{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T11:33:33Z","timestamp":1791804813712,"version":"4.3.4"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T00:00:00Z","timestamp":1785974400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,8,6]],"date-time":"2026-08-06T00:00:00Z","timestamp":1785974400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/100009392","name":"Prince Sattam bin Abdulaziz University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004242","name":"Princess Nourah Bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2026R528"],"award-info":[{"award-number":["PNURSP2026R528"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-026-51555-2","type":"journal-article","created":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T23:42:30Z","timestamp":1785973350000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Modelling the impact of awareness campaigns on HPV transmission dynamics: a neural network-enhanced mathematical model"],"prefix":"10.1038","volume":"16","author":[{"given":"Kamil","family":"Shah","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changqing","family":"Du","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fahad Sameer","family":"Alshammari","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sanaa Ahmed","family":"Bajri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamiden Abd","family":"El-Wahed Khalifa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,8,6]]},"reference":[{"issue":"1","key":"51555_CR1","doi-asserted-by":"publisher","first-page":"25","DOI":"10.3390\/pathogens12010025","volume":"12","author":"AG Mukherjee","year":"2022","unstructured":"Mukherjee, A. G. et al. Exploring the molecular pathogenesis, pathogen association, and therapeutic strategies against HPV infection. Pathogens 12 (1), 25 (2022).","journal-title":"Pathogens"},{"issue":"2","key":"51555_CR2","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1111\/j.1479-828X.2010.01269.x","volume":"51","author":"JR Carter","year":"2011","unstructured":"Carter, J. R., Ding, Z. & Rose, B. R. HPV infection and cervical disease: a review. Aust. N. Z. J. Obstet. Gynaecol. 51 (2), 103\u2013108 (2011).","journal-title":"Aust. N. Z. J. Obstet. Gynaecol."},{"key":"51555_CR3","unstructured":"Das, B. et al. Cancer of the uterine cervix and human papillomavirus infection. Curr. Sci., pp. 52\u201363. (2000)."},{"issue":"3","key":"51555_CR4","first-page":"570","volume":"55","author":"Y Liu","year":"2019","unstructured":"Liu, Y. et al. Current strategies against persistent human papillomavirus infection. Int. J. Oncol. 55 (3), 570\u2013584 (2019).","journal-title":"Int. J. Oncol."},{"issue":"9590","key":"51555_CR5","doi-asserted-by":"publisher","first-page":"890","DOI":"10.1016\/S0140-6736(07)61416-0","volume":"370","author":"M Schiffman","year":"2007","unstructured":"Schiffman, M. et al. Human papillomavirus and cervical cancer. Lancet 370 (9590), 890\u2013907 (2007).","journal-title":"lancet"},{"key":"51555_CR6","doi-asserted-by":"crossref","unstructured":"Viens, L. J. Human papillomavirus\u2013associated cancers\u2014United States, 2008\u20132012. MMWR. Morbid. Mortality Weekly Rep., 65. (2016).","DOI":"10.15585\/mmwr.mm6526a1"},{"issue":"10","key":"51555_CR7","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1038\/nrc1973","volume":"6","author":"R Roden","year":"2006","unstructured":"Roden, R. & Wu, T. C. How will HPV vaccines affect cervical cancer? Nat. Rev. Cancer. 6 (10), 753\u2013763 (2006).","journal-title":"Nat. Rev. Cancer"},{"issue":"3","key":"51555_CR8","first-page":"52","volume":"8","author":"A Yusupov","year":"2019","unstructured":"Yusupov, A. et al. The nonavalent vaccine: A review of high-risk HPVs and a plea to the CDC. Am. J. Stem Cells. 8 (3), 52 (2019).","journal-title":"Am. J. stem cells"},{"issue":"3","key":"51555_CR9","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1017\/S0950268815002198","volume":"144","author":"P Castle","year":"2016","unstructured":"Castle, P. & Maza, M. Prophylactic HPV vaccination: past, present, and future. Epidemiol. Infect. 144 (3), 449\u2013468 (2016).","journal-title":"Epidemiol. Infect."},{"key":"51555_CR10","doi-asserted-by":"crossref","unstructured":"Larson, C. P. et al. Infectious childhood diarrhea in developing countries, in Modern Infectious Disease Epidemiology: Concepts, Methods, Mathematical Models, and Public Health p. 291\u2013308 (Springer, 2009).","DOI":"10.1007\/978-0-387-93835-6_17"},{"issue":"1","key":"51555_CR11","doi-asserted-by":"publisher","first-page":"12616","DOI":"10.1038\/s41598-024-63263-w","volume":"14","author":"U Khan","year":"2024","unstructured":"Khan, U. et al. Optimal control strategies for toxoplasmosis disease transmission dynamics via harmonic mean-type incident rate. Sci. Rep. 14 (1), 12616 (2024).","journal-title":"Sci. Rep."},{"key":"51555_CR12","doi-asserted-by":"crossref","unstructured":"Shah, K. et al. Optimal control of COVID-19 through strategic mathematical modeling: Incorporating harmonic mean incident rate and vaccination. AIP Adv., 14(9). (2024).","DOI":"10.1063\/5.0229920"},{"key":"51555_CR13","doi-asserted-by":"crossref","unstructured":"Shah, K. et al. Modeling the dynamics of co-infection between COVID-19 and tuberculosis with quarantine strategies: A mathematical approach. AIP Adv., 14(7). (2024).","DOI":"10.1063\/5.0212904"},{"key":"51555_CR14","doi-asserted-by":"crossref","unstructured":"Khan, U. et al. Mathematical analysis of computer viruses through deterministic and stochastic models p. 104734 (Computers & Security, 2025).","DOI":"10.1016\/j.cose.2025.104734"},{"key":"51555_CR15","doi-asserted-by":"publisher","first-page":"100230","DOI":"10.1016\/j.health.2023.100230","volume":"4","author":"AI Alaje","year":"2023","unstructured":"Alaje, A. I. & Olayiwola, M. O. A fractional-order mathematical model for examining the spatiotemporal spread of COVID-19 in the presence of vaccine distribution. Healthc. Anal. 4, 100230 (2023).","journal-title":"Healthc. Analytics"},{"key":"51555_CR16","doi-asserted-by":"publisher","first-page":"100384","DOI":"10.1016\/j.rico.2024.100384","volume":"14","author":"MO Olayiwola","year":"2024","unstructured":"Olayiwola, M. O. et al. A mathematical modeling of COVID-19 treatment strategies utilizing the Laplace Adomian decomposition method. Results Control. Optim. 14, 100384 (2024).","journal-title":"Results Control Optim."},{"issue":"1","key":"51555_CR17","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1186\/s43088-022-00317-w","volume":"11","author":"AI Alaje","year":"2022","unstructured":"Alaje, A. I. et al. Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg\u2013de Vries equation. Beni-Suef Univ. J. Basic. Appl. Sci. 11 (1), 139 (2022).","journal-title":"Beni-Suef Univ. J. Basic. Appl. Sci."},{"key":"51555_CR18","doi-asserted-by":"publisher","first-page":"100117","DOI":"10.1016\/j.fraope.2024.100117","volume":"7","author":"MO Olayiwola","year":"2024","unstructured":"Olayiwola, M. O. & Alaje, A. I. Mathematical analysis of intrahost spread and control of dengue virus: unraveling the crucial role of antigenic immunity. Frankl. Open. 7, 100117 (2024).","journal-title":"Frankl. Open."},{"issue":"8","key":"51555_CR19","doi-asserted-by":"publisher","first-page":"e0290640","DOI":"10.1371\/journal.pone.0290640","volume":"18","author":"H Liu","year":"2023","unstructured":"Liu, H. et al. Impact of vaccine measures on the transmission dynamics of COVID-19. PLoS One. 18 (8), e0290640 (2023).","journal-title":"PLoS One"},{"key":"51555_CR20","doi-asserted-by":"publisher","first-page":"100179","DOI":"10.1016\/j.health.2023.100179","volume":"3","author":"MO Olayiwola","year":"2023","unstructured":"Olayiwola, M. O. et al. A Caputo fractional order epidemic model for evaluating the effectiveness of high-risk quarantine and vaccination strategies on the spread of COVID-19. Healthc. Anal. 3, 100179 (2023).","journal-title":"Healthc. Analytics"},{"key":"51555_CR21","doi-asserted-by":"crossref","unstructured":"OKWARE, F. O. Mathematical modelling of human papillomavirus dynamics with vaccination incorporating optimal control analysis. (2023).","DOI":"10.9734\/arjom\/2023\/v19i11751"},{"issue":"07","key":"51555_CR22","doi-asserted-by":"publisher","first-page":"1850092","DOI":"10.1142\/S1793524518500924","volume":"11","author":"A Omame","year":"2018","unstructured":"Omame, A. et al. Mathematical analysis of a two-sex Human Papillomavirus (HPV) model. Int. J. Biomathem. 11 (07), 1850092 (2018).","journal-title":"Int. J. Biomathematics"},{"issue":"1","key":"51555_CR23","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1007\/s12190-022-01767-2","volume":"69","author":"PK Rajan","year":"2023","unstructured":"Rajan, P. K., Kuppusamy, M. & Egbelowo, O. F. A mathematical model for human papillomavirus and its impact on cervical cancer in India. J. Appl. Math. Comput. 69 (1), 753\u2013770 (2023).","journal-title":"J. Appl. Math. Comput."},{"issue":"42","key":"51555_CR24","doi-asserted-by":"publisher","first-page":"6858","DOI":"10.1016\/j.vaccine.2010.08.030","volume":"28","author":"EH Elbasha","year":"2010","unstructured":"Elbasha, E. H. & Dasbach, E. J. Impact of vaccinating boys and men against HPV in the United States. Vaccine 28 (42), 6858\u20136867 (2010).","journal-title":"Vaccine"},{"issue":"1","key":"51555_CR25","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1007\/s10479-015-1834-4","volume":"251","author":"O Sharomi","year":"2017","unstructured":"Sharomi, O. & Malik, T. Optimal control in epidemiology. Ann. Oper. Res. 251 (1), 55\u201371 (2017).","journal-title":"Ann. Oper. Res."},{"issue":"1","key":"51555_CR26","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1186\/s13662-024-03838-z","volume":"2024","author":"C Liu","year":"2024","unstructured":"Liu, C. et al. Dynamic analysis of human papillomavirus transmission model under vaccine intervention: a case study of cervical cancer patients from Hungary. Adv. Continuous Discrete Models. 2024 (1), 36 (2024).","journal-title":"Adv. Continuous Discrete Models"},{"issue":"5\u20136","key":"51555_CR27","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1016\/j.mcm.2010.12.005","volume":"53","author":"AK Misra","year":"2011","unstructured":"Misra, A. K., Sharma, A. & Shukla, J. Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comput. Model. 53 (5\u20136), 1221\u20131228 (2011).","journal-title":"Math. Comput. Model."},{"key":"51555_CR28","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.mbs.2017.01.009","volume":"286","author":"GO Agaba","year":"2017","unstructured":"Agaba, G. O., Kyrychko, Y. N. & Blyuss, K. B. Mathematical model for the impact of awareness on the dynamics of infectious diseases. Math. Biosci. 286, 22\u201330 (2017).","journal-title":"Math. Biosci."},{"issue":"1","key":"51555_CR29","doi-asserted-by":"publisher","first-page":"1077","DOI":"10.1080\/17455030.2022.2028933","volume":"35","author":"MA Zahoor Raja","year":"2025","unstructured":"Zahoor Raja, M. A. et al. Heat transport in entropy-optimized flow of viscoelastic fluid due to Riga plate: analysis of artificial neural network. Waves Random Complex. Media. 35 (1), 1077\u20131096 (2025).","journal-title":"Waves Random Complex. Media"},{"issue":"1","key":"51555_CR30","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1007\/s44196-021-00057-2","volume":"15","author":"Z Sabir","year":"2022","unstructured":"Sabir, Z. et al. A novel design of morlet wavelet to solve the dynamics of nervous stomach nonlinear model. Int. J. Comput. Intell. Syst. 15 (1), 4 (2022).","journal-title":"Int. J. Comput. Intell. Syst."},{"issue":"05","key":"51555_CR31","doi-asserted-by":"publisher","first-page":"2240147","DOI":"10.1142\/S0218348X22401478","volume":"30","author":"B Wang","year":"2022","unstructured":"Wang, B. et al. Numerical computing to solve the nonlinear corneal system of eye surgery using the capability of Morlet wavelet artificial neural networks. Fractals 30 (05), 2240147 (2022).","journal-title":"Fractals"},{"issue":"2","key":"51555_CR32","doi-asserted-by":"publisher","first-page":"1607","DOI":"10.1016\/j.aej.2021.06.060","volume":"61","author":"M Shoaib","year":"2022","unstructured":"Shoaib, M. et al. Soft computing paradigm for Ferrofluid by exponentially stretched surface in the presence of magnetic dipole and heat transfer. Alexandria Eng. J. 61 (2), 1607\u20131623 (2022).","journal-title":"Alexandria Eng. J."},{"issue":"10","key":"51555_CR33","doi-asserted-by":"publisher","first-page":"1831","DOI":"10.1140\/epjs\/s11734-022-00457-1","volume":"231","author":"Z Sabir","year":"2022","unstructured":"Sabir, Z. et al. Applications of neural networks for the novel designed of nonlinear fractional seventh order singular system. Eur. Phys. J. Special Top. 231 (10), 1831\u20131845 (2022).","journal-title":"Eur. Phys. J. Special Top."},{"key":"51555_CR34","doi-asserted-by":"crossref","unstructured":"Raja, M. A. Z., Shah, F. H. & Syam, M. I. Intelligent computing approach to solve the nonlinear Van der Pol system for heartbeat model. Neural Comput. Appl. 30(12): pp. 3651\u20133675. (2018).","DOI":"10.1007\/s00521-017-2949-0"},{"key":"51555_CR35","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.jtice.2018.05.046","volume":"91","author":"A Mehmood","year":"2018","unstructured":"Mehmood, A. et al. Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery\u2013Hamel flow. J. Taiwan Inst. Chem. Eng. 91, 57\u201385 (2018).","journal-title":"J. Taiwan Inst. Chem. Eng."},{"key":"51555_CR36","doi-asserted-by":"publisher","first-page":"71326","DOI":"10.1109\/ACCESS.2020.2985763","volume":"8","author":"AH Bukhari","year":"2020","unstructured":"Bukhari, A. H. et al. Fractional neuro-sequential ARFIMA-LSTM for financial market forecasting. Ieee Access. 8, 71326\u201371338 (2020).","journal-title":"Ieee Access."},{"key":"51555_CR37","doi-asserted-by":"publisher","first-page":"111957","DOI":"10.1016\/j.knosys.2024.111957","volume":"296","author":"K Shah","year":"2024","unstructured":"Shah, K. et al. Neural network approach for cholera dynamics: Integrating deterministic and stochastic insights. Knowl. Based Syst. 296, 111957 (2024).","journal-title":"Knowl. Based Syst."},{"key":"51555_CR38","doi-asserted-by":"crossref","unstructured":"Shah, K. et al. Exploring the dynamics of smoking behavior using a neural network approach with high-order polynomial perturbations. Phys. Lett. A, p. 130929. (2025).","DOI":"10.1016\/j.physleta.2025.130929"},{"key":"51555_CR39","doi-asserted-by":"crossref","unstructured":"Shah, J. et al. Dynamics of Divorce Epidemic: A Nonlinear Mathematical Model With Stochastic Influences and Behavioral Factors, Math. Methods Appl. Sci., (2025).","DOI":"10.22541\/au.171476375.59771522\/v1"},{"key":"51555_CR40","doi-asserted-by":"crossref","unstructured":"Farooq, A. et al. Revolutionizing corruption dynamics: Integrating jury influence and fractional approaches with neural network and optimal control. Expert Syst. Appl., p. 129111. (2025).","DOI":"10.1016\/j.eswa.2025.129111"},{"issue":"1","key":"51555_CR41","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1140\/epjp\/s13360-025-06025-5","volume":"140","author":"J Shah","year":"2025","unstructured":"Shah, J. et al. Modeling scabies transmission dynamics: a stochastic approach with spectral collocation and neural network insights. Eur. Phys. J. Plus. 140 (1), 66 (2025).","journal-title":"Eur. Phys. J. Plus"},{"issue":"3","key":"51555_CR42","doi-asserted-by":"publisher","first-page":"3941","DOI":"10.1007\/s40808-024-01976-7","volume":"10","author":"MO Olayiwola","year":"2024","unstructured":"Olayiwola, M. O. & Alaje, A. I. Mathematical modelling of diphtheria transmission and vaccine efficacy using Nigeria. Model. Earth Syst. Environ. 10 (3), 3941\u20133967 (2024).","journal-title":"Model. Earth Syst. Environ."},{"issue":"6","key":"51555_CR43","first-page":"2097","volume":"45","author":"K Adewale Adedokun","year":"2025","unstructured":"Adewale Adedokun, K. et al. A Caputo fractional-order model of tuberculosis incorporating enlightenment and therapy using the Laplace-Adomian decomposition method. Int. J. Model. Simul. 45 (6), 2097\u20132111 (2025).","journal-title":"Int. J. Model. Simul."},{"issue":"1","key":"51555_CR44","doi-asserted-by":"publisher","first-page":"144","DOI":"10.1186\/s43088-022-00326-9","volume":"11","author":"AO Yunus","year":"2022","unstructured":"Yunus, A. O. et al. Mathematical analysis of fractional-order Caputo\u2019s derivative of coronavirus disease model via Laplace Adomian decomposition method. Beni-Suef Univ. J. Basic. Appl. Sci. 11 (1), 144 (2022).","journal-title":"Beni-Suef Univ. J. Basic. Appl. Sci."},{"issue":"1","key":"51555_CR45","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1186\/s43088-023-00343-2","volume":"12","author":"MK Kolawole","year":"2023","unstructured":"Kolawole, M. K. et al. Conceptual analysis of the combined effects of vaccination, therapeutic actions, and human subjection to physical constraint in reducing the prevalence of COVID-19 using the homotopy perturbation method. Beni-Suef Univ. J. Basic. Appl. Sci. 12 (1), 10 (2023).","journal-title":"Beni-Suef Univ. J. Basic. Appl. Sci."}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-51555-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-51555-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-51555-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T23:42:33Z","timestamp":1785973353000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-51555-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,6]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["51555"],"URL":"https:\/\/doi.org\/10.1038\/s41598-026-51555-2","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,6]]},"assertion":[{"value":"9 February 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 August 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","label":"Competing interests","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"24277"}}