{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,19]],"date-time":"2026-04-19T07:42:36Z","timestamp":1776584556546,"version":"3.51.2"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,2,1]],"date-time":"2020-02-01T00:00:00Z","timestamp":1580515200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,2,1]],"date-time":"2020-02-01T00:00:00Z","timestamp":1580515200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1815095"],"award-info":[{"award-number":["DMS-1815095"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11601392"],"award-info":[{"award-number":["11601392"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["GK201902005"],"award-info":[{"award-number":["GK201902005"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shaanxi Hundred-Talent Program"},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971285"],"award-info":[{"award-number":["11971285"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull Math Biol"],"published-print":{"date-parts":[[2020,2]]},"DOI":"10.1007\/s11538-020-00704-3","type":"journal-article","created":{"date-parts":[[2020,2,5]],"date-time":"2020-02-05T07:02:41Z","timestamp":1580886161000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Resonance of Periodic Combination Antiviral Therapy and Intracellular Delays in Virus Model"],"prefix":"10.1007","volume":"82","author":[{"given":"Cameron J.","family":"Browne","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuejun","family":"Pan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hongying","family":"Shu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0410-4643","authenticated-orcid":false,"given":"Xiang-Sheng","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,2,5]]},"reference":[{"issue":"1","key":"704_CR1","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.cam.2005.02.004","volume":"184","author":"B Adams","year":"2005","unstructured":"Adams B, Banks H, Davidian M, Kwon H-D, Tran H, Wynne S, Rosenberg E (2005) HIV dynamics: modeling, data analysis, and optimal treatment protocols. J Comput Appl Math 184(1):10\u201349","journal-title":"J Comput Appl Math"},{"issue":"3","key":"704_CR2","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1007\/s11538-006-9166-9","volume":"69","author":"N Baca\u00ebr","year":"2007","unstructured":"Baca\u00ebr N (2007) Approximation of the basic reproduction number $$R_0$$ for vector-borne diseases with a periodic vector population. Bull Math Biol 69(3):1067\u20131091","journal-title":"Bull Math Biol"},{"issue":"5","key":"704_CR3","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/s00285-008-0183-1","volume":"57","author":"N Baca\u00ebr","year":"2008","unstructured":"Baca\u00ebr N, Abdurahman X (2008) Resonance of the epidemic threshold in a periodic environment. J Math Biol 57(5):649","journal-title":"J Math Biol"},{"issue":"4","key":"704_CR4","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s00285-011-0479-4","volume":"65","author":"N Baca\u00ebr","year":"2012","unstructured":"Baca\u00ebr N, Dads EHA (2012) On the biological interpretation of a definition for the parameter $$R_0$$ in periodic population models. J Math Biol 65(4):601\u2013621","journal-title":"J Math Biol"},{"issue":"2","key":"704_CR5","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1016\/j.mbs.2007.07.005","volume":"210","author":"N Baca\u00ebr","year":"2007","unstructured":"Baca\u00ebr N, Ouifki R (2007) Growth rate and basic reproduction number for population models with a simple periodic factor. Math Biosci 210(2):647\u2013658","journal-title":"Math Biosci"},{"key":"704_CR6","doi-asserted-by":"crossref","first-page":"42765","DOI":"10.1038\/srep42765","volume":"7","author":"CA Beauchemin","year":"2017","unstructured":"Beauchemin CA, Miura T, Iwami S (2017) Duration of SHIV production by infected cells is not exponentially distributed: implications for estimates of infection parameters and antiviral efficacy. Sci Rep 7:42765","journal-title":"Sci Rep"},{"issue":"3","key":"704_CR7","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1007\/s11538-011-9677-x","volume":"74","author":"CJ Browne","year":"2012","unstructured":"Browne CJ, Pilyugin SS (2012) Periodic multidrug therapy in a within-host virus model. Bull Math Biol 74(3):562\u2013589","journal-title":"Bull Math Biol"},{"issue":"8","key":"704_CR8","first-page":"1999","volume":"18","author":"CJ Browne","year":"2013","unstructured":"Browne CJ, Pilyugin SS (2013) Global analysis of age-structured within-host virus model. Discrete Contin Dyn Syst Ser B 18(8):1999\u20132017","journal-title":"Discrete Contin Dyn Syst Ser B"},{"issue":"10","key":"704_CR9","doi-asserted-by":"crossref","first-page":"3315","DOI":"10.3934\/dcdsb.2016099","volume":"21","author":"CJ Browne","year":"2016","unstructured":"Browne CJ, Pilyugin SS (2016) Minimizing $$R_0$$ for in-host virus model with periodic combination antiviral therapy. Discrete Contin Dyn Syst Ser B 21(10):3315\u20133330","journal-title":"Discrete Contin Dyn Syst Ser B"},{"issue":"2","key":"704_CR10","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.jmaa.2011.07.006","volume":"385","author":"B Buonomo","year":"2012","unstructured":"Buonomo B, Vargas-De-Le\u00f3n C (2012) Global stability for an HIV-1 infection model including an eclipse stage of infected cells. J Math Anal Appl 385(2):709\u2013720","journal-title":"J Math Anal Appl"},{"key":"704_CR11","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/BF01190081","volume":"56","author":"L Burlando","year":"1991","unstructured":"Burlando L (1991) Monotonicity of spectral radius for positive operators on ordered Banach spaces. Arch Math 56:49\u201357","journal-title":"Arch Math"},{"issue":"1","key":"704_CR12","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/0022-247X(89)90287-4","volume":"137","author":"T Burton","year":"1989","unstructured":"Burton T, Hutson V (1989) Repellers in systems with infinite delay. J Math Anal Appl 137(1):240\u2013263","journal-title":"J Math Anal Appl"},{"issue":"1","key":"704_CR13","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/S0025-5564(00)00006-7","volume":"165","author":"RV Culshaw","year":"2000","unstructured":"Culshaw RV, Ruan S (2000) A delay-differential equation model of HIV infection of CD$$4^+$$ T-cells. Math Biosci 165(1):27\u201339","journal-title":"Math Biosci"},{"issue":"1","key":"704_CR14","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11538-008-9359-5","volume":"71","author":"P De Leenheer","year":"2009","unstructured":"De Leenheer P (2009) Within-host virus models with periodic antiviral therapy. Bull Math Biol 71(1):189\u2013210","journal-title":"Bull Math Biol"},{"key":"704_CR15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.jmaa.2007.05.011","volume":"338","author":"G Degla","year":"2008","unstructured":"Degla G (2008) An overview of semi-continuity results on the spectral radius and positivity. J Math Anal Appl 338:101\u2013110","journal-title":"J Math Anal Appl"},{"key":"704_CR16","doi-asserted-by":"crossref","DOI":"10.1142\/5999","volume-title":"Order structure and topological methods in nonlinear partial differential equations, vol. 1: maximum principles and applications, volume 2 of series in partial differential equations and applications","author":"Y Du","year":"2006","unstructured":"Du Y (2006) Order structure and topological methods in nonlinear partial differential equations, vol. 1: maximum principles and applications, volume 2 of series in partial differential equations and applications. World Scientific Publishing Co. Pte Ltd., Hackensack"},{"issue":"2","key":"704_CR17","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1137\/0520025","volume":"20","author":"JK Hale","year":"1989","unstructured":"Hale JK, Waltman P (1989) Persistence in infinite-dimensional systems. SIAM J Math Anal 20(2):388\u2013395","journal-title":"SIAM J Math Anal"},{"issue":"1","key":"704_CR18","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1137\/110826588","volume":"72","author":"G Huang","year":"2012","unstructured":"Huang G, Liu X, Takeuchi Y (2012) Lyapunov functions and global stability for age-structured HIV infection model. SIAM J Appl Math 72(1):25\u201338","journal-title":"SIAM J Appl Math"},{"issue":"2","key":"704_CR19","doi-asserted-by":"crossref","first-page":"e1005947","DOI":"10.1371\/journal.pcbi.1005947","volume":"14","author":"IA Neagu","year":"2018","unstructured":"Neagu IA, Olejarz J, Freeman M, Rosenbloom DI, Nowak MA, Hill AL (2018) Life cycle synchronization is a viral drug resistance mechanism. PLoS Comput Biol 14(2):e1005947","journal-title":"PLoS Comput Biol"},{"issue":"2","key":"704_CR20","doi-asserted-by":"crossref","first-page":"267","DOI":"10.3934\/mbe.2004.1.267","volume":"1","author":"PW Nelson","year":"2004","unstructured":"Nelson PW, Gilchrist MA, Coombs D, Hyman JM, Perelson AS (2004) An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells. Math Biosci Eng 1(2):267\u2013288","journal-title":"Math Biosci Eng"},{"issue":"1","key":"704_CR21","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0025-5564(02)00099-8","volume":"179","author":"PW Nelson","year":"2002","unstructured":"Nelson PW, Perelson AS (2002) Mathematical analysis of delay differential equation models of HIV-1 infection. Math Biosci 179(1):73\u201394","journal-title":"Math Biosci"},{"issue":"1","key":"704_CR22","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/S0036144598335107","volume":"41","author":"AS Perelson","year":"1999","unstructured":"Perelson AS, Nelson PW (1999) Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev 41(1):3\u201344","journal-title":"SIAM Rev"},{"issue":"5255","key":"704_CR23","doi-asserted-by":"crossref","first-page":"1582","DOI":"10.1126\/science.271.5255.1582","volume":"271","author":"AS Perelson","year":"1996","unstructured":"Perelson AS, Neumann AU, Markowitz M, Leonard JM, Ho DD (1996) HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271(5255):1582\u20131586","journal-title":"Science"},{"key":"704_CR24","first-page":"473","volume":"242","author":"D Posny","year":"2014","unstructured":"Posny D, Wang J (2014) Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments. Appl Math Comput 242:473\u2013490","journal-title":"Appl Math Comput"},{"key":"704_CR25","first-page":"1155","volume":"19","author":"C Rebelo","year":"2014","unstructured":"Rebelo C, Margheri A, Baca\u00ebr N (2014) Persistence in some periodic epidemic models with infection age or constant periods of infection. Discrete Contin Dyn Syst Ser B 19:1155\u20131170","journal-title":"Discrete Contin Dyn Syst Ser B"},{"issue":"3","key":"704_CR26","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1137\/060663945","volume":"67","author":"L Rong","year":"2007","unstructured":"Rong L, Feng Z, Perelson AS (2007) Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy. SIAM J Appl Math 67(3):731\u2013756","journal-title":"SIAM J Appl Math"},{"issue":"2","key":"704_CR27","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.jtbi.2009.06.011","volume":"260","author":"L Rong","year":"2009","unstructured":"Rong L, Perelson AS (2009) Modeling HIV persistence, the latent reservoir, and viral blips. J Theor Biol 260(2):308\u2013331","journal-title":"J Theor Biol"},{"key":"704_CR28","series-title":"Applied mathematical sciences","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-5977-0","volume-title":"Semidynamical systems in infinite-dimensional spaces","author":"SH Saperstone","year":"1981","unstructured":"Saperstone SH (1981) Semidynamical systems in infinite-dimensional spaces, vol 37. Applied mathematical sciences. Springer, New York"},{"issue":"7","key":"704_CR29","doi-asserted-by":"crossref","first-page":"762","DOI":"10.1038\/nm1777","volume":"14","author":"L Shen","year":"2008","unstructured":"Shen L, Peterson S, Sedaghat AR, McMahon MA, Callender M, Zhang H, Zhou Y, Pitt E, Anderson KS, Acosta EP et al (2008) Dose-response curve slope sets class-specific limits on inhibitory potential of anti-HIV drugs. Nat Med 14(7):762","journal-title":"Nat Med"},{"issue":"3","key":"704_CR30","doi-asserted-by":"crossref","first-page":"1280","DOI":"10.1137\/120896463","volume":"73","author":"H Shu","year":"2013","unstructured":"Shu H, Wang L, Watmough J (2013) Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses. SIAM J Appl Math 73(3):1280\u20131302","journal-title":"SIAM J Appl Math"},{"key":"704_CR31","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-7646-8","volume-title":"An introduction to delay differential equations with applications to the life sciences","author":"HL Smith","year":"2011","unstructured":"Smith HL (2011) An introduction to delay differential equations with applications to the life sciences, vol 57. Springer, New York"},{"key":"704_CR32","doi-asserted-by":"crossref","first-page":"6169","DOI":"10.1016\/S0362-546X(01)00678-2","volume":"47","author":"HL Smith","year":"2001","unstructured":"Smith HL, Zhao X-Q (2001) Global attractors and steady states for uniformly persistent dynamical systems. Nonlinear Anal 47:6169\u20136179","journal-title":"Nonlinear Anal"},{"issue":"5","key":"704_CR33","doi-asserted-by":"crossref","first-page":"1781","DOI":"10.1137\/16M1092003","volume":"77","author":"NK Vaidya","year":"2017","unstructured":"Vaidya NK, Rong L (2017) Modeling pharmacodynamics on HIV latent infection: choice of drugs is key to successful cure via early therapy. SIAM J Appl Math 77(5):1781\u20131804","journal-title":"SIAM J Appl Math"},{"issue":"3","key":"704_CR34","doi-asserted-by":"crossref","first-page":"569","DOI":"10.3934\/mbe.2018026","volume":"15","author":"J Wang","year":"2018","unstructured":"Wang J, Dong X (2018) Analysis of an HIV infection model incorporating latency age and infection age. Math Biosci Eng 15(3):569\u2013594","journal-title":"Math Biosci Eng"},{"issue":"6","key":"704_CR35","first-page":"1783","volume":"19","author":"X Wang","year":"2014","unstructured":"Wang X, Liu S, Rong L (2014) Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete Contin Dyn Syst Ser B 19(6):1783\u20131800","journal-title":"Discrete Contin Dyn Syst Ser B"},{"issue":"2","key":"704_CR36","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1007\/s11538-016-0145-5","volume":"78","author":"X Wang","year":"2016","unstructured":"Wang X, Song X, Tang S, Rong L (2016) Dynamics of an HIV model with multiple infection stages and treatment with different drug classes. Bull Math Biol 78(2):322\u2013349","journal-title":"Bull Math Biol"},{"issue":"3","key":"704_CR37","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s11538-013-9820-y","volume":"75","author":"Z Wang","year":"2013","unstructured":"Wang Z, Zhao X-Q (2013) A within-host virus model with periodic multidrug therapy. Bull Math Biol 75(3):543\u2013563","journal-title":"Bull Math Biol"},{"issue":"6510","key":"704_CR38","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1038\/373117a0","volume":"373","author":"X Wei","year":"1995","unstructured":"Wei X, Ghosh SK, Taylor ME, Johnson VA, Emini EA, Deutsch P, Lifson JD, Bonhoeffer S, Nowak MA, Hahn BH et al (1995) Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373(6510):117","journal-title":"Nature"},{"issue":"2","key":"704_CR39","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.jmaa.2005.02.062","volume":"311","author":"D Xu","year":"2005","unstructured":"Xu D, Zhao X-Q (2005) Dynamics in a periodic competitive model with stage structure. J Math Anal Appl 311(2):417\u2013438","journal-title":"J Math Anal Appl"},{"issue":"1","key":"704_CR40","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10884-015-9425-2","volume":"29","author":"X-Q Zhao","year":"2017","unstructured":"Zhao X-Q (2017) Basic reproduction ratios for periodic compartmental models with time delay. J Dyn Differ Equ 29(1):67\u201382","journal-title":"J Dyn Differ Equ"},{"key":"704_CR41","volume-title":"CMS books in mathematics\/Ouvrages de Math\u00e9matiques de la SMC","author":"X-Q Zhao","year":"2017","unstructured":"Zhao X-Q (2017) Dynamical systems in population biology. In: Sinclair N, Pimm D, Higginson W (eds) CMS books in mathematics\/Ouvrages de Math\u00e9matiques de la SMC, 2nd edn. Springer, Cham","edition":"2"},{"key":"704_CR42","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/0362-546X(94)90244-5","volume":"23","author":"X-Q Zhao","year":"1994","unstructured":"Zhao X-Q, Hutson V (1994) Permanence in Kolmogorov periodic predator-prey models with diffusion. Nonlinear Anal 23:651\u2013668","journal-title":"Nonlinear Anal"}],"container-title":["Bulletin of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-020-00704-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11538-020-00704-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11538-020-00704-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,4]],"date-time":"2021-02-04T00:53:22Z","timestamp":1612400002000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11538-020-00704-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2]]},"references-count":42,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,2]]}},"alternative-id":["704"],"URL":"https:\/\/doi.org\/10.1007\/s11538-020-00704-3","relation":{},"ISSN":["0092-8240","1522-9602"],"issn-type":[{"value":"0092-8240","type":"print"},{"value":"1522-9602","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2]]},"assertion":[{"value":"9 August 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 January 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 February 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"29"}}