853_CR35","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.biosystems.2006.09.038","volume":"90","author":"H Zhang","year":"2007","unstructured":"Zhang, H., Jiao, J. & Chen, L. Pest management through continuous and impulsive control strategies. Biosystems 90(2), 350\u2013361. https:\/\/doi.org\/10.1016\/j.biosystems.2006.09.038 (2007).","journal-title":"Biosystems"},{"issue":"2\u20133","key":"40853_CR36","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/S0096-3003(01)00004-2","volume":"127","author":"P \u00d6gren","year":"2002","unstructured":"\u00d6gren, P. & Martin, C. F. Vaccination strategies for epidemics in highly mobile populations. Appl. Math. Comput. 127(2\u20133), 261\u2013276. https:\/\/doi.org\/10.1016\/S0096-3003(01)00004-2 (2002).","journal-title":"Appl. Math. Comput."},{"key":"40853_CR37","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/781028","volume":"1","author":"D Goufo","year":"2014","unstructured":"Goufo, D., Franc, E., Noutchie, S. C. O. & Mugisha, S. A fractional SEIR epidemic model for spatial and temporal spread of measles in metapopulations. Abstr. Appl. Anal. 1, 781028. https:\/\/doi.org\/10.1155\/2014\/781028 (2014).","journal-title":"Abstr. Appl. Anal."},{"issue":"4","key":"40853_CR38","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1002\/oca.710","volume":"23","author":"HR Joshi","year":"2002","unstructured":"Joshi, H. R. Optimal control of an HIV immunology model. Optim. Control Appl. Methods 23(4), 199\u2013213. https:\/\/doi.org\/10.1002\/oca.710 (2002).","journal-title":"Optim. Control Appl. Methods"},{"key":"40853_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.23952\/jnfa.2018.33","volume":"33","author":"I Boudjema","year":"2018","unstructured":"Boudjema, I. & Touaoula, T. A. R. I. K. M. O. H. A. M. M. E. D. Global stability of an infection and vaccination age-structured model with general nonlinear incidence. J. Nonlinear Funct. Anal 33, 1\u201321. https:\/\/doi.org\/10.23952\/jnfa.2018.33 (2018).","journal-title":"J. Nonlinear Funct. Anal"},{"key":"40853_CR40","unstructured":"Yan, Mimi, Shi, Ruiqing & Ji, Zhuchao. Analysis of an SIR epidemic model with saturation incidence rate in an environmentally-driven infectious disease. J. Nonlinear Funct. Anal 2015(10), (2015)."},{"key":"40853_CR41","doi-asserted-by":"publisher","unstructured":"Humphries, T., Loreto, M., Halter, B. & W. O\u2019Keeffe, & L. Ramirez.,. Comparison of regularized and superiorized methods for tomographic image reconstruction. Journal of Applied & Numerical Optimization 2(1), (2020). https:\/\/doi.org\/10.23952\/jano.2.2020.1.06","DOI":"10.23952\/jano.2.2020.1.06"}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-40853-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-40853-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-40853-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T23:01:51Z","timestamp":1774652511000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-40853-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,24]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["40853"],"URL":"https:\/\/doi.org\/10.1038\/s41598-026-40853-4","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,24]]},"assertion":[{"value":"1 December 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 February 2026","order":3,"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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"10409"}}