{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T16:54:56Z","timestamp":1782924896473,"version":"3.54.5"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12571548"],"award-info":[{"award-number":["12571548"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["DMS-2324692"],"award-info":[{"award-number":["DMS-2324692"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,12]]},"DOI":"10.1007\/s40314-026-03837-8","type":"journal-article","created":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T16:21:16Z","timestamp":1782922876000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modeling HIV dynamics with two strains: stability analysis and optimal control"],"prefix":"10.1007","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1583-4118","authenticated-orcid":false,"given":"Xia","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xin","family":"Meng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Libin","family":"Rong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,1]]},"reference":[{"key":"3837_CR1","doi-asserted-by":"crossref","unstructured":"Acquired Immunodeficiency Syndrome Professional Group, Society of Infectious Diseases, Chinese Medical Association, Chinese Center for Disease Control and Prevention (2024) Chinese guidelines for the diagnosis and treatment of human immunodeficiency virus infection\/acquired immunodeficiency syndrome (2024 edition), Chin Med J 137(22):2654-2680","DOI":"10.1097\/CM9.0000000000003383"},{"issue":"4","key":"3837_CR2","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.epidem.2012.10.001","volume":"4","author":"HK Alexander","year":"2012","unstructured":"Alexander HK, Bonhoeffer S (2012) Pre-existence and emergence of drug resistance in a generalized model of intra-host viral dynamics. Epidemics 4(4):187\u2013202","journal-title":"Epidemics"},{"issue":"1","key":"3837_CR3","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/s12591-021-00572-w","volume":"32","author":"L Beilina","year":"2024","unstructured":"Beilina L, Eriksson M, Gainova I (2024) Time-adaptive determination of drug efficacy in mathematical model of HIV infection. Differ Equ Dyn Syst 32(1):313\u2013347","journal-title":"Differ Equ Dyn Syst"},{"key":"3837_CR4","doi-asserted-by":"publisher","first-page":"260","DOI":"10.3389\/fimmu.2018.00260","volume":"9","author":"L Bracq","year":"2018","unstructured":"Bracq L, Xie M, Benichou S et al (2018) Mechanisms for cell-to-cell transmission of HIV-1. Front Immunol 9:260","journal-title":"Front Immunol"},{"key":"3837_CR5","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/j.matcom.2023.04.029","volume":"212","author":"C Chen","year":"2023","unstructured":"Chen C, Zhou Y (2023) Dynamic analysis of HIV model with a general incidence, CTLs immune response and intracellular delays. Math Comput Simul 212:159\u2013181","journal-title":"Math Comput Simul"},{"key":"3837_CR6","volume":"406","author":"W Chen","year":"2021","unstructured":"Chen W, Teng Z, Zhang L (2021) Global dynamics for a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed delays. Appl Math Comput 406:126284","journal-title":"Appl Math Comput"},{"issue":"15","key":"3837_CR7","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1080\/10255842.2023.2275248","volume":"27","author":"C Chen","year":"2024","unstructured":"Chen C, Zhou Y, Ye Z (2024) Stability and optimal control of a cytokine-enhanced general HIV infection model with antibody immune response and CTLs immune response. Comput Methods Biomech Biomed Eng 27(15):2199\u20132230","journal-title":"Comput Methods Biomech Biomed Eng"},{"issue":"4","key":"3837_CR8","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1093\/imammb\/dqn023","volume":"25","author":"P De Leenheer","year":"2008","unstructured":"De Leenheer P, Pilyugin SS (2008) Multistrain virus dynamics with mutations: a global analysis. Math Med Biol 25(4):285\u2013322","journal-title":"Math Med Biol"},{"issue":"7","key":"3837_CR9","doi-asserted-by":"publisher","first-page":"3543","DOI":"10.3934\/dcdsb.2020245","volume":"26","author":"Q Deng","year":"2021","unstructured":"Deng Q, Qiu Z, Guo T et al (2021) Modeling within-host viral dynamics: the role of CTL immune responses in the evolution of drug resistance. Discrete Contin Dyn Syst -Ser B 26(7):3543\u20133562","journal-title":"Discrete Contin Dyn Syst -Ser B"},{"issue":"1","key":"3837_CR10","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1007\/s00285-024-02104-w","volume":"89","author":"Q Deng","year":"2024","unstructured":"Deng Q, Guo T, Qiu Z et al (2024) A mathematical model for HIV dynamics with multiple infections: implications for immune escape. J Math Biol 89(1):6","journal-title":"J Math Biol"},{"issue":"3","key":"3837_CR11","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1016\/j.immuni.2012.08.010","volume":"37","author":"E Eisele","year":"2012","unstructured":"Eisele E, Siliciano RF (2012) Redefining the viral reservoirs that prevent HIV-1 eradication. Immunity 37(3):377\u2013388","journal-title":"Immunity"},{"key":"3837_CR12","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1007\/s40840-018-0699-5","volume":"43","author":"T Guo","year":"2020","unstructured":"Guo T, Qiu Z, Rong L (2020) Analysis of an HIV model with immune responses and cell-to-cell transmission. Bull Malays Math Sci Soc 43:581\u2013607","journal-title":"Bull Malays Math Sci Soc"},{"issue":"1","key":"3837_CR13","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s00285-020-01513-x","volume":"81","author":"T Guo","year":"2020","unstructured":"Guo T, Qiu Z, Rong L (2020) Modeling the role of macrophages in HIV persistence during antiretroviral therapy. J Math Biol 81(1):369\u2013402","journal-title":"J Math Biol"},{"key":"3837_CR14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2020.110502","volume":"509","author":"T Guo","year":"2021","unstructured":"Guo T, Qiu Z, Kitagawa K et al (2021) Modeling HIV multiple infection. J Theor Biol 509:110502","journal-title":"J Theor Biol"},{"key":"3837_CR15","unstructured":"Hale JK, Lunel SMV (2013) Introduction to functional differential equations. Springer Science & Business Media"},{"issue":"6","key":"3837_CR16","doi-asserted-by":"publisher","first-page":"16280","DOI":"10.3934\/math.2024788","volume":"9","author":"L Hong","year":"2024","unstructured":"Hong L, Li J, Rong L et al (2024) Global dynamics of a delayed model with cytokine-enhanced viral infection and cell-to-cell transmission. AIMS Math 9(6):16280\u201316296","journal-title":"AIMS Math"},{"key":"3837_CR17","unstructured":"Kuang Y (1993) Delay differential equations: with applications in population dynamics. Academic Press"},{"issue":"3","key":"3837_CR18","doi-asserted-by":"publisher","first-page":"898","DOI":"10.1137\/130930145","volume":"74","author":"X Lai","year":"2014","unstructured":"Lai X, Zou X (2014) Modeling HIV-1 virus dynamics with both virus-to-cell infection and cell-to-cell transmission. SIAM J Appl Math 74(3):898\u2013917","journal-title":"SIAM J Appl Math"},{"issue":"1","key":"3837_CR19","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1016\/j.jmaa.2014.10.086","volume":"426","author":"X Lai","year":"2015","unstructured":"Lai X, Zou X (2015) Modeling cell-to-cell spread of HIV-1 with logistic target cell growth. J Math Anal Appl 426(1):563\u2013584","journal-title":"J Math Anal Appl"},{"key":"3837_CR20","doi-asserted-by":"crossref","unstructured":"Lenhart S, Workman JT (2007) Optimal control applied to biological models. Chapman and Hall\/CRC","DOI":"10.1201\/9781420011418"},{"key":"3837_CR21","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/j.cam.2016.02.051","volume":"307","author":"A Mallela","year":"2016","unstructured":"Mallela A, Lenhart S, Vaidya NK (2016) HIV-TB co-infection treatment: modeling and optimal control theory perspectives. J Comput Appl Math 307:143\u2013161","journal-title":"J Comput Appl Math"},{"issue":"4","key":"3837_CR22","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1016\/j.cell.2006.02.007","volume":"124","author":"M Marsh","year":"2006","unstructured":"Marsh M, Helenius A (2006) Virus entry: open sesame. Cell 124(4):729\u2013740","journal-title":"Cell"},{"issue":"2","key":"3837_CR23","doi-asserted-by":"publisher","DOI":"10.1371\/journal.ppat.1000788","volume":"6","author":"D Mazurov","year":"2010","unstructured":"Mazurov D, Ilinskaya A, Heidecker G et al (2010) Quantitative comparison of HTLV-1 and HIV-1 cell-to-cell infection with new replication dependent vectors. PLoS Pathog 6(2):e1000788","journal-title":"PLoS Pathog"},{"key":"3837_CR24","unstructured":"Miranda F (2014) Systems theory: perspectives, applications and developments. Nova Publishers"},{"issue":"7","key":"3837_CR25","doi-asserted-by":"publisher","first-page":"3924","DOI":"10.1128\/JVI.06478-11","volume":"86","author":"B Monel","year":"2012","unstructured":"Monel B, Beaumont E, Vendrame D et al (2012) HIV cell-to-cell transmission requires the production of infectious virus particles and does not proceed through env-mediated fusion pores. J Virol 86(7):3924\u20133933","journal-title":"J Virol"},{"issue":"17","key":"3837_CR26","doi-asserted-by":"publisher","first-page":"8360","DOI":"10.1128\/JVI.00443-10","volume":"84","author":"W Mothes","year":"2010","unstructured":"Mothes W, Sherer NM, Jin J et al (2010) Virus cell-to-cell transmission. J Virol 84(17):8360\u20138368","journal-title":"J Virol"},{"issue":"2","key":"3837_CR27","doi-asserted-by":"publisher","first-page":"619","DOI":"10.3934\/mbe.2019030","volume":"16","author":"D Olabode","year":"2019","unstructured":"Olabode D, Rong L, Wang X (2019) Optimal control in HIV chemotherapy with termination viral load and latent reservoir. Math Biosci Eng 16(2):619\u2013635","journal-title":"Math Biosci Eng"},{"issue":"1","key":"3837_CR28","first-page":"1890320","volume":"2020","author":"C Qin","year":"2020","unstructured":"Qin C, Wang X, Rong L (2020) An age-structured model of HIV latent infection with two transmission routes: analysis and optimal control. Complexity 2020(1):1890320","journal-title":"Complexity"},{"key":"3837_CR29","doi-asserted-by":"publisher","first-page":"2027","DOI":"10.1007\/s11538-007-9203-3","volume":"69","author":"L Rong","year":"2007","unstructured":"Rong L, Feng Z, Perelson AS (2007) Emergence of HIV-1 drug resistance during antiretroviral treatment. Bull Math Biol 69:2027\u20132060","journal-title":"Bull Math Biol"},{"issue":"217","key":"3837_CR30","first-page":"1","volume":"2015","author":"PK Roy","year":"2015","unstructured":"Roy PK, Saha S, Basir FA (2015) Effect of awareness programs in controlling the disease HIV\/AIDS: an optimal control theoretic approach. Adv Differ Equ 2015(217):1\u201318","journal-title":"Adv Differ Equ"},{"issue":"11","key":"3837_CR31","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1038\/nrmicro1972","volume":"6","author":"Q Sattentau","year":"2008","unstructured":"Sattentau Q (2008) Avoiding the void: cell-to-cell spread of human viruses. Nat Rev Microbiol 6(11):815\u2013826","journal-title":"Nat Rev Microbiol"},{"key":"3837_CR32","doi-asserted-by":"crossref","unstructured":"Sattentau QJ (2010) Cell-to-cell spread of retroviruses. Viruses 2(6):1306\u20131321","DOI":"10.3390\/v2061306"},{"issue":"5","key":"3837_CR33","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1016\/j.coviro.2011.09.004","volume":"1","author":"QJ Sattentau","year":"2011","unstructured":"Sattentau QJ (2011) The direct passage of animal viruses between cells. Curr Opin Virol 1(5):396\u2013402","journal-title":"Curr Opin Virol"},{"key":"3837_CR34","doi-asserted-by":"crossref","unstructured":"Sebastian B, Martin AN (1997) Pre-existence and emergence of drug resistance in HIV-1 infection. Proc R Soc London Ser B 264(1382):631\u2013637","DOI":"10.1098\/rspb.1997.0089"},{"issue":"11","key":"3837_CR35","doi-asserted-by":"publisher","DOI":"10.1101\/cshperspect.a006965","volume":"2","author":"GM Shaw","year":"2012","unstructured":"Shaw GM, Hunter E (2012) HIV transmission. Cold Spring Harb Perspect Med 2(11):a006965","journal-title":"Cold Spring Harb Perspect Med"},{"issue":"3","key":"3837_CR36","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1038\/ncb1544","volume":"9","author":"NM Sherer","year":"2007","unstructured":"Sherer NM, Lehmann MJ, Jimenez-Soto LF et al (2007) Retroviruses can establish filopodial bridges for efficient cell-to-cell transmission. Nat Cell Biol 9(3):310\u2013315","journal-title":"Nat Cell Biol"},{"issue":"7362","key":"3837_CR37","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1038\/nature10347","volume":"477","author":"A Sigal","year":"2011","unstructured":"Sigal A, Kim JT, Balazs AB et al (2011) Cell-to-cell spread of HIV permits ongoing replication despite antiretroviral therapy. Nature 477(7362):95\u201398","journal-title":"Nature"},{"issue":"2","key":"3837_CR38","doi-asserted-by":"publisher","first-page":"1000","DOI":"10.1128\/JVI.01629-06","volume":"81","author":"M Sourisseau","year":"2007","unstructured":"Sourisseau M, Sol-Foulon N, Porrot F et al (2007) Inefficient human immunodeficiency virus replication in mobile lymphocytes. J Virol 81(2):1000\u20131012","journal-title":"J Virol"},{"key":"3837_CR39","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1007\/s11538-010-9543-2","volume":"73","author":"MO Souza","year":"2011","unstructured":"Souza MO, Zubelli JP (2011) Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation. Bull Math Biol 73:609\u2013625","journal-title":"Bull Math Biol"},{"issue":"1","key":"3837_CR40","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","volume":"180","author":"P Van den Driessche","year":"2002","unstructured":"Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29\u201348","journal-title":"Math Biosci"},{"issue":"sup2","key":"3837_CR41","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1080\/17513758.2016.1242784","volume":"11","author":"X Wang","year":"2017","unstructured":"Wang X, Tang S, Song X et al (2017) Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission. J Biol Dyn 11(sup2):455\u2013483","journal-title":"J Biol Dyn"},{"issue":"4","key":"3837_CR42","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1142\/S021833901740006X","volume":"25","author":"S Wang","year":"2017","unstructured":"Wang S, Xu F, Rong L (2017) Bistability analysis of an HIV model with immune response. J Biol Syst 25(4):677\u2013695","journal-title":"J Biol Syst"},{"key":"3837_CR43","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1016\/j.apm.2020.03.040","volume":"84","author":"P Wu","year":"2020","unstructured":"Wu P, Zhao H (2020) Dynamics of an HIV infection model with two infection routes and evolutionary competition between two viral strains. Appl Math Modell 84:240\u2013264","journal-title":"Appl Math Modell"},{"issue":"2","key":"3837_CR44","doi-asserted-by":"publisher","first-page":"2350019","DOI":"10.1142\/S0218127423500190","volume":"33","author":"R Xu","year":"2023","unstructured":"Xu R, Yang Y (2023) Stability and Hopf bifurcation analysis of an HIV infection model in the evolution of drug resistance. Int J Bifurcation Chaos 33(2):2350019","journal-title":"Int J Bifurcation Chaos"},{"issue":"18","key":"3837_CR45","doi-asserted-by":"publisher","first-page":"10123","DOI":"10.1128\/JVI.01102-12","volume":"86","author":"A Yakimovich","year":"2012","unstructured":"Yakimovich A, Gumpert H, Burckhardt CJ et al (2012) Cell-free transmission of human adenovirus by passive mass transfer in cell culture simulated in a computer model. J Virol 86(18):10123\u201310137","journal-title":"J Virol"},{"issue":"9","key":"3837_CR46","doi-asserted-by":"publisher","first-page":"3491","DOI":"10.3934\/dcdsb.2020070","volume":"25","author":"H Zhao","year":"2020","unstructured":"Zhao H, Wu P, Ruan S (2020) Dynamic analysis and optimal control of a three-age-class HIV\/AIDS epidemic model in China. Discrete Contin Dyn Syst -Ser B 25(9):3491\u20133521","journal-title":"Discrete Contin Dyn Syst -Ser B"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03837-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03837-8","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03837-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T16:21:32Z","timestamp":1782922892000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03837-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,1]]},"references-count":46,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,12]]}},"alternative-id":["3837"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03837-8","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,1]]},"assertion":[{"value":"8 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 June 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 July 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"449"}}