{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T01:20:07Z","timestamp":1773019207571,"version":"3.50.1"},"reference-count":59,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,21]],"date-time":"2022-10-21T00:00:00Z","timestamp":1666310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Fund of Khon Kaen University"},{"name":"National Science, Research and Innovation Fund or NSRF"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study deals with the numerical solution of the human immunodeficiency virus (HIV) infection model, which is a significant problem for global public health. Acquired immunodeficiency syndrome (AIDS) is a communicable disease, and HIV is the causative agent for AIDS, which damages the ability of the body to fight against disease and easily usual innocuous infections attack the body. On entering the body, HIV infects a large amount of CD4+ T-cells and disturbs the supply rate of these cells from the thymus. Herein, we consider the model with variable source terms in which the production of these cells is a monotonically decreasing function of viral load. Based on the reproduction number, we describe the stability of free equilibrium. The continuous Galerkin\u2013Petrov method, in particular the cGP(2)-method, is implemented to determine the numerical solutions of the model. The influence of different parameters on the population dynamics of healthy\/infected CD4+ T-cells and free HIV particles are examined, and the results are presented graphically. On the other hand, the model is solved using the fourth-order Runge\u2013Kutta method, and briefly, the RK4-method, and the results of the proposed schemes are compared with those obtained from other classical schemes such as the Bessel collocation method (BCM), Laplace Adomian decomposition method (LADM), perturbation iteration algorithm (PIA), modified variational iteration method (MVIM), differential transform method (DTM), and exponential Galerkin method (EGM), numerically. Furthermore, absolute errors relative to the RK4 method are computed to describe the accuracy of the proposed scheme. It is presented that the cGP(2)-method gains accurate results at larger time step sizes in comparison with the results of the aforementioned methods. The numerical and graphical comparison reveals that the proposed scheme yields more accurate results relative to other traditional schemes from the literature.<\/jats:p>","DOI":"10.3390\/axioms11100578","type":"journal-article","created":{"date-parts":[[2022,10,23]],"date-time":"2022-10-23T20:43:50Z","timestamp":1666557830000},"page":"578","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Computational Approach to a Model for HIV and the Immune System Interaction"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2640-2377","authenticated-orcid":false,"family":"Attaullah","sequence":"first","affiliation":[{"name":"Department of Mathematics & Statistics, Bacha Khan University, Charsadda 24461, Pakistan"}]},{"family":"Zeeshan","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, Bacha Khan University, Charsadda 24461, Pakistan"}]},{"given":"Muhammad","family":"Tufail Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Statistics, Bacha Khan University, Charsadda 24461, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6364-3690","authenticated-orcid":false,"given":"Sultan","family":"Alyobi","sequence":"additional","affiliation":[{"name":"King Abdulaziz University, College of Science & Arts, Department of Mathematics, Rabigh, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3952-4341","authenticated-orcid":false,"given":"Mansour F.","family":"Yassen","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Aflaj, Prince Sattam Bin Abdulaziz University, Al-Aflaj 11912, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5485-8956","authenticated-orcid":false,"given":"Din","family":"Prathumwan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.mcm.2009.04.019","article-title":"A Fractional Order Differential Equation Model of HIV Infection of CD4+ T-cells","volume":"50","author":"Ding","year":"2009","journal-title":"Math. Comput. Model."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1038\/ni.3158","article-title":"Anti Body Responses to Envelope Glycoprotein\u2019s in HIV-1 Infection","volume":"16","author":"Burton","year":"2015","journal-title":"J. Nat. Immunol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1016\/j.nonrwa.2010.09.010","article-title":"Permanence and Extinction of a Non Autonomous HIV\/AIDS Epidemic Model with Distributed Time Delay","volume":"12","author":"Samanta","year":"2011","journal-title":"J. Non. Linea. Anal. Real World Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1007\/s002850050076","article-title":"Optimal Control of the Chemotherapy of HIV","volume":"35","author":"Kirschner","year":"1997","journal-title":"J. Math. Biol."},{"key":"ref_5","first-page":"15","article-title":"The Rloe of Sexually Transmitted Infections in HIV-1 Progression","volume":"2013","author":"Chun","year":"2013","journal-title":"J. Sex. Trans. Dis."},{"key":"ref_6","first-page":"14","article-title":"Dynamic Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate","volume":"2014","author":"Sum","year":"2014","journal-title":"J. Com. Math. Meth."},{"key":"ref_7","first-page":"538","article-title":"A Fractional Order Model of HIV Dynamics of HIV Infection with Drug Therapy Effect","volume":"22","author":"Arafa","year":"2014","journal-title":"J. Non. Linea. Biol. Phys."},{"key":"ref_8","first-page":"2947","article-title":"Global Stability of an HIV Pathogenesis Model with Care Rate","volume":"12","author":"Liu","year":"2011","journal-title":"J. Non. Linea. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1781","DOI":"10.4236\/am.2015.610158","article-title":"Stability Analysis of Delayed HIV\/AIDS Epidemic Model with Treatment and Vertical Transmission","volume":"6","author":"Osman","year":"2015","journal-title":"J. App. Math."},{"key":"ref_10","first-page":"157","article-title":"On the Behavior of Solutions in Viral Dynamical Models","volume":"73","author":"Tuckwell","year":"2004","journal-title":"J. Biol. Syst."},{"key":"ref_11","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":"J. Math. Biosci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1142\/S0218339009002764","article-title":"Modeling the Drug Therapy for HIV Infection","volume":"17","author":"Srivastava","year":"2009","journal-title":"J. Bio. Syst."},{"key":"ref_13","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":"J. Math. Biosci."},{"key":"ref_14","unstructured":"World Health Organization (2015, May 01). Global Health Observatory (GHO) Data. Available online: http:\/\/www.who.int\/gho\/tb\/en."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0025-5564(02)00099-8","article-title":"Mathematical Analysis of Delay Differential Equation Models of HIV-1 Infection","volume":"179","author":"Nelson","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.1007\/s11538-007-9203-3","article-title":"Emergence of HIV-1 Drug Resistance During Anti Retroviral Treatment","volume":"69","author":"Ronga","year":"2007","journal-title":"Bull. Math. Biol."},{"key":"ref_18","first-page":"215","article-title":"Mathematical Models of the Complete Course of HIV Infection and AIDS","volume":"4","author":"Duffin","year":"2002","journal-title":"J. Theo. Med."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.4134\/JKMS.2005.42.5.1071","article-title":"A Delay-Differential Equation Model of HIV Infection of CD4+ T-cells","volume":"42","author":"Song","year":"2005","journal-title":"J. Korean Math. Soc."},{"key":"ref_20","first-page":"5069","article-title":"Application of Lie Symmetry for Mathematical Model of HIV Infection of CD4+ T-cells","volume":"13","author":"Mechee","year":"2018","journal-title":"J. Appl. Eng. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1342","DOI":"10.1016\/j.jmaa.2008.01.008","article-title":"A Differential Equation Model of HIV Infection of CD4+ T-cells with Cure Rate","volume":"342","author":"Zhou","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","first-page":"1313","article-title":"Virus Dynamics: A Global Analysis","volume":"4","author":"Leenheer","year":"2003","journal-title":"J. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1016\/j.nonrwa.2008.10.037","article-title":"Modeling the Dynamics of HIVand CD4+ T-cells during Primary Infection","volume":"11","author":"Srivastava","year":"2010","journal-title":"J. Nonlinear Anal."},{"key":"ref_24","first-page":"1","article-title":"A Class Age-Structured HIV\/AIDS Model with Impulsive Drug Treatment Strategy","volume":"2010","author":"Liu","year":"2010","journal-title":"J. Disc. Dyna. Nat. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1038\/373123a0","article-title":"Rapid Turnover of Plasma Virion and CD4 Lymphocytes in HIV-1 Infection","volume":"373","author":"Ho","year":"1995","journal-title":"Nature"},{"key":"ref_26","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_27","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1515\/jnum.2010.002","article-title":"A Stable Discontinuous Galerkin-Petrov Time Discretization of Higher Order","volume":"18","author":"Schieweck","year":"2010","journal-title":"J. Numer. Math."},{"key":"ref_28","unstructured":"Kuang, Y. (2004). Delay Differential Equation with Applications in Population Dynamics, Academic Press."},{"key":"ref_29","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+ T-cells","volume":"53","author":"Ongun","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5876","DOI":"10.1016\/j.apm.2011.12.021","article-title":"A Numerical Approach to Solve the Model for HIV Infection of CD4 T-cell","volume":"36","author":"Yuzbasi","year":"2012","journal-title":"J. Appl. Math. Mod."},{"key":"ref_31","first-page":"79","article-title":"A Numerical Solution of a Model for HIV Infection of CD4 T-Cells","volume":"16","author":"Khalid","year":"2015","journal-title":"J. Inno. Sci. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.camwa.2011.04.058","article-title":"On the Numerical Solution of the Model for HIV Infection of CD4 T-Cells","volume":"62","author":"Merdan","year":"2011","journal-title":"J. Comput. Math. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"17507","DOI":"10.3934\/math.2022965","article-title":"A study on the transmission and dynamical behavior of an HIV\/AIDS epidemic model with a cure rate","volume":"7","author":"Attaullah","year":"2022","journal-title":"AIMS Math."},{"key":"ref_34","first-page":"4217548","article-title":"Mathematical Model for an Effective Management of HIV Infection","volume":"2016","author":"Ogunlaran","year":"2016","journal-title":"J. Biomed. Res. Int."},{"key":"ref_35","first-page":"1","article-title":"A Discrete Model for HIV Infection with Distributed Delay","volume":"2014","author":"Boukari","year":"2014","journal-title":"J. Diff. Equa."},{"key":"ref_36","first-page":"1","article-title":"Global Dynamics of a Virus Immune System with Virus Guided Therapy and Saturation Growth of Virus","volume":"2018","author":"Li","year":"2018","journal-title":"J. Math. Probl. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.bjid.2016.10.004","article-title":"Classical and Alternative Macrophages have Impaired Function during Acute and Chronic HIV-1 Infection","volume":"21","author":"Espindola","year":"2017","journal-title":"J. Braz. Infect. Dis."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.jadohealth.2017.09.030","article-title":"Age-Specific Global Prevalence of Hepatitis B, Hepatitis C, HIV and Tuberculosis Among Incarcerated People: A Systematic Review","volume":"62","author":"Kinner","year":"2018","journal-title":"J. Adolesc. Health"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.bjid.2018.12.002","article-title":"A Systematic Review on the Influence of HLA-B Polymorphisms on HIV-1 Mother to Child Transmission","volume":"23","author":"Angulo","year":"2019","journal-title":"J. Braz. Infect. Dis."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.coviro.2017.12.001","article-title":"The Impact of HIV-1 within Host Evolution on Transmission Dynamics","volume":"28","author":"Theys","year":"2018","journal-title":"J. Curr. Opin. Viro."},{"key":"ref_41","first-page":"49","article-title":"Factors Affecting HIV Disclosure among Partners in Morongo, Tanzania","volume":"10","author":"Hallberg","year":"2019","journal-title":"J. Inter. J. Afri. Nurs. Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.ssmph.2018.05.007","article-title":"Social Capital and HIV\/AIDS in the United States: Knowledge, Gaps and Future Directions","volume":"5","author":"Ransome","year":"2018","journal-title":"J. SSM. Popu. Health"},{"key":"ref_43","first-page":"100","article-title":"Quality of TB Care among People Living with HIV: Gaps and Solutions","volume":"17","author":"Naidoo","year":"2019","journal-title":"J. Clin. Tube. Myco. Dis."},{"key":"ref_44","first-page":"83","article-title":"A Mathematical Modeling Study of HIV Infection in two Heterosexual Age Groups in Kenya","volume":"4","author":"Omondi","year":"2019","journal-title":"J. Infect. Dis. Model."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/j.jmii.2016.09.003","article-title":"Routine CD4 Monitoring in HIV Patients with Viral Suppression: Is it Really Necessary? A Portuguese Cohort","volume":"51","author":"Duro","year":"2018","journal-title":"J. Microbio. Immun. Infect."},{"key":"ref_46","first-page":"103708","article-title":"Stochastic Model for In-Host HIV Dynamics with Therapeutic Intervention","volume":"2013","author":"Mbogo","year":"2013","journal-title":"Int. Sch. Res. Not."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3007","DOI":"10.1016\/j.mcm.2011.07.029","article-title":"Application of the Hemotopy Analysis Method for Solving a Model for HIV Infection of CD4+ T-cells","volume":"54","author":"Ghoreishi","year":"2011","journal-title":"J. Math. Comput. Model."},{"key":"ref_48","first-page":"13","article-title":"Global Dynamics of an HIV Infection Model with two Classes of Target Cells and Distributed Delayes","volume":"2012","author":"Elaiw","year":"2012","journal-title":"J. Discret. Dyn. Nat. Soc."},{"key":"ref_49","first-page":"5","article-title":"The Adomian Decomposition Method for Solving HIV Infection Model of Latently Infected Cells","volume":"3","author":"Ali","year":"2019","journal-title":"J. MSMK"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.compbiolchem.2016.12.006","article-title":"An exponential Galerkin method for solutions of HIV infection model of CD4+ T-cells","volume":"67","year":"2017","journal-title":"Comput. Biol. Chem."},{"key":"ref_51","first-page":"191","article-title":"Using Mathematics to Understand HIV Immune Dynamics","volume":"43","author":"Kirschner","year":"1996","journal-title":"J. Math. Biosci."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/BF02458312","article-title":"A Model for HIV Treatment Strategy in the Chemotherapy of AIDS","volume":"58","author":"Webb","year":"1996","journal-title":"J. Math. Biol."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"100118","DOI":"10.1016\/j.rinam.2020.100118","article-title":"Mathematical modeling and numerical simulation of HIV infection model","volume":"7","author":"Attaullah","year":"2020","journal-title":"Results Appl. Math."},{"key":"ref_54","first-page":"435","article-title":"Beitrag zur naerungsweisen integration totaler differentialgleichungen","volume":"46","author":"Kutta","year":"1901","journal-title":"Z. Math. Phy."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Butcher, J.C. (2008). Numerical Methods for Ordinary Differential Equations, Wiley.","DOI":"10.1002\/9780470753767"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.matcom.2022.02.024","article-title":"Local radial basis function-finite difference based algorithms for singularly perturbed Burgers\u2019 model","volume":"198","author":"Jiwari","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1375","DOI":"10.1007\/s00366-020-01278-3","article-title":"A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models","volume":"38","author":"Mittal","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"102846","DOI":"10.1016\/j.wavemoti.2021.102846","article-title":"Local radial basis functions and scale-3 Haar wavelets operational matrices based numerical algorithms for generalized regularized long wave model","volume":"109","author":"Pandit","year":"2022","journal-title":"Wave Motion"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s40010-018-0538-y","article-title":"New scale-3 haar wavelets algorithm for numerical simulation of second order ordinary differential equations","volume":"89","author":"Mittal","year":"2019","journal-title":"Proc. Natl. Acad. Sci. India Sect. A Phys. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/578\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:59:05Z","timestamp":1760144345000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/578"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,21]]},"references-count":59,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["axioms11100578"],"URL":"https:\/\/doi.org\/10.3390\/axioms11100578","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,21]]}}}