{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T17:36:24Z","timestamp":1768498584387,"version":"3.49.0"},"reference-count":49,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,24]],"date-time":"2024-02-24T00:00:00Z","timestamp":1708732800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Natural Science Foundation of Xinjiang Uygur Autonomous Region","award":["2022TSYCCX0015"],"award-info":[{"award-number":["2022TSYCCX0015"]}]},{"name":"the Natural Science Foundation of Xinjiang Uygur Autonomous Region","award":["2021D01E12"],"award-info":[{"award-number":["2021D01E12"]}]},{"name":"the Natural Science Foundation of Xinjiang Uygur Autonomous Region","award":["12361103"],"award-info":[{"award-number":["12361103"]}]},{"name":"the National Natural Science Foundation of China","award":["2022TSYCCX0015"],"award-info":[{"award-number":["2022TSYCCX0015"]}]},{"name":"the National Natural Science Foundation of China","award":["2021D01E12"],"award-info":[{"award-number":["2021D01E12"]}]},{"name":"the National Natural Science Foundation of China","award":["12361103"],"award-info":[{"award-number":["12361103"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS\/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R\u02dc0h, is rigorously investigated. More specifically, if R\u02dc0h&lt;1, the disease-free equilibrium is globally asymptotically stable; if R\u02dc0h&gt;1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.<\/jats:p>","DOI":"10.3390\/axioms13030147","type":"journal-article","created":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T03:34:04Z","timestamp":1708918444000},"page":"147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Analysis of a Delayed Multiscale AIDS\/HIV-1 Model Coupling Between-Host and Within-Host Dynamics"],"prefix":"10.3390","volume":"13","author":[{"given":"Miao","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"given":"Yaping","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1233-3676","authenticated-orcid":false,"given":"Lin","family":"Hu","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5960-8078","authenticated-orcid":false,"given":"Linfei","family":"Nie","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,24]]},"reference":[{"key":"ref_1","first-page":"3","article-title":"National Institute of Allergy and Infectious Disease","volume":"1","author":"Westem","year":"2008","journal-title":"Reseach Conf. Natl. Inst. Allergy Infect. Dis."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Whiteside, A. (2016). HIV and AIDS: A Very Short Introduction, Oxford University Press.","DOI":"10.1093\/actrade\/9780198727491.001.0001"},{"key":"ref_3","unstructured":"(2023, July 18). HIV Statistic Globally and by WHO Region, 2023. Available online: https:\/\/www.who.int\/teams\/global-hiv-hepatitis-and-stis-programmes\/hiv\/strategic-information\/hiv-data-and-statistics."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0025-5564(02)00149-9","article-title":"A model of HIV\/AIDS with staged progression and amelioration","volume":"181","author":"Mccluskey","year":"2003","journal-title":"Math. Biosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1016\/j.nonrwa.2010.09.010","article-title":"Permanence and extinction of a nonautonomous HIV\/AIDS epidemic model with distributed time delay","volume":"12","author":"Samanta","year":"2011","journal-title":"Nonlinear Anal. Real"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1582","DOI":"10.1126\/science.271.5255.1582","article-title":"HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time","volume":"271","author":"Perelson","year":"1996","journal-title":"Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"511","DOI":"10.3934\/dcdsb.2009.12.511","article-title":"Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay","volume":"12","author":"Zhu","year":"2009","journal-title":"Discrete Cont. Dyn. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1038\/nature08898","article-title":"Immunology and the elusive AIDS vaccine","volume":"464","author":"Virgin","year":"2010","journal-title":"Nature"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1038\/nature12893","article-title":"Immunological and virological mechanisms of vaccine-mediated protection against SIV and HIV","volume":"505","author":"Roederer","year":"2014","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1313","DOI":"10.1016\/j.apm.2011.07.086","article-title":"Global stability of in-host viral models with humoral immunity and intracellular delays","volume":"36","author":"Wang","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s00285-005-0321-y","article-title":"Stability analysis of pathogen-immune interaction dynamics","volume":"51","author":"Murase","year":"2005","journal-title":"J. Math. Biol."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Huang, G., Takeuchi, Y., Ma, W., and Wei, D. (2010). Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. B. Math. Biol., 72.","DOI":"10.1007\/s11538-009-9487-6"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.matcom.2012.10.004","article-title":"Global dynamics of an SEIS epidemiological model with time delay describing a latent period","volume":"785","author":"Xu","year":"2012","journal-title":"Math. Comput. Simulat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"111527","DOI":"10.1016\/j.chaos.2021.111527","article-title":"Dynamics of a time-delayed two-strain epidemic model with general incidence rates","volume":"153","author":"Farah","year":"2021","journal-title":"Chaos Soliton. Fract."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.chaos.2015.02.017","article-title":"The effect of time delay on the dynamics of an SEIR model with nonlinear incidence","volume":"75","author":"Tipsri","year":"2015","journal-title":"Chaos Soliton. Fract."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.matcom.2015.11.002","article-title":"Global hopf bifurcation and permanence of a delayed SEIRS epidemic model","volume":"122","author":"Jiang","year":"2016","journal-title":"Math. Comput. Simulat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s00285-002-0191-5","article-title":"A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay","volume":"46","author":"Culshaw","year":"2003","journal-title":"J. Math. Biol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2799","DOI":"10.1016\/j.camwa.2011.03.050","article-title":"Global stability of an HIV-1 infection model with saturation infection and intracellular delay","volume":"61","author":"Xu","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.physa.2018.09.087","article-title":"Dynamics of two time delays differential equation model to HIV latent infection","volume":"541","author":"Liu","year":"2019","journal-title":"Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.mbs.2011.11.002","article-title":"A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data","volume":"235","author":"Pawelek","year":"2012","journal-title":"Math. Biosci."},{"key":"ref_21","first-page":"516","article-title":"Threshold dynamics of an HIV-1 virus model with both virus-to-cell and cell-to-cell transmissions, tntracellular delay, and humoral immunity","volume":"315","author":"Lin","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1938","DOI":"10.3934\/mbe.2023089","article-title":"Stability and Hopf bifurcation of an HIV infection model with two time delays","volume":"20","author":"Yang","year":"2023","journal-title":"Math. Biosci. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s11071-011-0291-0","article-title":"A model for coupling within-host and between-host dynamics in an infectious disease","volume":"68","author":"Feng","year":"2011","journal-title":"Nonlinear Dynam."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Lingappa, J.R., Hughes, J.P., Wang, R.S., Baeten, J.M., Connie, C., Gray, G.E., Stevens, W.S., Deborah, D., Campbell, M.S., and Carey, F. (2010). Estimating the impact of plasma HIV-1 RNA reductions on heterosexual HIV-1 transmission risk. PLoS ONE, 5.","DOI":"10.1371\/journal.pone.0012598"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1056\/NEJMoa1105243","article-title":"Prevention of HIV-1 infection with early antiretroviral therapy","volume":"365","author":"Cohen","year":"2011","journal-title":"New Engl. J. Med."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1006\/jtbi.2002.3076","article-title":"Modeling host-parasite coevolution: A nested spproach based on mechanistic models","volume":"218","author":"Gilchrist","year":"2002","journal-title":"J. Theor. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.tpb.2005.07.002","article-title":"Evolution of virulence: Interdependence, constraints, and selection using nested models","volume":"69","author":"Gilchrist","year":"2006","journal-title":"Theor. Popul. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1007\/s00285-007-0135-1","article-title":"Superinfections can induce evolutionarily stable coexistence of pathogens","volume":"56","author":"Boldin","year":"2008","journal-title":"J. Math. Biol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.mbs.2012.09.004","article-title":"A mathematical model for coupling within-host and between-host dynamics in an environmentally infectious disease","volume":"241","author":"Feng","year":"2013","journal-title":"Math. Biosci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.jtbi.2014.07.030","article-title":"Emerging disease dynamics in a model coupling within-host and between-host systems","volume":"361","author":"Cen","year":"2014","journal-title":"J. Theor. Biol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.mbs.2015.02.012","article-title":"Coupled within-host and between-host dynamics and evolution of virulence","volume":"270","author":"Feng","year":"2015","journal-title":"Math. Biosci."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wang, X.Y., Wang, S.P., Wang, J., and Rong, L.B. (2022). A multiscale model of COVID-19 dynamics. Bull. Math. Biol., 84.","DOI":"10.1007\/s11538-022-01058-8"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"6720","DOI":"10.3934\/mbe.2020350","article-title":"Analysis of a multiscale HIV-1 model coupling within-host viral dynamics and between-host transmission dynamics","volume":"17","author":"Xue","year":"2020","journal-title":"Math. Biosci. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.nonrwa.2016.08.001","article-title":"Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission","volume":"34","author":"Wang","year":"2017","journal-title":"Nonlinear Anal.-Real"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Hale, J.K., and Lunel, S.M.V. (1993). Introduction to Functional Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-4342-7"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.chaos.2005.01.019","article-title":"On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays","volume":"26","author":"Li","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Hale, J.K. (1977). Theory of Functional Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/cpa.20154","article-title":"Asymptotic speeds of spread and traveling waves for monotone semiflows with applications","volume":"60","author":"Liang","year":"2007","journal-title":"Commun. Pur. Appl. Math."},{"key":"ref_39","unstructured":"Hassard, B.D., Kazarinoff, N.D., and Wan, Y.H. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.chaos.2003.12.075","article-title":"Bifurcation analysis for Chen\u2019s system with delayed feedback and its application to control of chaos","volume":"22","author":"Song","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10884-015-9425-2","article-title":"Basic reproduction ratios for periodic compartmental models with time delay","volume":"29","author":"Zhao","year":"2017","journal-title":"J. Dyn. Differ. Equ."},{"key":"ref_42","unstructured":"Smith, H.L. (1995). Monotone Dynamical Systems: An Introduction To the Theory of Competitive and Cooperative Systems, American Mathematical Society."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Zhao, X.Q. (2003). Dynamical Systems in Population Biology, Springer.","DOI":"10.1007\/978-0-387-21761-1"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1137\/15M1046277","article-title":"A periodic vector-bias malaria model with incubation period","volume":"77","author":"Wang","year":"2017","journal-title":"SIAM. J. Appl. Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"853","DOI":"10.1137\/16M1087916","article-title":"Dynamics of a time-delayed Lyme disease model with seasonality","volume":"16","author":"Wang","year":"2017","journal-title":"SIAM. J. Appl. Dyn. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Hale, J.K. (1988). Asymptotic Behavior of Dissipative Systems, American Mathematical Society.","DOI":"10.1007\/978-3-642-86458-2_14"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1126\/science.272.5258.74","article-title":"Population dynamics of immune responses to persistent viruses","volume":"272","author":"Nowak","year":"1996","journal-title":"Science"},{"key":"ref_48","first-page":"6771","article-title":"Remarks on basic reproduction ratios for periodic abstract functional differential equations","volume":"24","author":"Yang","year":"2019","journal-title":"Discrete Cont. Dyn. B"},{"key":"ref_49","unstructured":"(2023, July 26). World Health Organization. Available online: https:\/\/www.who.int\/director-general\/speeches\/detail\/who-director-general-s-opening-remarks-at-the-media-briefing---26-july-2023."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/147\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:04:21Z","timestamp":1760105061000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/147"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,24]]},"references-count":49,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030147"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030147","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,24]]}}}