{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:27:12Z","timestamp":1760146032436,"version":"build-2065373602"},"reference-count":63,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T00:00:00Z","timestamp":1727654400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye\u2019s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam\u2013Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.<\/jats:p>","DOI":"10.3390\/sym16101284","type":"journal-article","created":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T07:19:37Z","timestamp":1727680777000},"page":"1284","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On a Symmetry-Based Structural Deterministic Fractal Fractional Order Mathematical Model to Investigate Conjunctivitis Adenovirus Disease"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8812-2859","authenticated-orcid":false,"given":"Mdi Begum","family":"Jeelani","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 5701, Riyadh 11432, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0342-491X","authenticated-orcid":false,"given":"Nadiyah Hussain","family":"Alharthi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 5701, Riyadh 11432, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,30]]},"reference":[{"key":"ref_1","unstructured":"Center for Disease Control (CDC) (2023, December 31). Conjunctivitis (Pink Eye), Available online: https:\/\/www.cdc.gov."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Fehily, S.R., Cross, G.B., and Fuller, A.J. (2015). Bilateral conjunctivitis in a returned traveller. PLoS Neglected Trop., 9.","DOI":"10.1371\/journal.pntd.0003351"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1136\/bmj.1.3340.12-a","article-title":"Conjunctivitis in the tropics","volume":"1","author":"Elliot","year":"1925","journal-title":"Br. Med. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"166","DOI":"10.4103\/0300-1652.129664","article-title":"Allergic conjunctivitis in Jos-Nigeria","volume":"55","author":"Malu","year":"2014","journal-title":"Niger. Med. J. J. Niger. Med. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kimberlin, D.W. (2018). Red Book: 2018\u20132021 Report of the Committee on Infectious Diseases, American Academy of Pediatrics. No. Ed. 31.","DOI":"10.1542\/9781610025225"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1840","DOI":"10.1002\/sim.2352","article-title":"Modelling the transmission dynamics of acute haemorrhagic conjunctivitis: Application to the 2003 outbreak in Mexico","volume":"25","author":"Chowell","year":"2006","journal-title":"Stat. Med."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Murray, J.D. (2003). Mathematical Biology I, Springer.","DOI":"10.1007\/b98869"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"20200204","DOI":"10.1098\/rsif.2020.0204","article-title":"Symmetry structures in dynamic models of biochemical systems","volume":"17","author":"Ohlsson","year":"2020","journal-title":"J. R. Soc. Interface"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/S0042-6989(01)00299-1","article-title":"Symmetry perception: A novel approach for biological shapes","volume":"42","author":"Wilson","year":"2002","journal-title":"Vis. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1002\/mma.4617","article-title":"Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications","volume":"41","author":"Almeida","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Das, S., and Pan, I. (2011). Fractional Order Signal Processing: Introductory Concepts and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-23117-9"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.aej.2020.09.058","article-title":"Analysis of a COVID-19 model: Optimal control, stability and simulations","volume":"60","author":"Araz","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.9734\/JAMCS\/2018\/43054","article-title":"Modeling exponential growth and exponential decay real phenomena by \u03c8-Caputo fractional derivative","volume":"28","author":"Awadalla","year":"2018","journal-title":"J. Adv. Math. Comput. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1186\/s13662-020-03095-w","article-title":"Mathematical model of COVID-19 spread in Turkey and South Africa: Theory, methods, and applications","volume":"2020","author":"Atangana","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s40998-020-00364-y","article-title":"Output feedback adaptive fractional-order super-twisting sliding mode control of robotic manipulator","volume":"45","author":"Ahmed","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. Electr. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2305","DOI":"10.1016\/j.aej.2020.02.022","article-title":"On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative","volume":"59","author":"Shah","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2193","DOI":"10.1007\/s12555-018-0767-5","article-title":"Robust adaptive control of robotic manipulator with input time-varying delay","volume":"17","author":"Ahmed","year":"2019","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1016\/j.aej.2021.08.030","article-title":"Dynamical behaviour of HIV\/AIDS model using fractional derivative with Mittag-Leffler kernel","volume":"61","author":"Shaikh","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_19","first-page":"1823","article-title":"Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using atangana-baleanu operator","volume":"66","author":"Peter","year":"2021","journal-title":"Comput. Mater. Contin."},{"key":"ref_20","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. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.jcp.2019.03.008","article-title":"A review of definitions of fractional derivatives and other operators","volume":"388","author":"Teodoro","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2014.01.002","article-title":"A new definition of fractional derivative","volume":"264","author":"Khalil","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_24","first-page":"1191","article-title":"Hadamard-type fractional calculus","volume":"38","author":"Kilbas","year":"2001","journal-title":"J. Korean Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"108675","DOI":"10.1016\/j.knosys.2022.108675","article-title":"Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAM fuzzy neural networks via exponential Euler technique","volume":"246","author":"Zhang","year":"2022","journal-title":"Knowl.-Based Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Khan, M., Ahmad, Z., Ali, F., Khan, N., Khan, I., and Nisar, K.S. (2023). Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel. PLoS ONE, 18.","DOI":"10.1371\/journal.pone.0277806"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18576\/pfda\/020101","article-title":"Applications of new time and spatial fractional derivatives with exponential kernels","volume":"2","author":"Caputo","year":"2016","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_28","first-page":"73","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Progr. Fract. Differ. Appl."},{"key":"ref_29","first-page":"87","article-title":"Properties of a new fractional derivative without singular kernel","volume":"1","author":"Losada","year":"2015","journal-title":"Progr. Fract. Differ. Appl."},{"key":"ref_30","first-page":"4714032","article-title":"Qualitative analysis of implicit Dirichlet boundary value problem for Caputo-Fabrizio fractional differential equations","volume":"2020","author":"Gul","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"112662","DOI":"10.1016\/j.chaos.2022.112662","article-title":"Oscillatory, crossover behavior and chaos analysis of HIV-1 infection model using piece-wise Atangana\u2013Baleanu fractional operator: Real data approach","volume":"164","author":"Xu","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6858592","DOI":"10.1155\/2021\/6858592","article-title":"Time-fractional Klein\u2013Gordon equation with solitary\/shock waves solutions","volume":"2021","author":"Saifullah","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"e22699","DOI":"10.1002\/num.22699","article-title":"Numerical solutions of fractional parabolic equations with generalized Mittag-Leffler kernels","volume":"40","author":"Alomari","year":"2024","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Saad Alshehry, A., Imran, M., Shah, R., and Weera, W. (2022). Fractional-View Analysis of Fokker-Planck equations by ZZ Transform with Mittag-Leffler Kernel. Symmetry, 14.","DOI":"10.3390\/sym14081513"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.chaos.2017.04.027","article-title":"Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system","volume":"102","author":"Atangana","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.rinp.2018.06.011","article-title":"Fractal calculus and its geometrical explanation","volume":"10","author":"He","year":"2018","journal-title":"Results Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"773","DOI":"10.2298\/TSCI1603773H","article-title":"On fractal space-time and fractional calculus","volume":"20","author":"Hu","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"109812","DOI":"10.1016\/j.chaos.2020.109812","article-title":"Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data","volume":"136","author":"Qureshi","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Saad, K.M. (2020). Numerical simulation of the fractal-fractional Ebola virus. Fractal Fract., 4.","DOI":"10.3390\/fractalfract4040049"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2150017","DOI":"10.1142\/S0218348X21500171","article-title":"A fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media","volume":"29","author":"Xiao","year":"2021","journal-title":"Fractals"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"17880","DOI":"10.1016\/j.ijhydene.2018.07.186","article-title":"An analytical model for the transverse permeability of gas diffusion layer with electrical double layer effects in proton exchange membrane fuel cells","volume":"43","author":"Liang","year":"2018","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"129499","DOI":"10.1016\/j.fuel.2023.129499","article-title":"Characterization of water migration behavior during spontaneous imbibition in coal: From the perspective of fractal theory and NMR","volume":"355","author":"Yu","year":"2024","journal-title":"Fuel"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"100386","DOI":"10.1016\/j.rico.2024.100386","article-title":"Some appropriate results for the existence theory and numerical solutions of fractals-fractional order malaria disease mathematical model","volume":"14","author":"Ahmad","year":"2024","journal-title":"Results Control. Optim."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.euromechflu.2020.09.002","article-title":"Role of fractal\u2013fractional derivative on ferromagnetic fluid via fractal Laplace transform: A first problem via fractal-fractional differential operator","volume":"85","author":"Abro","year":"2021","journal-title":"Eur. J. Mech.-B\/Fluids"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2040043","DOI":"10.1142\/S0218348X20400435","article-title":"Cauchy problems with fractal-fractional operators and applications to groundwater dynamics","volume":"28","author":"Atangana","year":"2020","journal-title":"Fractals"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.chaos.2019.04.020","article-title":"Modeling attractors of chaotic dynamical systems with fractal\u2013fractional operators","volume":"123","author":"Atangana","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_48","first-page":"189","article-title":"Local stability analysis of mathematical model for hemorrhagic conjunctivitis disease","volume":"12","author":"Sangsawang","year":"2012","journal-title":"Curr. Appl. Sci. Technol."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"085253","DOI":"10.1088\/1402-4896\/ad62a5","article-title":"Investigation of conjunctivitis adenovirus spread in human eyes by using bifurcation tool and numerical treatment approach","volume":"99","author":"Javed","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_50","unstructured":"Fatunla, S.O. (2014). Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Academic Press."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Khan, M.A., and Atangana, A. (2023). Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling, CRC Press.","DOI":"10.1201\/9781003359258"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Watmough","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_53","first-page":"159","article-title":"Further notes on the basic reproduction number","volume":"2008","author":"Watmough","year":"2008","journal-title":"Math. Epidemiol."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"5288","DOI":"10.1103\/PhysRevA.35.5288","article-title":"Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations","volume":"35","author":"DeJesus","year":"1987","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., and Yakubu, A.A. (2002). Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, Springer Science & Business Media.","DOI":"10.1007\/978-1-4613-0065-6"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1109\/MCSE.2007.27","article-title":"Being sensitive to uncertainty","volume":"9","author":"Arriola","year":"2007","journal-title":"Comput. Sci. Eng."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"103836","DOI":"10.1016\/j.rinp.2021.103836","article-title":"Modeling and sensitivity analysis of HBV epidemic model with convex incidence rate","volume":"22","author":"Khan","year":"2021","journal-title":"Results Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1073\/pnas.27.4.222","article-title":"On the stability of the linear functional equation","volume":"27","author":"Hyers","year":"1941","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_59","first-page":"297","article-title":"On the stability of the linear mapping in Banach spaces","volume":"72","author":"Rassias","year":"1978","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"6464","DOI":"10.1002\/mma.6390","article-title":"On qualitative theory of fractional order delay evolution equation via the prior estimate method","volume":"43","author":"Sher","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_61","unstructured":"Yanagiwara, H.I.R.O.K.I. (1995, January 18\u201323). On the Stability of a Multistep Method. Proceedings of the Sixth International Colloquim on Differential Equations, Plovdiv, Bulgaria."},{"key":"ref_62","first-page":"26","article-title":"Stability and convergence of numerical computations","volume":"3","year":"2011","journal-title":"Inf. Sci. Technol. Bull. Acm Slovak."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1016\/j.camwa.2009.07.050","article-title":"On the fractional Adams method","volume":"58","author":"Li","year":"2009","journal-title":"Comput. Math. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1284\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:07:40Z","timestamp":1760112460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1284"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,30]]},"references-count":63,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101284"],"URL":"https:\/\/doi.org\/10.3390\/sym16101284","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,9,30]]}}}