{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T23:20:15Z","timestamp":1787613615651,"version":"build-2736575974"},"reference-count":37,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,3,23]],"date-time":"2021-03-23T00:00:00Z","timestamp":1616457600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Molecules"],"abstract":"<jats:p>Human immunodeficiency virus (HIV) is a life life-threatening and serious infection caused by a virus that attacks CD4+ T-cells, which fight against infections and make a person susceptible to other diseases. It is a global public health problem with no cure; therefore, it is highly important to study and understand the intricate phenomena of HIV. In this article, we focus on the numerical study of the path-tracking damped oscillatory behavior of a model for the HIV infection of CD4+ T-cells. We formulate fractional dynamics of HIV with a source term for the supply of new CD4+ T-cells depending on the viral load via the Caputo\u2013Fabrizio derivative. In the formulation of fractional HIV dynamics, we replaced the constant source term for the supply of new CD4+ T-cells from the thymus with a variable source term depending on the concentration of the viral load, and introduced a term that describes the incidence of the HIV infection of CD4+ T-cells. We present a novel numerical scheme for fractional view analysis of the proposed model to highlight the solution pathway of HIV. We inspect the periodic and chaotic behavior of HIV for the given values of input factors using numerical simulations.<\/jats:p>","DOI":"10.3390\/molecules26061806","type":"journal-article","created":{"date-parts":[[2021,3,23]],"date-time":"2021-03-23T14:08:16Z","timestamp":1616508496000},"page":"1806","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["Fractional Dynamics of HIV with Source Term for the Supply of New CD4+ T-Cells Depending on the Viral Load via Caputo\u2013Fabrizio Derivative"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5539-4225","authenticated-orcid":false,"given":"Zahir","family":"Shah","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Lakki Marwat, Lakki Marwat 28420, Pakistan"},{"name":"Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut\u2019s, University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rashid","family":"Jan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5463-4581","authenticated-orcid":false,"given":"Poom","family":"Kumam","sequence":"additional","affiliation":[{"name":"Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut\u2019s, University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand"},{"name":"KMUTT Fixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group, SCL 802 Fixed Point Laboratory, Department of Mathematics, Faculty of Science, King Mongkut\u2019s, University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5022-2558","authenticated-orcid":false,"given":"Wejdan","family":"Deebani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science &amp; Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1454-2962","authenticated-orcid":false,"given":"Meshal","family":"Shutaywi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science &amp; Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.mbs.2005.12.026","article-title":"Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells","volume":"200","author":"Wang","year":"2006","journal-title":"Math. Biosci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Antar, A.A., Jenike, K.M., Jang, S., Rigau, D.N., Reeves, D.B., Hoh, R., Krone, M.R., Keruly, J.C., Moore, R.D., and Schiffer, J.T. (2020). Longitudinal study reveals HIV-1\u2013infected CD4+ T cell dynamics during long-term antiretroviral therapy. J. Clin. Investig., 130.","DOI":"10.1172\/JCI135953"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1967","DOI":"10.1002\/mma.5491","article-title":"Effect of partial immunity on transmission dynamics of dengue disease with optimal control","volume":"42","author":"Jan","year":"2019","journal-title":"Math. Method. Appl. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"113064","DOI":"10.1016\/j.cam.2020.113064","article-title":"Dynamics of fractional order HIV infection model with antibody and cytotoxic T-lymphocyte immune responses","volume":"382","author":"Tamilalagan","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1186\/s13662-019-2314-y","article-title":"Effect of pulse vaccination on dynamics of dengue with periodic transmission functions","volume":"2019","author":"Jan","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1186\/s13662-020-03116-8","article-title":"Modeling the effect of delay strategy on transmission dynamics of HIV\/AIDS disease","volume":"2020","author":"Raza","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0025-5564(93)90043-A","article-title":"Dynamics of HIV infection of CD4+ T cells","volume":"114","author":"Perelson","year":"1993","journal-title":"Math. Biosci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/S0036144598335107","article-title":"Mathematical analysis of HIV-1 dynamics in vivo","volume":"41","author":"Perelson","year":"1999","journal-title":"Siam. Rev."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0025-5564(00)00006-7","article-title":"A delay-differential equation model of HIV infection of CD4+ T-cells","volume":"165","author":"Culshaw","year":"2000","journal-title":"Math. Biosci."},{"key":"ref_10","first-page":"16","article-title":"Existence theory of HIV-1 infection model by using arbitrary order derivative of without singular kernel type","volume":"9","author":"Bushnaq","year":"2018","journal-title":"J. Math. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3007","DOI":"10.1016\/j.mcm.2011.07.029","article-title":"Application of the homotopy analysis method for solving a model for HIV infection of CD4+ T-cells","volume":"54","author":"Ghoreishi","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1016\/j.mcm.2010.09.009","article-title":"The laplace adomian decomposition method for solving a model for HIV infection of CD4+ 220 T-cells","volume":"53","author":"Ongun","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Vazquez-Leal, H., Hernandez-Martinez, L., Khan, Y., Jimenez-Fernandez, V.M., Filobello-Nino, U., Diaz-Sanchez, A., Herrera-May, A.L., Castaneda-Sheissa, R., Marin-Hernandez, A., and Rabago-Bernal, F. (2014). Multistage HPM applied to path tracking damped oscillations of a model for HIV infection of CD4+ T cells. J. Adv. Math. Comput. Sci., 1035\u20131047.","DOI":"10.9734\/BJMCS\/2014\/7714"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"110092","DOI":"10.1016\/j.chaos.2020.110092","article-title":"Computational solutions of the HIV-1 infection of CD4+ T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy","volume":"139","author":"Khater","year":"2020","journal-title":"Chaos. Soliton. Fract."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"125061","DOI":"10.1016\/j.amc.2020.125061","article-title":"A second order accurate approximation for fractional derivatives with singular and non-singular kernel applied to a HIV model","volume":"374","author":"Arshad","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_16","first-page":"79","article-title":"Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination","volume":"2020","author":"Jan","year":"2020","journal-title":"Commun. Math. Biol. Neurosci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1186\/s13662-020-02544-w","article-title":"Analysis of the model of HIV-1 infection of CD4+ CD4+ T-cell with a new approach of fractional derivative","volume":"2020","author":"Baleanu","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/j.chaos.2019.07.002","article-title":"Modeling the transmission of dengue infection through fractional derivatives","volume":"127","author":"Jan","year":"2019","journal-title":"Chaos. Soliton. Fract."},{"key":"ref_19","unstructured":"Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier."},{"key":"ref_20","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Yverdon-les-Bains, Gordon and Breach Science Publishers."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3029","DOI":"10.1016\/j.aej.2020.05.007","article-title":"On modelling of epidemic childhood diseases with the Caputo-Fabrizio derivative by using the Laplace Adomian decomposition method","volume":"59","author":"Baleanu","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"109833","DOI":"10.1016\/j.chaos.2020.109833","article-title":"SIR epidemic model with Mittag-Leffler fractional derivative","volume":"137","author":"Sene","year":"2020","journal-title":"Chaos. Soliton. Fract."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1432521","DOI":"10.1080\/23311835.2018.1432521","article-title":"Mathematical analysis of HIV\/AIDS infection model with Caputo-Fabrizio fractional derivative","volume":"5","author":"Bushnaq","year":"2018","journal-title":"Cogent. Math. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Brandibur, O., Kaslik, E., Mozyrska, D., and Wyrwas, M. (2020). Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations. Mathematics, 8.","DOI":"10.3390\/math8101751"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Saad, K.M. (2020). A comparative study of the fractional-order clock chemical model. Mathematics, 8.","DOI":"10.3390\/math8091436"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2503","DOI":"10.1016\/j.camwa.2017.07.040","article-title":"Magnetohydrodynamic electroosmotic flow of Maxwell fluids with Caputo-Fabrizio derivatives through circular tubes","volume":"74","author":"Abdulhameed","year":"2017","journal-title":"Comput. Math. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5267","DOI":"10.3934\/mbe.2020285","article-title":"A new model of dengue fever in terms of fractional derivative","volume":"17","author":"Fatmawati","year":"2020","journal-title":"Math. Biosci. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kumar, S., Ahmadian, A., Kumar, R., Kumar, D., Singh, J., Baleanu, D., and Salimi, M. (2020). An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Mathematics, 8.","DOI":"10.3390\/math8040558"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Muresan, C.I., Birs, I.R., and Dulf, E.H. (2020). Event-Based Implementation of Fractional Order IMC Controllers for Simple FOPDT Processes. Mathematics, 8.","DOI":"10.3390\/math8081378"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1515\/fca-2015-0034","article-title":"A biomathematical view on the fractional dynamics of cellulose degradation","volume":"18","author":"Goufo","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_31","first-page":"1","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_32","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1126\/science.274.5289.985","article-title":"Quantitative image analysis of HIV-1 infection in lymphoid tissue","volume":"274","author":"Haase","year":"1996","journal-title":"Science"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1038\/387188a0","article-title":"Decay characteristics of HIV-1-infected compartments during combination therapy","volume":"387","author":"Perelson","year":"1997","journal-title":"Nature"},{"key":"ref_34","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_35","doi-asserted-by":"crossref","first-page":"1729","DOI":"10.3934\/math.2020117","article-title":"Fast algorithm based on the novel approximation formula for the Caputo-Fabrizio fractional derivative","volume":"5","author":"Liu","year":"2020","journal-title":"Aims. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1080\/00207160.2017.1290434","article-title":"A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative","volume":"95","author":"Liu","year":"2018","journal-title":"Int. J. Comput. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1051\/mmnp\/2018010","article-title":"New numerical approach for fractional differential equations","volume":"13","author":"Atangana","year":"2018","journal-title":"Math. Model. Nat. Phenom."}],"container-title":["Molecules"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1420-3049\/26\/6\/1806\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:39:46Z","timestamp":1760161186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1420-3049\/26\/6\/1806"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,23]]},"references-count":37,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["molecules26061806"],"URL":"https:\/\/doi.org\/10.3390\/molecules26061806","relation":{},"ISSN":["1420-3049"],"issn-type":[{"value":"1420-3049","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,23]]}}}