{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T04:09:43Z","timestamp":1790741383310,"version":"4.1.0"},"reference-count":40,"publisher":"Public Library of Science (PLoS)","issue":"8","license":[{"start":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T00:00:00Z","timestamp":1724889600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS ONE"],"abstract":"<jats:p>In this research, we developed an epidemic model with a combination of Atangana-Baleanu Caputo derivative and classical operators for the hybrid operator\u2019s memory effects, allowing us to observe the dynamics and treatment effects at different time phases of syphilis infection caused by sex. The developed model properties, which take into account linear growth and Lipschitz requirements relating the rate of effects within its many sub-compartments according to the equilibrium points, include positivity, unique solution, exitance, and boundedness in the feasible domain. After conducting sensitivity analysis with various parameters influencing the model for the piecewise fractional operator, the reproductive number <jats:italic>R<\/jats:italic><jats:sub>0<\/jats:sub> for the biological viability of the model is determined. Generalized Ulam-Hyers stability results are employed to preserve global stability. The investigated model thus has a unique solution in the specified subinterval in light of the Banach conclusion, and contraction as a consequence holds for the Atangana-Baleanu Caputo derivative with classical operators. The piecewise model that has been suggested has a maximum of one solution. For numerical solutions, piecewise fractional hybrid operators at various fractional order values are solved using the Newton polynomial interpolation method. A comparison is also made between Caputo operator and the piecewise derivative proposed operator. This work improves our knowledge of the dynamics of syphilis and offers a solid framework for assessing the effectiveness of interventions for planning and making decisions to manage the illness.<\/jats:p>","DOI":"10.1371\/journal.pone.0307732","type":"journal-article","created":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T18:58:20Z","timestamp":1724957900000},"page":"e0307732","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":10,"title":["Investigation and application of a classical piecewise hybrid with a fractional derivative for the epidemic model: Dynamical transmission and modeling"],"prefix":"10.1371","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2263-3373","authenticated-orcid":true,"given":"Muhammad Umer","family":"Saleem","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Farman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kottakkaran Sooppy","family":"Nisar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7754-9355","authenticated-orcid":true,"given":"Aqeel","family":"Ahmad","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zainab","family":"Munir","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Evren","family":"Hincal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2024,8,29]]},"reference":[{"issue":"4","key":"pone.0307732.ref001","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1137\/S0036144500371907","article-title":"The mathematics of infectious diseases","volume":"42","author":"HW Hethcote","year":"2000","journal-title":"SIAM review"},{"key":"pone.0307732.ref002","doi-asserted-by":"crossref","first-page":"107098","DOI":"10.1016\/j.rinp.2023.107098","article-title":"A fractal-fractional sex structured syphilis model with three stages of infection and loss of immunity with analysis and modeling","volume":"54","author":"M Farman","year":"2023","journal-title":"Results in Physics"},{"issue":"4","key":"pone.0307732.ref003","doi-asserted-by":"crossref","first-page":"550","DOI":"10.5114\/ada.2021.108930","article-title":"Syphilis: then and now","volume":"38","author":"K Plagens-Rotman","year":"2021","journal-title":"Advances in Dermatology and Allergology\/Postepy Dermatologii i Alergologii"},{"issue":"2","key":"pone.0307732.ref004","first-page":"122","article-title":"Neurosyphilis, ocular syphilis, and otosyphilis: detection and treatment","volume":"106","author":"JG Jones-Vanderleest","year":"2022","journal-title":"American Family Physician"},{"key":"pone.0307732.ref005","doi-asserted-by":"crossref","DOI":"10.15585\/mmwr.mm6922a1","article-title":"Missed opportunities for prevention of congenital syphilis\u2014United States, 2018","volume":"69","author":"A Kimball","year":"2020","journal-title":"MMWR. Morbidity and Mortality Weekly Report"},{"key":"pone.0307732.ref006","unstructured":"Luise CD, Blank S, Brown J, Rubin S, Meyers A, Neylans L, et al. Primary and secondary syphilis among men who have sex with men-New York City, 2001."},{"issue":"8","key":"pone.0307732.ref007","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1017\/S0950268814002830","article-title":"Syphilis: antibiotic treatment and resistance","volume":"143","author":"LV Stamm","year":"2015","journal-title":"Epidemiology & Infection"},{"issue":"6","key":"pone.0307732.ref008","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1051\/mmnp\/20105605","article-title":"A new mathematical model of syphilis","volume":"5","author":"FA Milner","year":"2010","journal-title":"Mathematical Modelling of Natural Phenomena"},{"issue":"1","key":"pone.0307732.ref009","doi-asserted-by":"crossref","first-page":"5472","DOI":"10.1038\/s41598-020-62208-3","article-title":"The Spectrum-STI Groups model: syphilis prevalence trends across high-risk and lower-risk populations in Yunnan, China","volume":"10","author":"EL Korenromp","year":"2020","journal-title":"Scientific Reports"},{"issue":"11","key":"pone.0307732.ref010","doi-asserted-by":"crossref","first-page":"1364","DOI":"10.3390\/pathogens10111364","article-title":"A 2021 update on syphilis: Taking stock from pathogenesis to vaccines","volume":"10","author":"G Tiecco","year":"2021","journal-title":"Pathogens"},{"key":"pone.0307732.ref011","doi-asserted-by":"crossref","first-page":"112356","DOI":"10.1016\/j.chaos.2022.112356","article-title":"Mathematical analysis of the Cauchy type dynamical system under piecewise equations with Caputo fractional derivative","volume":"161","author":"K Shah","year":"2022","journal-title":"Chaos, Solitons & Fractals"},{"key":"pone.0307732.ref012","doi-asserted-by":"crossref","unstructured":"Abdeljawad T, Baleanu D. Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. arXiv preprint arXiv:1607.00262. 2016 Jun 25.","DOI":"10.22436\/jnsa.010.03.20"},{"key":"pone.0307732.ref013","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1016\/j.cnsns.2017.12.003","article-title":"On some new properties of fractional derivatives with Mittag-Leffler kernel","volume":"59","author":"D Baleanu","year":"2018","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"issue":"3","key":"pone.0307732.ref014","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1051\/mmnp\/201712302","article-title":"Numerical computation of a fractional derivative with non-local and non-singular kernel","volume":"12","author":"JD Djida","year":"2017","journal-title":"Mathematical Modelling of Natural Phenomena"},{"key":"pone.0307732.ref015","doi-asserted-by":"crossref","unstructured":"Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arXiv preprint arXiv:1602.03408. 2016 Jan 20.","DOI":"10.2298\/TSCI160111018A"},{"key":"pone.0307732.ref016","doi-asserted-by":"crossref","first-page":"107447","DOI":"10.1016\/j.cnsns.2023.107447","article-title":"Saturated delayed impulsive effects for fractional order nonlinear system with piecewise Caputo derivative and its application","volume":"126","author":"F Wang","year":"2023","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"issue":"2","key":"pone.0307732.ref017","doi-asserted-by":"crossref","first-page":"162","DOI":"10.3390\/fractalfract7020162","article-title":"Nonlinear piecewise Caputo fractional pantograph system with respect to another function","volume":"7","author":"MS Abdo","year":"2023","journal-title":"Fractal and Fractional"},{"issue":"7","key":"pone.0307732.ref018","doi-asserted-by":"crossref","first-page":"13383","DOI":"10.3934\/math.2022739","article-title":"Analysis of HIV\/AIDS model with Mittag-Leffler kernel","volume":"7","author":"MM Akram","year":"2022","journal-title":"Aims Math"},{"issue":"3","key":"pone.0307732.ref019","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1080\/03602532.2020.1770782","article-title":"The broad-spectrum antiviral recommendations for drug discovery against COVID-19","volume":"52","author":"A Hazafa","year":"2020","journal-title":"Drug metabolism reviews"},{"issue":"12","key":"pone.0307732.ref020","doi-asserted-by":"crossref","first-page":"4584","DOI":"10.1172\/JCI57173","article-title":"Syphilis: using modern approaches to understand an old disease","volume":"121","author":"EL Ho","year":"2011","journal-title":"The Journal of clinical investigation"},{"issue":"5","key":"pone.0307732.ref021","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1097\/DAD.0b013e3181e8b587","article-title":"The immunopathobiology of syphilis: the manifestations and course of syphilis are determined by the level of delayed-type hypersensitivity","volume":"33","author":"JA Carlson","year":"2011","journal-title":"The American journal of dermatopathology"},{"issue":"13","key":"pone.0307732.ref022","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1002\/pd.5728","article-title":"Congenital syphilis: a contemporary update on an ancient disease","volume":"40","author":"MW Rac","year":"2020","journal-title":"Prenatal Diagnosis"},{"issue":"4","key":"pone.0307732.ref023","first-page":"705","article-title":"Syphilis in the modern era: an update for physicians","volume":"27","author":"SE Cohen","year":"2013","journal-title":"Infectious Disease Clinics"},{"issue":"4","key":"pone.0307732.ref024","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1097\/00007435-199704000-00002","article-title":"The natural history of syphilis: implications for the transmission dynamics and control of infection","volume":"24","author":"GP Garnett","year":"1997","journal-title":"Sexually transmitted diseases"},{"issue":"18","key":"pone.0307732.ref025","doi-asserted-by":"crossref","first-page":"2350179","DOI":"10.1142\/S0217979223501795","article-title":"Adaptive the Dirichlet model of mobile\/immobile advection\/dispersion in a time-fractional sense with the reproducing kernel computational approach: Formulations and approximations","volume":"37","author":"OA Arqub","year":"2023","journal-title":"International Journal of Modern Physics B"},{"key":"pone.0307732.ref026","doi-asserted-by":"crossref","first-page":"100092","DOI":"10.1016\/j.csfx.2023.100092","article-title":"Hilbert approximate solutions and fractional geometric behaviors of a dynamical fractional model of social media addiction affirmed by the fractional Caputo differential operator","volume":"10","author":"B Maayah","year":"2023","journal-title":"Chaos, Solitons & Fractals: X"},{"key":"pone.0307732.ref027","first-page":"1","article-title":"Studying of the Covid-19 model by using the finite element method: theoretical and numerical simulation","author":"W Alhejili","year":"2023","journal-title":"Soft Computing"},{"issue":"1","key":"pone.0307732.ref028","doi-asserted-by":"crossref","first-page":"7","DOI":"10.17512\/jamcm.2024.1.01","article-title":"On the stability analysis for a semi-analytical scheme for solving the fractional order blood ethanol concentration system using LVIM","volume":"23","author":"M Adel","year":"2024","journal-title":"Journal of Applied Mathematics and Computational Mechanics"},{"key":"pone.0307732.ref029","doi-asserted-by":"crossref","first-page":"105994","DOI":"10.1016\/j.bspc.2024.105994","article-title":"Computational dynamics of a fractional order model of chickenpox spread in Phuket province","volume":"91","author":"SA Jose","year":"2024","journal-title":"Biomedical Signal Processing and Control"},{"issue":"5","key":"pone.0307732.ref030","doi-asserted-by":"crossref","first-page":"5060","DOI":"10.1002\/mma.8818","article-title":"Computational dynamics of a fractional order substance addictions transfer model with Atangana-Baleanu-Caputo derivative","volume":"46","author":"SA Jose","year":"2023","journal-title":"Mathematical Methods in the Applied Sciences"},{"issue":"08","key":"pone.0307732.ref031","doi-asserted-by":"crossref","first-page":"2040007","DOI":"10.1142\/S0218348X20400071","article-title":"Piecewise optimal fractional reproducing kernel solution and convergence analysis for the Atangana-Baleanu-Caputo model of the Lienard\u2019s equation","volume":"28","author":"S Momani","year":"2020","journal-title":"Fractals"},{"key":"pone.0307732.ref032","doi-asserted-by":"crossref","first-page":"109910","DOI":"10.1016\/j.chaos.2020.109910","article-title":"An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model","volume":"138","author":"B Ghanbari","year":"2020","journal-title":"Chaos, Solitons & Fractals"},{"issue":"3","key":"pone.0307732.ref033","doi-asserted-by":"crossref","first-page":"292","DOI":"10.3390\/axioms12030292","article-title":"On using piecewise fractional differential operator to study a dynamical system","volume":"12","author":"S Khan","year":"2023","journal-title":"Axioms"},{"key":"pone.0307732.ref034","doi-asserted-by":"crossref","first-page":"104694","DOI":"10.1016\/j.rinp.2021.104694","article-title":"Modeling third waves of Covid-19 spread with piecewise differential and integral operators: Turkey, Spain and Czechia","volume":"29","author":"A Atangana","year":"2021","journal-title":"Results in Physics"},{"issue":"11","key":"pone.0307732.ref035","doi-asserted-by":"crossref","first-page":"27268","DOI":"10.3934\/math.20231395","article-title":"On a SEIR-type model of COVID-19 using piecewise and stochastic differential operators undertaking management strategies","volume":"8","author":"MB Jeelani","year":"2023","journal-title":"AIMS Mathematics"},{"issue":"7","key":"pone.0307732.ref036","doi-asserted-by":"crossref","first-page":"5649","DOI":"10.1016\/j.aej.2021.11.039","article-title":"A robust study of a piecewise fractional order COVID-19 mathematical model","volume":"61","author":"A Zeb","year":"2022","journal-title":"Alexandria Engineering Journal"},{"issue":"7","key":"pone.0307732.ref037","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1086\/529147","article-title":"Clinical presentation of nipah virus infection in Bangladesh","volume":"46","author":"MJ Hossain","year":"2008","journal-title":"Clinical infectious diseases"},{"key":"pone.0307732.ref038","doi-asserted-by":"crossref","unstructured":"Zhao T, Liao Y, Jiao H. Mathematical Analysis of the Transmission Dynamics of Syphilis in China. 2022.","DOI":"10.21203\/rs.3.rs-1973748\/v1"},{"key":"pone.0307732.ref039","doi-asserted-by":"crossref","first-page":"112879","DOI":"10.1016\/j.chaos.2022.112879","article-title":"Dynamics of HIV-TB coinfection model using classical and Caputo piecewise operator: A dynamic approach with real data from South-East Asia, European and American regions","volume":"165","author":"C Xu","year":"2022","journal-title":"Chaos, Solitons & Fractals"},{"key":"pone.0307732.ref040","doi-asserted-by":"crossref","first-page":"110638","DOI":"10.1016\/j.chaos.2020.110638","article-title":"New concept in calculus: Piecewise differential and integral operators","volume":"145","author":"A Atangana","year":"2021","journal-title":"Chaos, Solitons & Fractals"}],"container-title":["PLOS ONE"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0307732","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T18:58:48Z","timestamp":1724957928000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0307732"}},"subtitle":[],"editor":[{"given":"Muhammad","family":"Aqeel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2024,8,29]]},"references-count":40,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8,29]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0307732","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,29]]}}}