{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T04:00:41Z","timestamp":1780286441935,"version":"3.54.0"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T00:00:00Z","timestamp":1776643200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"vor","delay-in-days":42,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["BMC Infect Dis"],"DOI":"10.1186\/s12879-026-13288-5","type":"journal-article","created":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T04:05:32Z","timestamp":1776657932000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fractional-order modeling of HIV\/AIDS dynamics in the working-class population using the ABC operator"],"prefix":"10.1186","volume":"26","author":[{"given":"Abdulsamad Engida","family":"Sado","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sisay Fikadu","family":"Jaleta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chernet Tuge","family":"Deressa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,20]]},"reference":[{"key":"13288_CR1","doi-asserted-by":"publisher","DOI":"10.1097\/coh.0b013e328343acad","volume-title":"Interaction of mathematical modeling and social and behavioral HIV \/ AIDS research","author":"S Cassels","year":"2011","unstructured":"Cassels S, Goodreau SM. Interaction of mathematical modeling and social and behavioral HIV \/ AIDS research. 2011. https:\/\/doi.org\/10.1097\/coh.0b013e328343acad."},{"issue":"1","key":"13288_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41598-022-24459-0","volume":"12","author":"RB De Oliveira","year":"2022","unstructured":"De Oliveira RB, Rubio FA, Anderle R, Sanchez M, de Souza LE, Macinko J, et al. Incorporating social determinants of health into the mathematical modeling of HIV \/ AIDS. Sci Rep. 2022;12(1):1\u20139. https:\/\/doi.org\/10.1038\/s41598-022-24459-0.","journal-title":"Sci Rep"},{"key":"13288_CR3","doi-asserted-by":"publisher","unstructured":"Buot M-LG, Docena JP, Ratemo BK, Bittner MJ, Burlew JT, Nuritdinov AR, et al. Beyond race and place: distal sociological determinants of HIV disparities. PLoS One. 2014;9(4):e91711. https:\/\/doi.org\/10.1371\/journal.pone.0091711.","DOI":"10.1371\/journal.pone.0091711"},{"issue":"1","key":"13288_CR4","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s11904-021-00595-6","volume":"19","author":"RM Beltran","year":"2022","unstructured":"Beltran RM, Holloway IW, Hong C, Miyashita A, Cordero L, Wu E, et al. Social determinants of disease: HIV and COVID-19 experiences. Curr HIV\/AIDS Rep. 2022;19(1):101\u201312. https:\/\/doi.org\/10.1007\/s11904-021-00595-6.","journal-title":"Curr HIV\/AIDS Rep"},{"key":"13288_CR5","doi-asserted-by":"publisher","unstructured":"Id DR, et al. Evaluating the impact of social determinants, conditional cash transfers, and primary health care on HIV\/AIDS: study protocol of a retrospective and forecasting approach based on the data integration with a cohort of 100 million Brazilians. 2022;1\u201315. https:\/\/doi.org\/10.1371\/journal.pone.0265253.","DOI":"10.1371\/journal.pone.0265253"},{"issue":"7","key":"13288_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/ofid\/ofab330","volume":"8","author":"TW Menza","year":"2021","unstructured":"Menza TW, Hixson LK, Lipira L, Drach L. Social determinants of health and care outcomes among people with HIV in the United States. Open Forum Infect Dis. 2021;8(7):1\u201310. https:\/\/doi.org\/10.1093\/ofid\/ofab330.","journal-title":"Open Forum Infect Dis"},{"key":"13288_CR7","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.2000.7194","volume-title":"Analysis of fractional differential equations","author":"M Analysis","year":"2002","unstructured":"Analysis M, Diethelm K, Universit T. Analysis of fractional differential equations, December 2016. 2002. https:\/\/doi.org\/10.1006\/jmaa.2000.7194."},{"key":"13288_CR8","unstructured":"Rahimy M. Applications of fractional differential equations. 2010;4(50):2453\u201361."},{"issue":"3","key":"13288_CR9","first-page":"997","volume":"7","author":"AE Sado","year":"2025","unstructured":"Sado AE, Duressa GF, Deressa CT. The fractal-fractional mathematical model analysis of the impact of HIV\/AIDS on the working-class in Ethiopia. J Educ Chang Nonlinear Modeling Anal. 2025;7(3):997\u20131036. https:\/\/journal.global-sci.orgjnma.","journal-title":"J Educ Chang Nonlinear Modeling Anal"},{"issue":"1","key":"13288_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s12982-025-00959-y","volume":"22","author":"AE Sado","year":"2025","unstructured":"Sado AE, Duressa GF, Deressa CT. Piecewise combination of fractional and stochastic mathematical modeling to analyses the effect of HIV\/AIDS on working-class population. Discov Public Health. 2025;22(1):1\u201323. https:\/\/doi.org\/10.1186\/s12982-025-00959-y.","journal-title":"Discov Public Health"},{"key":"13288_CR11","doi-asserted-by":"publisher","unstructured":"Cheneke KR. Optimal control and bifurcation analysis of HIV model. Comput Math Method M. 2023;2023(1). https:\/\/doi.org\/10.1155\/2023\/4754426.","DOI":"10.1155\/2023\/4754426"},{"key":"13288_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2021\/7471290","volume":"2021","author":"KR Cheneke","year":"2021","unstructured":"Cheneke KR, Rao KP, Edessa GK. Bifurcation and stabillity analysis of HIV transmission model with optimal control. Pac J Math. 2021;2021:1\u201314. https:\/\/doi.org\/10.1155\/2021\/7471290.","journal-title":"Pac J Math"},{"key":"13288_CR13","doi-asserted-by":"publisher","first-page":"101498","DOI":"10.1016\/j.imu.2024.101498","volume":"47","author":"AE Sado","year":"2024","unstructured":"Sado AE, Kotola BS. A mathematical model based on ABC fractional order for TB transmission with treatment interruptions in case of Bule Hora town, Ethiopia. Inf Med Unlocked. 2024;47:101498. https:\/\/doi.org\/10.1016\/j.imu.2024.101498.","journal-title":"Inf Med Unlocked"},{"key":"13288_CR14","unstructured":"Victor AO, Oduwole HK. Stability analysis of disease free equilibrium state (E 0) for an endemic deterministic model for HIV\/AIDS, medRxiv. 2020;2020\u201323."},{"issue":"1","key":"13288_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pone.0244543","volume":"16","author":"S Das","year":"2021","unstructured":"Das S, Ghosh P, Banerjee S, Pyne S, Chattopadhyay J, Mukhopadhyay I. Determination of critical community size from an HIV\/AIDS model. PLoS One. 2021, January;16(1):1\u201316. https:\/\/doi.org\/10.1371\/journal.pone.0244543.","journal-title":"PLoS One"},{"key":"13288_CR16","doi-asserted-by":"publisher","unstructured":"Mutiara AS, Kasbawati AKJ, Anisa, and Samsir R. A comparison of deterministic and stochastic models on the dynamics of HIV and Cd4+ T-Cell interactions. Commun Math Biol Neurosci. 2022;2022:1\u201325. https:\/\/doi.org\/10.28919\/cmbn\/7147.","DOI":"10.28919\/cmbn\/7147"},{"key":"13288_CR17","doi-asserted-by":"publisher","unstructured":"Luchko Y, Yamamoto M. The general fractional derivative and related fractional differential equations. 2020;(1):1\u201320. https:\/\/doi.org\/10.3390\/math8122115.","DOI":"10.3390\/math8122115"},{"key":"13288_CR18","doi-asserted-by":"publisher","unstructured":"Li1 Z-B, He J-H. Fractional complex transform for fractional differential equations. MCA. 2010;15(5):970\u201373. https:\/\/doi.org\/10.3390\/mca15050970.","DOI":"10.3390\/mca15050970"},{"key":"13288_CR19","doi-asserted-by":"publisher","unstructured":"Boukhouima A, Lotfi EM, Mahrouf M, Rosa S, Torres DFM, Yousfi N. Stability analysis and optimal control of a fractional HIV-AIDS epidemic model with memory and general incidence rate. Eur Phys J Plus. 2021;136(1). https:\/\/doi.org\/10.1140\/epjp\/s13360-020-01013-3.","DOI":"10.1140\/epjp\/s13360-020-01013-3"},{"key":"13288_CR20","doi-asserted-by":"publisher","unstructured":"Aslam M, Murtaza R, Abdeljawad T, Rahman G, Khan A, Khan H, et al. A fractional order HIV\/AIDS epidemic model with Mittag-Leffler kernel. Adv Differ Equations. 2021;2021(1). https:\/\/doi.org\/10.1186\/s13662-021-03264-5.","DOI":"10.1186\/s13662-021-03264-5"},{"issue":"3","key":"13288_CR21","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s10441-020-09407-7","volume":"69","author":"B Seidu","year":"2021","unstructured":"Seidu B, Makinde ODMCS, Bornaa CS. Mathematical analysis of an industrial HIV\/AIDS model that incorporates carefree attitude towards sex. Acta Biotheor. 2021;69(3):257\u201376. https:\/\/doi.org\/10.1007\/s10441-020-09407-7.","journal-title":"Acta Biotheor"},{"key":"13288_CR22","doi-asserted-by":"publisher","unstructured":"Sado AE, Duressa GF, Deressa CT. Mathematical modeling of HIV\/AIDS dynamics with optimal control analysis. Sci Afr. 2025;30:e02972. https:\/\/doi.org\/10.1016\/j.sciaf.2025.e02972.","DOI":"10.1016\/j.sciaf.2025.e02972"},{"issue":"2","key":"13288_CR23","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10441-015-9255-y","volume":"63","author":"B Seidu","year":"2015","unstructured":"Seidu B, Makinde OD, Seini I. Seini,mathematical analysis of the effects of HIV-Malaria Co-infection on workplace productivity. Acta Biotheor. 2015;63(2):151\u201382. https:\/\/doi.org\/10.1007\/s10441-015-9255-y.","journal-title":"Acta Biotheor"},{"key":"13288_CR24","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/j.chaos.2019.03.022","volume":"122","author":"A Khan","year":"2019","unstructured":"Khan A, Gomez-Aguilar JF, Khan TS, Khan H. Stability analysis and numerical solutions of fractional order HIV\/AIDS model. Chaos, Solitons Fractals. 2019;122:119\u201328. https:\/\/doi.org\/10.1016\/j.chaos.2019.03.022.","journal-title":"Chaos, Solitons Fractals"},{"issue":"5","key":"13288_CR25","doi-asserted-by":"publisher","first-page":"12109","DOI":"10.3934\/math.2023610","volume":"8","author":"& Boutiara","year":"2023","unstructured":"Boutiara A, Matar MM, Alzabut J, Esmael M, Samei M, Khan H. On ABC coupled Langevin fractional differential equations constrained by Perov\u2019s fixed point in generalized Banach spaces. AIMS Math. 2023;8(5):12109\u201332. https:\/\/doi.org\/10.3934\/math.2023610.","journal-title":"Aims Math"},{"issue":"1","key":"13288_CR26","doi-asserted-by":"publisher","first-page":"9291","DOI":"10.1038\/s41598-025-93436-0","volume":"15","author":"M Khan","year":"2025","unstructured":"Khan M, Khan N, Ullah I, Shah K, Abdeljawad T, Abdalla B. A novel fractal fractional mathematical model for HIV\/AIDS transmission stability and sensitivity with numerical analysis. Sci Rep. 2025;15(1):9291. https:\/\/doi.org\/10.1038\/s41598-025-93436-0.","journal-title":"Sci Rep"},{"issue":"1","key":"13288_CR27","doi-asserted-by":"publisher","first-page":"1000","DOI":"10.1186\/s12879-025-11383-7","volume":"25","author":"BC Agbata","year":"2025","unstructured":"Agbata BC, Cenaj E, Dervishi R, Danjuma YJ, Shior MM-A, Abah E, et al. Fractional-order mathematical model for Monkeypox transmission dynamics using the Atangana-Baleanu Caputo operator. BMC Infect Dis. 2025;25(1):1000. https:\/\/doi.org\/10.1186\/s12879-025-11383-7.","journal-title":"BMC Infect Dis"},{"issue":"103515","key":"13288_CR28","doi-asserted-by":"publisher","first-page":"103515","DOI":"10.1016\/j.rinp.2020.103515","volume":"19","author":"A Atangana","year":"2020","unstructured":"Atangana A. Extension of rate of change concept: from local to nonlocal operators with applications. Results Phys. 2020;19(103515):103515. https:\/\/doi.org\/10.1016\/j.rinp.2020.103515.","journal-title":"Results Phys"},{"key":"13288_CR29","first-page":"340","volume":"198","author":"I Podlubny","year":"1999","unstructured":"Podlubny I. An introduction to fractional derivatives, fractional differential equations, to methods of their solution, and some of their applications. Math Sci Eng. 1999;198:340.","journal-title":"Math Sci Eng"},{"key":"13288_CR30","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-4036-9","volume-title":"Topics in fractional differential equations","author":"S Abbas","year":"2012","unstructured":"Abbas S, Benchohra M, N\u2019Gu\u00e9r\u00e9kata GM. Topics in fractional differential equations. Vol. 27. Springer Science & Business Media; 2012."},{"key":"13288_CR31","doi-asserted-by":"crossref","unstructured":"Rassias TM, Brzdek J, Eds. Functional equations in mathematical analysis. Vol. 86. New York: Springer; 2012.","DOI":"10.1007\/978-1-4614-0055-4"},{"key":"13288_CR32","doi-asserted-by":"publisher","unstructured":"Kai D. The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type. In: Lecture notes in mathematics. Springer; 2010. https:\/\/doi.org\/10.1007\/978-3-642-14574-2.","DOI":"10.1007\/978-3-642-14574-2"},{"issue":"84","key":"13288_CR33","first-page":"191","volume":"1","author":"C Castillo-Chavez","year":"2013","unstructured":"Castillo-Chavez C. Dynamical models of tuberculosis and applications. CBMS-NSF Reg Conf Ser Appl Math. 2013;1(84):191\u2013217.","journal-title":"CBMS-NSF Reg Conf Ser Appl Math"},{"key":"13288_CR34","doi-asserted-by":"crossref","unstructured":"Martcheva M. Introduction to epidemic modeling, in an Introduction to mathematical epidemiology. 2015.","DOI":"10.1007\/978-1-4899-7612-3"},{"issue":"1","key":"13288_CR35","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1016\/j.jmaa.2012.04.077","volume":"395","author":"AB Gumel","year":"2012","unstructured":"Gumel AB. Causes of backward bifurcations in some epidemiological models. J Math Anal Appl. 2012;395(1):355\u201365. https:\/\/doi.org\/10.1016\/j.jmaa.2012.04.077.","journal-title":"J Math Anal Appl"},{"key":"13288_CR36","doi-asserted-by":"publisher","unstructured":"Oyovwevotu SO. Mathematical modelling for assessing the impact of intervention strategies on HIV\/AIDS high risk group population dynamics. Heliyon. 2021;7(10):e07991. https:\/\/doi.org\/10.1016\/j.heliyon.2021.e07991.","DOI":"10.1016\/j.heliyon.2021.e07991"},{"issue":"4","key":"13288_CR37","doi-asserted-by":"publisher","first-page":"33","DOI":"10.5539\/ijsp.v9n4p33","volume":"9","author":"PE Tengaa","year":"2020","unstructured":"Tengaa PE, Mwalili S, Orwa GO. Stochastic modeling for HIV and AIDS epidemics with viral load detectability. IJSP. 2020;9(4):33. https:\/\/doi.org\/10.5539\/ijsp.v9n4p33.","journal-title":"IJSP"},{"key":"13288_CR38","doi-asserted-by":"publisher","unstructured":"Strategies C. A Caputo-Fabrizio fractional-order model of HIV\/AIDS with a treatment compartment: sensitivity analysis and optimal control strategies. 2021. p. e23050610. https:\/\/doi.org\/10.3390\/e23050610.","DOI":"10.3390\/e23050610"},{"issue":"1","key":"13288_CR39","doi-asserted-by":"publisher","first-page":"1714","DOI":"10.1038\/s41598-025-31376-5","volume":"16","author":"AE Sado","year":"2025","unstructured":"Sado AE, Duressa GF, Deressa CT. Mathematical modeling and neural network based fitting of HIV\/AIDS data in the workingclass population case study from Ethiopia. Sci Rep. 2025;16(1):1714. https:\/\/doi.org\/10.1038\/s41598-025-31376-5.","journal-title":"Sci Rep"},{"issue":"1","key":"13288_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13662-021-03334-8","volume":"2021","author":"CT Deressa","year":"2021","unstructured":"Deressa CT, Duressa GF. Analysis of Atangana-Baleanu fractional-order SEAIR epidemic model with optimal control. Adv Differ Equ. 2021;2021(1):1\u201325. https:\/\/doi.org\/10.1186\/s13662-021-03334-8.","journal-title":"Adv Differ Equ"},{"issue":"4","key":"13288_CR41","doi-asserted-by":"publisher","first-page":"2355","DOI":"10.1016\/j.aej.2020.02.031","volume":"59","author":"A Atangana","year":"2020","unstructured":"Atangana A, Araz SI. Extension of Atangana-seda numerical method to partial differential equations with integer and non-integer order. Alex Eng J. 2020;59(4):2355\u201370. https:\/\/doi.org\/10.1016\/j.aej.2020.02.031.","journal-title":"Alex Eng J"},{"issue":"10","key":"13288_CR42","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1140\/epjp\/i2017-11717-0","volume":"132","author":"M Toufik","year":"2017","unstructured":"Toufik M, Atangana A. New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models. Eur Phys J Plus. 2017;132(10):1\u201316. https:\/\/doi.org\/10.1140\/epjp\/i2017-11717-0.","journal-title":"Eur Phys J Plus"},{"key":"13288_CR43","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1016\/j.chaos.2019.07.026","volume":"127","author":"A Khan","year":"2019","unstructured":"Khan A, Khan H, Gomez-Aguilar JF, Abdeljawad T. Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos, Solitons Fractals. 2019;127:422\u201327. https:\/\/doi.org\/10.1016\/j.chaos.2019.07.026.","journal-title":"Chaos, Solitons Fractals"},{"issue":"5","key":"13288_CR44","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3390\/pr6050048","volume":"6","author":"TD Awoke","year":"2018","unstructured":"Awoke TD, Kassa SM. Optimal control strategy for TB-HIV\/AIDS co-infection model in the presence of behaviour modification. Processes. 2018;6(5):1\u201325. https:\/\/doi.org\/10.3390\/pr6050048.","journal-title":"Processes"},{"key":"13288_CR45","doi-asserted-by":"publisher","first-page":"1462701","DOI":"10.3389\/fams.2024.1462701","volume":"10","author":"HT Alemneh","year":"2024","unstructured":"Alemneh HT, Teklu SW, Kotola BS, Mekonen KG. Investigation of an optimal control strategy for a cholera disease transmission model with programs. Front Appl Math Stat. 2024;10:1462701. https:\/\/doi.org\/10.3389\/fams.2024.1462701.","journal-title":"Front Appl Math Stat"},{"key":"13288_CR46","first-page":"152","volume":"16","author":"FAHAD Al Basir","year":"2017","unstructured":"Al Basir FAHAD, Roy PK. Stability analysis and optimal control of a fractional-order model for HIV infection. WSEAS Trans Math. 2017;16:152\u201362.","journal-title":"WSEAS Trans Math"}],"container-title":["BMC Infectious Diseases"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s12879-026-13288-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s12879-026-13288-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s12879-026-13288-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T03:04:57Z","timestamp":1780283097000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1186\/s12879-026-13288-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,20]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["13288"],"URL":"https:\/\/doi.org\/10.1186\/s12879-026-13288-5","relation":{},"ISSN":["1471-2334"],"issn-type":[{"value":"1471-2334","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,20]]},"assertion":[{"value":"10 December 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 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":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare no competing interests.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"1049"}}