{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T11:25:38Z","timestamp":1750677938497},"reference-count":31,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["DCDS-S"],"published-print":{"date-parts":[[2022]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;p style='text-indent:20px;'&gt;We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.&lt;\/p&gt;<\/jats:p>","DOI":"10.3934\/dcdss.2021148","type":"journal-article","created":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T10:42:30Z","timestamp":1638528150000},"page":"501","source":"Crossref","is-referenced-by-count":6,"title":["Optimal control of an HIV model with a trilinear antibody growth function"],"prefix":"10.3934","volume":"15","author":[{"given":"Karam","family":"Allali","sequence":"first","affiliation":[]},{"given":"Sanaa","family":"Harroudi","sequence":"additional","affiliation":[]},{"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[]}],"member":"2321","reference":[{"key":"key-10.3934\/dcdss.2021148-1","doi-asserted-by":"publisher","unstructured":"K. Allali, S. Harroudi, D. F. M. Torres.Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, <i>Math. Comput. Sci.<\/i>, <b>12<\/b> (2018), 111-127.","DOI":"10.1007\/s11786-018-0333-9"},{"key":"key-10.3934\/dcdss.2021148-2","doi-asserted-by":"publisher","unstructured":"K. Allali, Y. Tabit, S. Harroudi.On HIV model with adaptive immune response, two saturated rates and therapy, <i>Math. Model. Nat. Phenom.<\/i>, <b>12<\/b> (2017), 1-14.","DOI":"10.1051\/mmnp\/201712501"},{"key":"key-10.3934\/dcdss.2021148-3","doi-asserted-by":"publisher","unstructured":"M. S. Ciupe, B. L. Bivort, D. M. Bortz, P. W. Nelson.Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, <i>Math. Biosci.<\/i>, <b>200<\/b> (2006), 1-27.","DOI":"10.1016\/j.mbs.2005.12.006"},{"key":"key-10.3934\/dcdss.2021148-4","doi-asserted-by":"publisher","unstructured":"R. Culshaw, S. Ruan, R. J. Spiteri.Optimal HIV treatment by maximising immune response, <i>J. Math. Biol.<\/i>, <b>48<\/b> (2004), 545-562.","DOI":"10.1007\/s00285-003-0245-3"},{"key":"key-10.3934\/dcdss.2021148-5","doi-asserted-by":"publisher","unstructured":"M. P. Davenport, R. M. Ribeiro, A. S. Perelson.Kinetics of virus-specific CD8+ T cells and the control of human immunodeficiency virus infection, <i>J. Vir.<\/i>, <b>78<\/b> (2004), 10096-10103.","DOI":"10.1128\/JVI.78.18.10096-10103.2004"},{"key":"key-10.3934\/dcdss.2021148-6","doi-asserted-by":"publisher","unstructured":"R. J. De Boer, A. S. Perelson.Quantifying T lymphocyte turnover, <i>J. Theoret. Biol.<\/i>, <b>327<\/b> (2013), 45-87.","DOI":"10.1016\/j.jtbi.2012.12.025"},{"key":"key-10.3934\/dcdss.2021148-7","doi-asserted-by":"publisher","unstructured":"R. Denysiuk, C. J. Silva, D. F. M. Torres.Multiobjective optimization to a TB-HIV\/AIDS coinfection optimal control problem, <i>Comput. Appl. Math.<\/i>, <b>37<\/b> (2018), 2112-2128.","DOI":"10.1007\/s40314-017-0438-9"},{"key":"key-10.3934\/dcdss.2021148-8","doi-asserted-by":"publisher","unstructured":"J. Djordjevic, C. J. Silva, D. F. M. Torres.A stochastic SICA epidemic model for HIV transmission, <i>Appl. Math. Lett.<\/i>, <b>84<\/b> (2018), 168-175.","DOI":"10.1016\/j.aml.2018.05.005"},{"key":"key-10.3934\/dcdss.2021148-9","doi-asserted-by":"crossref","unstructured":"W. H. Fleming and R. W. Rishel, <i>Deterministic and Stochastic Optimal Control<\/i>, Springer-Verlag, Berlin, 1975.","DOI":"10.1007\/978-1-4612-6380-7"},{"key":"key-10.3934\/dcdss.2021148-10","doi-asserted-by":"crossref","unstructured":"I. S. Gradshte\u01d0n, I. M. Ryzhik.Table of integrals, series, and products, <i>Math. Comp.<\/i>, <b>39<\/b> (1982), 747-757.","DOI":"10.1090\/S0025-5718-82-99823-4"},{"key":"key-10.3934\/dcdss.2021148-11","doi-asserted-by":"publisher","unstructured":"O. Kostylenko, H. S. Rodrigues, D. F. M. Torres.The spread of a financial virus through Europe and beyond, <i>AIMS Math.<\/i>, <b>4<\/b> (2019), 86-98.","DOI":"10.3934\/Math.2019.1.86"},{"key":"key-10.3934\/dcdss.2021148-12","unstructured":"D. L. Lukes, <i>Differential Equations<\/i>, Mathematics in Science and Engineering, 162, Academic Press, Inc. 1982."},{"key":"key-10.3934\/dcdss.2021148-13","doi-asserted-by":"publisher","unstructured":"C. C. McCluskey, M. Santoprete.A bare-bones mathematical model of radicalization, <i>J. Dyn. Games<\/i>, <b>5<\/b> (2018), 243-264.","DOI":"10.3934\/jdg.2018016"},{"key":"key-10.3934\/dcdss.2021148-14","doi-asserted-by":"publisher","unstructured":"M. A. Nowak, R. M. May.Mathematical biology of HIV infections: Antigenic variation and diversity threshold, <i>Math. Biosci.<\/i>, <b>106<\/b> (1991), 1-21.","DOI":"10.1016\/0025-5564(91)90037-J"},{"key":"key-10.3934\/dcdss.2021148-15","doi-asserted-by":"crossref","unstructured":"M. A. Nowak, R. M. May., <i>Virus Dynamics<\/i>, <b>${ref.volume}<\/b> (2000).","DOI":"10.1093\/oso\/9780198504184.001.0001"},{"key":"key-10.3934\/dcdss.2021148-16","doi-asserted-by":"publisher","unstructured":"K. A. Pawelek, S. Liu, F. Pahlevani, L. Rong.A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, <i>Math. Biosci.<\/i>, <b>235<\/b> (2012), 98-109.","DOI":"10.1016\/j.mbs.2011.11.002"},{"key":"key-10.3934\/dcdss.2021148-17","doi-asserted-by":"publisher","unstructured":"A. S. Perelson, P. W. Nelson.Mathematical analysis of HIV-1 dynamics in vivo, <i>SIAM Rev.<\/i>, <b>41<\/b> (1999), 3-44.","DOI":"10.1137\/S0036144598335107"},{"key":"key-10.3934\/dcdss.2021148-18","doi-asserted-by":"publisher","unstructured":"A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, D. D. Ho.HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, <i>Science<\/i>, <b>271<\/b> (1996), 1582-1586.","DOI":"10.1126\/science.271.5255.1582"},{"key":"key-10.3934\/dcdss.2021148-19","unstructured":"L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, <i>The Mathematical Theory of Optimal Processes<\/i>, Interscience Publishers John Wiley &amp; Sons, Inc., New York, 1962."},{"key":"key-10.3934\/dcdss.2021148-20","doi-asserted-by":"publisher","unstructured":"D. Rocha, C. J. Silva, D. F. M. Torres.Stability and optimal control of a delayed HIV model, <i>Math. Methods Appl. Sci.<\/i>, <b>41<\/b> (2018), 2251-2260.","DOI":"10.1002\/mma.4207"},{"key":"key-10.3934\/dcdss.2021148-21","doi-asserted-by":"publisher","unstructured":"F. Rodrigues, C. J. Silva, D. F. M. Torres, H. Maurer.Optimal control of a delayed HIV model, <i>Discrete Contin. Dyn. Syst. Ser. B<\/i>, <b>23<\/b> (2018), 443-458.","DOI":"10.3934\/dcdsb.2018030"},{"key":"key-10.3934\/dcdss.2021148-22","doi-asserted-by":"publisher","unstructured":"S. Saha, G. P. Samanta.Modelling and optimal control of HIV\/AIDS prevention through PrEP and limited treatment, <i>Phys. A<\/i>, <b>516<\/b> (2019), 280-307.","DOI":"10.1016\/j.physa.2018.10.033"},{"key":"key-10.3934\/dcdss.2021148-23","doi-asserted-by":"publisher","unstructured":"C. J. Silva and D. F. M. Torres, Modeling TB-HIV syndemic and treatment, <i>J. Appl. Math.<\/i>, <b>2014<\/b> (2014), 248407, 14 pp.","DOI":"10.1155\/2014\/248407"},{"key":"key-10.3934\/dcdss.2021148-24","doi-asserted-by":"publisher","unstructured":"C. J. Silva, D. F. M. Torres.A TB-HIV\/AIDS coinfection model and optimal control treatment, <i>Discrete Contin. Dyn. Syst.<\/i>, <b>35<\/b> (2015), 4639-4663.","DOI":"10.3934\/dcds.2015.35.4639"},{"key":"key-10.3934\/dcdss.2021148-25","doi-asserted-by":"publisher","unstructured":"C. J. Silva, D. F. M. Torres.A SICA compartmental model in epidemiology with application to HIV\/AIDS in Cape Verde, <i>Ecological Complexity<\/i>, <b>30<\/b> (2017), 70-75.","DOI":"10.1016\/j.ecocom.2016.12.001"},{"key":"key-10.3934\/dcdss.2021148-26","doi-asserted-by":"publisher","unstructured":"C. J. Silva, D. F. M. Torres.Global stability for a HIV\/AIDS model, <i>Commun. Fac. Sci. Univ. Ank. S\u00e9r. A1 Math. Stat.<\/i>, <b>67<\/b> (2018), 93-101.","DOI":"10.1501\/Commua1_0000000833"},{"key":"key-10.3934\/dcdss.2021148-27","doi-asserted-by":"publisher","unstructured":"C. J. Silva, D. F. M. Torres.Modeling and optimal control of HIV\/AIDS prevention through PrEP, <i>Discrete Contin. Dyn. Syst. Ser. S<\/i>, <b>11<\/b> (2018), 119-141.","DOI":"10.3934\/dcdss.2018008"},{"key":"key-10.3934\/dcdss.2021148-28","doi-asserted-by":"publisher","unstructured":"P. van den Driessche.Reproduction numbers of infectious disease models, <i>Infect. Dis. Model.<\/i>, <b>2<\/b> (2017), 288-303.","DOI":"10.1016\/j.idm.2017.06.002"},{"key":"key-10.3934\/dcdss.2021148-29","doi-asserted-by":"publisher","unstructured":"Y. Wang, Y. Zhou, F. Brauer, J. M. Heffernan.Viral dynamics model with CTL immune response incorporating antiretroviral therapy, <i>J. Math. Biol.<\/i>, <b>67<\/b> (2013), 901-934.","DOI":"10.1007\/s00285-012-0580-3"},{"key":"key-10.3934\/dcdss.2021148-30","doi-asserted-by":"publisher","unstructured":"D. Wodarz, <i>Killer Cell Dynamics<\/i>, Interdisciplinary Applied Mathematics, 32, Springer-Verlag, New York, 2007.","DOI":"10.1007\/978-0-387-68733-9"},{"key":"key-10.3934\/dcdss.2021148-31","doi-asserted-by":"publisher","unstructured":"H. Zhu, X. Zou.Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, <i>Discrete Contin. Dyn. Syst. Ser. B<\/i>, <b>12<\/b> (2009), 511-524.","DOI":"10.3934\/dcdsb.2009.12.511"}],"container-title":["Discrete &amp; Continuous Dynamical Systems - S"],"original-title":[],"deposited":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T15:22:50Z","timestamp":1699888970000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/dcdss.2021148"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022]]}},"alternative-id":["1937-1632_2022_3_501"],"URL":"https:\/\/doi.org\/10.3934\/dcdss.2021148","relation":{},"ISSN":["1937-1632","1937-1179"],"issn-type":[{"value":"1937-1632","type":"print"},{"value":"1937-1179","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022]]}}}