{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T00:58:37Z","timestamp":1771635517245,"version":"3.50.1"},"reference-count":49,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T00:00:00Z","timestamp":1675123200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["GRANT2486"],"award-info":[{"award-number":["GRANT2486"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this manuscript, we formulate a mathematical model of the deadly COVID-19 pandemic to understand the dynamic behavior of COVID-19. For the dynamic study, a new SEIAPHR fractional model was purposed in which infectious individuals were divided into three sub-compartments. The purpose is to construct a more reliable and realistic model for a complete mathematical and computational analysis and design of different control strategies for the proposed Caputo\u2013Fabrizio fractional model. We prove the existence and uniqueness of solutions by employing well-known theorems of fractional calculus and functional analyses. The positivity and boundedness of the solutions are proved using the fractional-order properties of the Laplace transformation. The basic reproduction number for the model is computed using a next-generation technique to handle the future dynamics of the pandemic. The local\u2013global stability of the model was also investigated at each equilibrium point. We propose basic fixed controls through manipulation of quarantine rates and formulate an optimal control problem to find the best controls (quarantine rates) employed on infected, asymptomatic, and \u201csuperspreader\u201d humans, respectively, to restrict the spread of the disease. For the numerical solution of the fractional model, a computationally efficient Adams\u2013Bashforth method is presented. A fractional-order optimal control problem and the associated optimality conditions of Pontryagin maximum principle are discussed in order to optimally reduce the number of infected, asymptomatic, and superspreader humans. The obtained numerical results are discussed and shown through graphs.<\/jats:p>","DOI":"10.3390\/sym15020380","type":"journal-article","created":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T03:22:47Z","timestamp":1675221767000},"page":"380","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Theoretical Analysis of a COVID-19 CF-Fractional Model to Optimally Control the Spread of Pandemic"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0846-1640","authenticated-orcid":false,"given":"Azhar Iqbal Kashif","family":"Butt","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia"},{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"}]},{"given":"Muhammad","family":"Imran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"}]},{"given":"Saira","family":"Batool","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Lahore 54000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9076-9945","authenticated-orcid":false,"given":"Muneerah AL","family":"Nuwairan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,31]]},"reference":[{"key":"ref_1","unstructured":"World of Health Organization (2023, January 01). Novel Coronavirus (2019-nCoV)-SITUATION REPORT. Available online: https:\/\/apps.who.int\/iris\/bitstream\/handle\/10665\/330760\/nCoVsitrep21Jan2020-eng.pdf?sequence=3&isAllowed=y."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2950","DOI":"10.1016\/j.jacc.2020.04.031","article-title":"COVID-19 and thrombotic or thromboembolic disease: Implications for prevention, antithrombotic therapy, and follow-up","volume":"75","author":"Bikdeli","year":"2020","journal-title":"J. Am. Cardiol."},{"key":"ref_3","first-page":"9","article-title":"The effect of control measures on COVID-19 transmission in Italy: Comparison with Guangdong province in China","volume":"130","author":"Liu","year":"2020","journal-title":"Infect. Dis. Poverty"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2449710","DOI":"10.1155\/2022\/2449710","article-title":"Mathematical Modeling and Analysis of TB and COVID- 19 Coinfection","volume":"2022","author":"Mekonen","year":"2022","journal-title":"J. Appl. Math."},{"key":"ref_5","unstructured":"Liang, T. (2020). Handbook of COVID-19 prevention and treatment, The First Affiliated Hospital, Zhejiang University School of Medicine. Compil. Accord. Clin. Exp., 68."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"104917","DOI":"10.1016\/j.rinp.2021.104917","article-title":"Mathematical analysis for the effect of voluntary vaccination on the propagation of Corona virus pandemic","volume":"31","author":"Ahmad","year":"2021","journal-title":"Results Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.ijid.2020.01.009","article-title":"The continuing 2019-nCoV epidemic threat of novel corona viruses to global health: The latest 2019 novel coronavirus outbreak in Wuhan, China","volume":"91","author":"Hui","year":"2020","journal-title":"Int. J. Infect. Dis."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1051\/mmnp\/2020006","article-title":"On a quarantine model of Coronavirus infection and data analysis","volume":"15","author":"Volpert","year":"2020","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"587","DOI":"10.3934\/publichealth.2020047","article-title":"Global stability of COVID-19 model involving the quarantine strategy and media coverage effects","volume":"7","author":"Mohsen","year":"2020","journal-title":"AIMS Public Health"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"De la Sen, M., Ibeas, A., and Agarwal, R.P. (2020). On confinement and quarantine concerns on an SEIAR epidemic model with simulated parameterizations for the COVID-19 pandemic. Symmetry, 12.","DOI":"10.3390\/sym12101646"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chowdhury, A., Kabir, K.M.A., and Tanimoto, J. (2020). How quarantine and social distancing policy can suppress the outbreak of novel coronavirus in developing or under poverty level countries: A mathematical and statistical analysis. Res. Sq.","DOI":"10.21203\/rs.3.rs-20294\/v2"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/S1473-3099(18)30127-0","article-title":"Middle East respiratory syndrome coronavirus: Risk factors and determinants of primary, household, and nosocomial transmission","volume":"18","author":"Hui","year":"2018","journal-title":"Lancet Infect. Dis."},{"key":"ref_13","first-page":"1","article-title":"Isolation, quarantine, social distancing and community containment: Pivotal role for old-style public health measures in the novel coronavirus (2019-ncov) outbreak","volume":"27","author":"Freedman","year":"2020","journal-title":"J. Travel Med."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1835","DOI":"10.1016\/j.apm.2020.08.082","article-title":"Design of a nonlinear model for the propagation of COVID-19 and its efficient nonstandard computational implementation","volume":"89","author":"Rafiq","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1142\/S0218202521500147","article-title":"Transmission dynamics and quarantine control of COVID-19 in cluster community: A new transmission-quarantine model with case study for diamond princess","volume":"31","author":"Gao","year":"2021","journal-title":"Math. Model. Methods Appl. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"20200038","DOI":"10.1515\/em-2020-0038","article-title":"The impact of quarantine on COVID-19 infections","volume":"10","author":"Marshall","year":"2021","journal-title":"Epidemiol. Methods"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3618","DOI":"10.3934\/mbe.2020204","article-title":"Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK","volume":"17","author":"Feng","year":"2020","journal-title":"Math. Biosci. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"100437","DOI":"10.1016\/j.epidem.2021.100437","article-title":"A model for COVID-19 with isolation, quarantine and testing as control measures","volume":"34","author":"Aronna","year":"2021","journal-title":"Epidemics"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Acuna-Zegarra, M.A., Diaz-Infanteb, S., Baca-Carrasco, D., and Liceaga, D.O. (2020). COVID-19 optimal vaccination policies: A modeling study on efficacy, natural and vaccine-induced immunity responses. medRxiv.","DOI":"10.1101\/2020.11.19.20235176"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.vacun.2020.05.002","article-title":"A review on Promising vaccine development progress for COVID-19 disease","volume":"21","author":"Belete","year":"2020","journal-title":"Vacunas"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"198114","DOI":"10.1016\/j.virusres.2020.198114","article-title":"COVID-19 Vaccine: A comprehensive status report","volume":"288","author":"Kaur","year":"2020","journal-title":"Virus Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1001\/jama.2020.15725","article-title":"Unwavering Regulatory Safeguards for COVID-19 Vaccines","volume":"324","author":"Shah","year":"2020","journal-title":"J. Am. Med. Assoc."},{"key":"ref_23","unstructured":"Ivanova, P. (2023, January 01). Russia Says Its Sputnik v Covid-19 Vaccine is 92% Effective. Press Release 11 November 2020. Available online: https:\/\/www.reuters.com\/article\/us-health-coronavirus-russia-vaccine-idCAKBN27R0Z6."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Cohen, E. (2023, January 01). Moderna\u2019s Coronavirus Vaccine is 94.5% Effective, According to Company Data. Press Release Updated 16 November 2020. Available online: https:\/\/edition.cnn.com\/2020\/11\/16\/health\/moderna-vaccine-results-coronavirus\/index.html.","DOI":"10.1126\/science.368.6486.16"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1007\/s13235-021-00382-3","article-title":"Modeling COVID-19 pandemic with hierarchical quarantine and time delay","volume":"11","author":"Yang","year":"2021","journal-title":"Dyn. Games Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ajbar, A., Alqahtani, R.T., and Boumaza, M. (2021). Dynamics of a COVID-19 model with a nonlinear incidence rate, quarantine, media effects, and number of hospital beds. Symmetry, 13.","DOI":"10.3390\/sym13060947"},{"key":"ref_27","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier."},{"key":"ref_28","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1016\/j.aej.2020.09.057","article-title":"A new fractional model for vector-host disease with saturated treatment function via singular and non-singular operators","volume":"60","author":"Khan","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"084005","DOI":"10.1088\/1402-4896\/ac7ebc","article-title":"Fractional order mathematical model of monkeypox transmission dynamics","volume":"97","author":"Peter","year":"2022","journal-title":"Phys. Scr."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Hadi, M.S., and Bilgehan, B. (2022). Fractional COVID-19 Modeling and Analysis on Successive Optimal Control Policies. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6100533"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1186\/s13662-020-02834-3","article-title":"A mathematical model of COVID-19 using fractional derivative: Outbreak in India with dynamics of transmission and control","volume":"2020","author":"Shaikh","year":"2020","journal-title":"Adv. Differ. Equations"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1140\/epjp\/s13360-022-02603-z","article-title":"A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions","volume":"137","author":"Ali","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1007\/s13398-022-01345-y","article-title":"Solution estimates to Caputo proportional fractional derivative delay integro-differential equations","volume":"117","year":"2023","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM"},{"key":"ref_35","first-page":"2577","article-title":"On the new qualitative results in integro-differential equations with Caputo fractional derivative and multiple kernels and delays","volume":"23","author":"Yao","year":"2022","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1528","DOI":"10.1002\/mma.8593","article-title":"Optimal control strategies for the reliable and competitive mathematical analysis of COVID-19 pandemic model","volume":"46","author":"Butt","year":"2023","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_37","first-page":"87","article-title":"Properties of a new fractional derivative without singular kernel","volume":"1","author":"Losada","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_38","first-page":"73","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.1140\/epjp\/s13360-021-02178-1","article-title":"A new fuzzy fractional-order model of transmission of COVID-19 with quarantine class","volume":"136","author":"Hanif","year":"2021","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1140\/epjp\/s13360-021-01360-9","article-title":"Effect of quarantine on transmission dynamics of Ebola virus epidemic: A mathematical analysis","volume":"136","author":"Ahmad","year":"2021","journal-title":"Eur. J. Plus"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1140\/epjp\/s13360-020-00683-3","article-title":"Mathematical analysis to control the spread of Ebola virus epidemic through voluntary vaccination","volume":"135","author":"Ahmad","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1186\/s13662-020-02994-2","article-title":"A reliable and competitive mathematical analysis of Ebola epidemic model","volume":"2020","author":"Rafiq","year":"2020","journal-title":"Adv. Differ. Equations"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3419","DOI":"10.1016\/j.aej.2020.05.021","article-title":"A mathematical analysis of an isothermal tube drawing process","volume":"59","author":"Butt","year":"2020","journal-title":"Alexanderia Eng. J."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Castillo-Chavez, C., Feng, Z., Huanz, W., Driessche, P.V.D., and Kirschner, D.E. (2002). Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer.","DOI":"10.1007\/978-1-4757-3667-0"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1186\/s13662-021-03316-w","article-title":"Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo-Fabrizio derivative","volume":"2021","author":"Thabet","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.cnsns.2014.12.013","article-title":"Volterra-type Lyapunov functions for fractional-order epidemic systems","volume":"24","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1186\/s13662-019-2138-9","article-title":"A Caputo-Fabrizio fractional differential equation model for HIV\/AIDS with treatment compartment","volume":"2019","author":"Moore","year":"2019","journal-title":"Adv. Differ. Equations"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Lenhart, S., and Workman, J.T. (2007). Optimal Control Applied to Biological Models, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420011418"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Wang, H., Jahanshahi, H., Wang, M.K., Bekiros, S., Liu, J., and Aly, A.A. (2021). A Caputo-Fabrizio fractional-order model of HIV\/AIDS with a treatment compartment: Sensitivity analysis and optimal control strategies. Entropy, 23.","DOI":"10.3390\/e23050610"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/380\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:19:59Z","timestamp":1760120399000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/380"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,31]]},"references-count":49,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020380"],"URL":"https:\/\/doi.org\/10.3390\/sym15020380","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,31]]}}}