{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:18:28Z","timestamp":1774437508142,"version":"3.50.1"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2018,11,28]],"date-time":"2018-11-28T00:00:00Z","timestamp":1543363200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2019,6]]},"DOI":"10.1007\/s12190-018-01226-x","type":"journal-article","created":{"date-parts":[[2018,11,28]],"date-time":"2018-11-28T04:04:19Z","timestamp":1543377859000},"page":"537-565","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Global stability and optimal control for a hepatitis B virus infection model with immune response and drug therapy"],"prefix":"10.1007","volume":"60","author":[{"given":"Pensiri","family":"Yosyingyong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1261-8214","authenticated-orcid":false,"given":"Ratchada","family":"Viriyapong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,11,28]]},"reference":[{"key":"1226_CR1","unstructured":"WHO (World Health Organiaztion): Hepatitis B fact sheet no. 204. The World Health Organisation, Geneva (2017). Retrieved January 2 (2017), from. http:\/\/www.who.int\/mediacentre\/factsheets\/fs204\/en\/"},{"key":"1226_CR2","doi-asserted-by":"publisher","first-page":"1573","DOI":"10.1155\/2008\/743690","volume":"38","author":"C Long","year":"2008","unstructured":"Long, C., Qi, H., Huang, S.H.: Mathematical modeling of cytotoxic lymphocyte-mediated immune responses to hepatitis B virus infection. J. Biomed. Biotechnol. 38, 1573\u20131585 (2008). https:\/\/doi.org\/10.1155\/2008\/743690","journal-title":"J. Biomed. Biotechnol."},{"key":"1226_CR3","doi-asserted-by":"publisher","first-page":"1754","DOI":"10.1136\/gutjnl-2011-301073","volume":"61","author":"A Bertoletti","year":"2012","unstructured":"Bertoletti, A., Ferrari, C.: Innate and adaptive immune responses in chronic hepatitis B virus infections: towards restoration of immune control of viral infection. Gut 61, 1754\u20131764 (2012). https:\/\/doi.org\/10.1136\/gutjnl-2011-301073","journal-title":"Gut"},{"key":"1226_CR4","doi-asserted-by":"publisher","first-page":"i6","DOI":"10.1136\/gutjnl-2012-302056","volume":"61","author":"M Dandri","year":"2012","unstructured":"Dandri, M., Locarnini, S.: New insight in the pathobiology of hepatitis B virus infection. Gut 61, i6\u2013i17 (2012). https:\/\/doi.org\/10.1136\/gutjnl-2012-302056","journal-title":"Gut"},{"issue":"11","key":"1226_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.3390\/v9110350","volume":"9","author":"A Goyal","year":"2017","unstructured":"Goyal, A., Ribeiro, R.M., Perelson, A.S.: The role of infected cell proliferation in the clearance of acute HBV infection in humans. Viruses 9(11), 1\u201317 (2017). https:\/\/doi.org\/10.3390\/v9110350","journal-title":"Viruses"},{"key":"1226_CR6","doi-asserted-by":"publisher","first-page":"189196","DOI":"10.2217\/17460794.1.2.189","volume":"1","author":"M Lannacone","year":"2006","unstructured":"Lannacone, M., Sitia, G., Guidotti, L.G.: Pathogenetic and antiviral immune responses against hepatitis B virus. Future Virol. 1, 189196 (2006). https:\/\/doi.org\/10.2217\/17460794.1.2.189","journal-title":"Future Virol."},{"key":"1226_CR7","doi-asserted-by":"publisher","first-page":"1778","DOI":"10.1053\/j.gastro.2018.01.034","volume":"154","author":"A Suslov","year":"2018","unstructured":"Suslov, A., Boldanova, T., Wang, X., Wieland, S., Heim, M.H.: Hepatitis B virus does not interfere with innate immune responses in the human liver. Gastroenterology 154, 1778\u20131790 (2018)","journal-title":"Gastroenterology"},{"key":"1226_CR8","doi-asserted-by":"publisher","first-page":"12398","DOI":"10.1073\/pnas.92.26.12398","volume":"92","author":"LV Tsui","year":"1995","unstructured":"Tsui, L.V., Guidotti, L.G., Ishikawa, T., Chisari, F.V.: Posttranscriptional clearance of hepatitis B virus RNA by cytotoxic T lymphocyte-activated hepatocytes. Proc. Natl. Acad. Sci. USA 92, 12398\u201312402 (1995)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"1226_CR9","doi-asserted-by":"publisher","first-page":"2536","DOI":"10.1016\/S1074-7613(00)80295-2","volume":"4","author":"LG Guidotti","year":"1996","unstructured":"Guidotti, L.G., Ishikawa, T., Hobbs, M.V., Matzke, B., Schreiber, R., Chisari, F.V.: Intracellular inactivation of the hepatitis B virus by cytotoxic T lymphocytes. Immunity 4, 2536 (1996)","journal-title":"Immunity"},{"key":"1226_CR10","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1126\/science.284.5415.825","volume":"284","author":"LG Guidotti","year":"1999","unstructured":"Guidotti, L.G., Rochford, R., Chung, J., Shapiro, M., Purcell, R., Chisari, F.V.: Viral clearance without destruction of infected cells during acute HBV infection. Science 284, 825\u2013829 (1999)","journal-title":"Science"},{"key":"1226_CR11","doi-asserted-by":"publisher","first-page":"287","DOI":"10.4049\/jimmunol.0902761","volume":"184","author":"S Phillips","year":"2010","unstructured":"Phillips, S., Chokshi, S., Riva, A., Evans, A., Williams, R., Naoumov, N.V.: CD8(+) T cell control of hepatitis B virus replication: direct comparison between cytolytic and noncytolytic functions. J. Immunol. 184, 287\u2013295 (2010). https:\/\/doi.org\/10.4049\/jimmunol.0902761","journal-title":"J. Immunol."},{"key":"1226_CR12","doi-asserted-by":"publisher","first-page":"1161811629","DOI":"10.3748\/wjg.v20.i33.11618","volume":"20","author":"RJ Pei","year":"2014","unstructured":"Pei, R.J., Chen, X.W., Lu, M.J.: Control of hepatitis B virus replication by interferons and toll-like receptor signaling pathways. World J. Gastroenterol. 20, 1161811629 (2014). https:\/\/doi.org\/10.3748\/wjg.v20.i33.11618","journal-title":"World J. Gastroenterol."},{"key":"1226_CR13","doi-asserted-by":"publisher","first-page":"8","DOI":"10.3390\/v9010018","volume":"9","author":"Y Xia","year":"2017","unstructured":"Xia, Y., Protzer, U.: Control of hepatitis B virus by cytokines. Viruses 9, 8 (2017). https:\/\/doi.org\/10.3390\/v9010018","journal-title":"Viruses"},{"key":"1226_CR14","doi-asserted-by":"publisher","first-page":"478","DOI":"10.1016\/S0952-7915(96)80034-3","volume":"8","author":"LG Guidotti","year":"1996","unstructured":"Guidotti, L.G., Chisari, F.V.: To kill or to cure: options in host defense against viral infection. Curr. Opin. Immunol. 8, 478\u2013483 (1996)","journal-title":"Curr. Opin. Immunol."},{"key":"1226_CR15","doi-asserted-by":"publisher","first-page":"1472","DOI":"10.1172\/JCI119308","volume":"99","author":"FV Chisari","year":"1997","unstructured":"Chisari, F.V.: Cytotoxic T cells and viral hepatitis. J. Clin. Invest. 99, 1472\u20131477 (1997). https:\/\/doi.org\/10.1172\/JCI119308","journal-title":"J. Clin. Invest."},{"key":"1226_CR16","doi-asserted-by":"publisher","first-page":"10304","DOI":"10.1128\/JVI.74.22.10304-10311.2000","volume":"74","author":"C Bartholdy","year":"2000","unstructured":"Bartholdy, C., Christensen, J.P., Wodarz, D., Thomsen, A.R.: Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in Gamma interferon-deficient mice infected with lymphocytic chroriomeningitis virus. J. Virol. 74, 10304\u201310311 (2000)","journal-title":"J. Virol."},{"key":"1226_CR17","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/S1471-4906(02)02189-0","volume":"23","author":"D Wodarz","year":"2002","unstructured":"Wodarz, D., Christensen, J.P., Thomsen, A.R.: The importance of lytic and nonlytic immune responses in viral infections. Trends Immunol. 23, 194\u2013200 (2002)","journal-title":"Trends Immunol."},{"issue":"5","key":"1226_CR18","doi-asserted-by":"publisher","first-page":"2247","DOI":"10.1128\/JVI.78.5.2247-2254.2004","volume":"78","author":"G Bocharov","year":"2004","unstructured":"Bocharov, G., Ludewig, B., Bertoletti, A., Klenerman, P., Junt, T., Krebs, P., Luzyanina, T., Fraser, G., Anderson, R.M.: Underwhelming the immune response: effect of slow virus growth on CD8+-T-lymphocytes responses. J. Virol. 78(5), 2247\u20132254 (2004)","journal-title":"J. Virol."},{"key":"1226_CR19","doi-asserted-by":"publisher","first-page":"1593","DOI":"10.1016\/j.camwa.2005.07.020","volume":"51","author":"K Wang","year":"2006","unstructured":"Wang, K., Wang, W., Liu, X.: Global stability in a viral infection model with lytic and nonlytic immune responses. Comput. Math. Appl. 51, 1593\u20131610 (2006). https:\/\/doi.org\/10.1016\/j.camwa.2005.07.020","journal-title":"Comput. Math. Appl."},{"key":"1226_CR20","doi-asserted-by":"publisher","first-page":"484","DOI":"10.3390\/v1030484","volume":"1","author":"P Lampertico","year":"2009","unstructured":"Lampertico, P., Aghemo, A., Vigan, M., Colombo, M.: HBV and HCV therapy. Viruses 1, 484\u2013509 (2009). https:\/\/doi.org\/10.3390\/v1030484","journal-title":"Viruses"},{"issue":"23","key":"1226_CR21","doi-asserted-by":"publisher","first-page":"2427","DOI":"10.4254\/wjh.v7.i23.2427","volume":"7","author":"S Hagiwara","year":"2015","unstructured":"Hagiwara, S., Nishida, N., Kudo, M.: Antiviral therapy for chronic hepatitis B: combination of nucleoside analogs and interferon. World J. Hepatol. 7(23), 2427\u20132431 (2015). https:\/\/doi.org\/10.4254\/wjh.v7.i23.2427","journal-title":"World J. Hepatol."},{"key":"1226_CR22","doi-asserted-by":"publisher","first-page":"800","DOI":"10.1056\/NEJMoa021812","volume":"348","author":"SJ Hadziyannis","year":"2003","unstructured":"Hadziyannis, S.J., Tassopoulos, N.C., Heathcote, E.J., Chang, T.T., Kitis, G., Rizzetto, M., Marcellin, P., Lim, S.G., Goodman, Z., Wulfsohn, M.S., et al.: Adefovir dipivoxil for the treatment of hepatitis Be antigenNegative chronic hepatitis B. N. Eng. J. Med. 348, 800\u2013807 (2003). https:\/\/doi.org\/10.1056\/NEJMoa021812","journal-title":"N. Eng. J. Med."},{"issue":"6","key":"1226_CR23","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.3390\/v2061279","volume":"2","author":"DC Erik","year":"2010","unstructured":"Erik, D.C., Geoffrey, F., Suzanne, K., Johan, N.: Antiviral treatment of chronic hepatitis B virus (HBV) infections. Viruses 2(6), 1279\u20131305 (2010)","journal-title":"Viruses"},{"key":"1226_CR24","doi-asserted-by":"publisher","first-page":"4398","DOI":"10.1073\/pnas.93.9.4398","volume":"93","author":"MA Nowak","year":"1996","unstructured":"Nowak, M.A., Bonhoeffer, S., Hill, A., Boehme, R., Thomas, H., McDade, H.: Viral dynamics in hepatitis B infection. Proc. Natl. Acad. Sci. USA 93, 4398\u20134402 (1996)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"1226_CR25","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1111\/imr.12686","volume":"285","author":"SM Ciupe","year":"2018","unstructured":"Ciupe, S.M.: Modeling the dynamics of hepatitis B infection, immunity, and drug therapy. Immunol. Rev. 285, 38\u201354 (2018)","journal-title":"Immunol. Rev."},{"key":"1226_CR26","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1126\/science.272.5258.74","volume":"272","author":"MA Nowak","year":"1996","unstructured":"Nowak, M.A., Bangham, C.R.M.: Population dynamics of immune responses to persistent viruses. Science 272, 74\u201379 (1996). https:\/\/doi.org\/10.1126\/science.272.5258.74","journal-title":"Science"},{"key":"1226_CR27","first-page":"2297","volume":"12","author":"S Koonprasert","year":"2016","unstructured":"Koonprasert, S., Moore, E.J., Banyatlersthaworn, S.: Sensitivity and stability analysis of hepatitis B virus model with non-cytolytic cure process and logistic hepatocyte growth. Glob. J. Pure Appl. Math. 12, 2297\u20132312 (2016)","journal-title":"Glob. J. Pure Appl. Math."},{"key":"1226_CR28","doi-asserted-by":"publisher","first-page":"5050","DOI":"10.1073\/pnas.0603626104","volume":"104","author":"SM Ciupe","year":"2007","unstructured":"Ciupe, S.M., Ribeiro, R.M., Nelson, P.W., Dusheiko, G., Perelson, A.S.: The role of cells refractory to productive infection in acute hepatitis B viral dynamics. Proc. Natl. Acad. Sci. USA 104, 5050\u20135055 (2007). https:\/\/doi.org\/10.1073\/pnas.0603626104","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"1226_CR29","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.jtbi.2007.02.017","volume":"247","author":"SM Ciupe","year":"2007","unstructured":"Ciupe, S.M., Ribeiro, R.M., Nelson, P.W.: Modeling the mechanisms of acute hepatitis B virus infection. J. Theor. Biol. 247, 23\u201335 (2007). https:\/\/doi.org\/10.1016\/j.jtbi.2007.02.017","journal-title":"J. Theor. Biol."},{"key":"1226_CR30","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1007\/s00285-009-0278-3","volume":"60","author":"S Hews","year":"2010","unstructured":"Hews, S., Eikenberry, S., Nagy, J.D., Kuang, Y.: Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth. J. Math. Biol. 60, 573\u2013590 (2010). https:\/\/doi.org\/10.1007\/s00285-009-0278-3","journal-title":"J. Math. Biol."},{"issue":"5","key":"1226_CR31","doi-asserted-by":"publisher","first-page":"933","DOI":"10.1007\/s00285-010-0397-x","volume":"63","author":"N Yous","year":"2011","unstructured":"Yous, N., Hattaf, K., Tridane, A.: Modeling the adaptive immune response in HBV infection. J. Math. Biol. 63(5), 933\u2013957 (2011). https:\/\/doi.org\/10.1007\/s00285-010-0397-x","journal-title":"J. Math. Biol."},{"key":"1226_CR32","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1016\/j.amc.2015.06.011","volume":"265","author":"AM Elaiw","year":"2015","unstructured":"Elaiw, A.M., Almuallem, N.A.: Global properties of delayed-HIV dynamics models with differential drug efficacy in co-circulating target cells. Appl. Math. Comput. 265, 1067\u20131089 (2015). https:\/\/doi.org\/10.1016\/j.amc.2015.06.011","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"1226_CR33","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1155\/2018\/6710575","volume":"4","author":"K Mboya","year":"2015","unstructured":"Mboya, K., Makinde, D.O., Massawe, E.S.: Cytotoxic cells and control strategies are effective in reducing the HBV infection through a mathematical modelling. Int. J. Prevent. Treat. 2015 4(3), 48\u201357 (2015). https:\/\/doi.org\/10.1155\/2018\/6710575","journal-title":"Int. J. Prevent. Treat. 2015"},{"key":"1226_CR34","unstructured":"Tridane, A., Hattaf, K., Yafia, R., Rihan, F.A.: Mathematical modeling of HBV with the antiviral therapy for the immunocompromised patients. Commun. Math. Biol. Neurosci. ISSN:2052\u20132541 (2016)"},{"key":"1226_CR35","doi-asserted-by":"publisher","first-page":"1012","DOI":"10.1053\/jhep.2001.28509","volume":"34","author":"S Lewin","year":"2001","unstructured":"Lewin, S., Ribeiro, R., Walters, T., Lau, G., Bowden, S., Locarnini, S., Perelson, A.: Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed. Hepatology 34, 1012\u20131020 (2001). https:\/\/doi.org\/10.1053\/jhep.2001.28509","journal-title":"Hepatology"},{"issue":"8","key":"1226_CR36","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1016\/S1286-4579(02)01603-9","volume":"4","author":"RM Ribeiro","year":"2002","unstructured":"Ribeiro, R.M., Lo, A., Perelson, A.S.: Dynamics of hepatitis B virus infection. Microbes Infect. 4(8), 829\u2013835 (2002)","journal-title":"Microbes Infect."},{"issue":"17\u201320","key":"1226_CR37","first-page":"949","volume":"3","author":"K Hattaf","year":"2009","unstructured":"Hattaf, K., Rachik, M., Saadi, S., Yousfi, N.: Optimal control of treatment in a basic virus infection model. Appl. Math. Sci. 3(17\u201320), 949\u2013958 (2009)","journal-title":"Appl. Math. Sci."},{"key":"1226_CR38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2013\/712829","volume":"2013","author":"AM Elaiw","year":"2013","unstructured":"Elaiw, A.M., Alghamdi, M.A., Aly, S.: Hepatitis B virus dynamics: modeling, analysis, and optimal treatment scheduling. Discrete Dyn. Nat. Soc. 2013, 1\u20139 (2013). https:\/\/doi.org\/10.1155\/2013\/712829","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"1226_CR39","doi-asserted-by":"publisher","first-page":"219","DOI":"10.3390\/app6080219","volume":"6","author":"JE Forde","year":"2016","unstructured":"Forde, J.E., Ciupe, S.M., Cintron-Arias, A., Lenhart, S.: Optimal control of drug therapy in a hepatitis B model. Appl. Sci. 6, 219 (2016). https:\/\/doi.org\/10.3390\/app6080219","journal-title":"Appl. Sci."},{"key":"1226_CR40","doi-asserted-by":"crossref","unstructured":"Allali, K., Meskaf, A., Tridane, A.: Mathematical modeling of the adaptive immune responses in the early stage of the HBV infection. Int. J. Differ. Equ. Article ID 6710575 (2018)","DOI":"10.1155\/2018\/6710575"},{"key":"1226_CR41","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1016\/j.jtbi.2018.03.025","volume":"447","author":"FF Chenar","year":"2108","unstructured":"Chenar, F.F., Kyrychko, Y.N., Blyuss, K.B.: Mathematical model of immune response to hepatitis B. J. Theor. Biol. 447, 98\u2013110 (2108)","journal-title":"J. Theor. Biol."},{"key":"1226_CR42","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","volume":"180","author":"P Driessche van den","year":"2002","unstructured":"van den Driessche, P., Watmough, J.: Reproductive numbers and sub-threshold endemic equilibria for compartment models of disease transmission. Math. Biosci. 180, 29\u201348 (2002). https:\/\/doi.org\/10.1016\/S0025-5564(02)00108-6","journal-title":"Math. Biosci."},{"key":"1226_CR43","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970432","volume-title":"The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics","author":"JP LaSalle","year":"1976","unstructured":"LaSalle, J.P.: The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1976)"},{"issue":"4","key":"1226_CR44","doi-asserted-by":"publisher","first-page":"10701083","DOI":"10.1137\/S0036141094266449","volume":"27","author":"MY Li","year":"1996","unstructured":"Li, M.Y., Muldowney, J.S.: A geometric approach to global-stability problems. SIAM J. Math. Anal. 27(4), 10701083 (1996). https:\/\/doi.org\/10.1137\/S0036141094266449","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"1226_CR45","doi-asserted-by":"publisher","first-page":"2739","DOI":"10.1006\/jdeq.1993.1097","volume":"106","author":"MY Li","year":"1993","unstructured":"Li, M.Y., Muldowney, J.S.: On Bendixsons criterion. J. Differ. Equ. 106(1), 2739 (1993). https:\/\/doi.org\/10.1006\/jdeq.1993.1097","journal-title":"J. Differ. Equ."},{"issue":"4","key":"1226_CR46","doi-asserted-by":"publisher","first-page":"583600","DOI":"10.1007\/BF02218848","volume":"6","author":"HI Freedman","year":"1994","unstructured":"Freedman, H.I., Ruan, S., Tang, M.: Uniform persistence and flows near a closed positively invariant set. J. Dyn. Differ. Equ. 6(4), 583600 (1994)","journal-title":"J. Dyn. Differ. Equ."},{"issue":"3","key":"1226_CR47","doi-asserted-by":"publisher","first-page":"42530","DOI":"10.1090\/S0002-9939-1986-0822433-4","volume":"96","author":"G Butler","year":"1986","unstructured":"Butler, G., Freedman, H.I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96(3), 42530 (1986)","journal-title":"Proc. Am. Math. Soc."},{"key":"1226_CR48","doi-asserted-by":"publisher","first-page":"903","DOI":"10.1016\/j.apm.2012.03.029","volume":"37","author":"MD Samsuzzoha","year":"2013","unstructured":"Samsuzzoha, M.D., Singh, M., Lucy, D.: Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza. Appl. Math. Model. 37, 903\u2013915 (2013). https:\/\/doi.org\/10.1016\/j.apm.2012.03.029","journal-title":"Appl. Math. Model."},{"issue":"5","key":"1226_CR49","doi-asserted-by":"publisher","first-page":"363","DOI":"10.11648\/j.acm.20150405.15","volume":"4","author":"FN Ngoteya","year":"2015","unstructured":"Ngoteya, F.N., Gyekye, Y.N.: Sensitivity analysis of parameters in a competition model. Appl. Comput. Math. 4(5), 363\u2013368 (2015). https:\/\/doi.org\/10.11648\/j.acm.20150405.15","journal-title":"Appl. Comput. Math."},{"issue":"7","key":"1226_CR50","doi-asserted-by":"publisher","first-page":"e1003730","DOI":"10.1371\/journal.pcbi.1003730","volume":"10","author":"SM Ciupe","year":"2014","unstructured":"Ciupe, S.M., Ribeiro, R.M., Perelson, A.S.: Antibody responses during hepatitis B viral infection. PLOS Comput. Biol. 10(7), e1003730 (2014). https:\/\/doi.org\/10.1371\/journal.pcbi.1003730","journal-title":"PLOS Comput. Biol."},{"key":"1226_CR51","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1001\/archinte.1961.03620030023003","volume":"107","author":"RA MacDonald","year":"1961","unstructured":"MacDonald, R.A.: Lifespan of liver cells. Autoradio-graphic study using tritiated thymidine in normal, cirrhotic, and partially hepatectomized rats. Arch. Intern. Med. 107, 335\u2013343 (1961)","journal-title":"Arch. Intern. Med."},{"key":"1226_CR52","first-page":"896","volume":"144","author":"MP Bralet","year":"1994","unstructured":"Bralet, M.P., Branchereau, S., Brechot, C., Ferry, N.: Cell lineage study in the liver using retroviral mediated gene transfer. Am. J. Pathol. 144, 896\u2013905 (1994)","journal-title":"Am. J. Pathol."},{"key":"1226_CR53","doi-asserted-by":"publisher","first-page":"283","DOI":"10.3934\/mbe.2009.6.283","volume":"6","author":"S Eikenberry","year":"2009","unstructured":"Eikenberry, S., Hews, S., Nagy, J.D., Kuang, Y.: The dynamics of a delay model of HBV infection with logistic hepatocyte growth. Math. Biosci. Eng. 6, 283\u2013299 (2009)","journal-title":"Math. Biosci. Eng."},{"key":"1226_CR54","doi-asserted-by":"publisher","first-page":"847","DOI":"10.1084\/jem.193.7.847","volume":"193","author":"SA Whalley","year":"2001","unstructured":"Whalley, S.A., Murray, J.M., Brown, D., Webster, G.J.M., Emery, V.C., Dusheiko, G.M., Perelson, A.S.: Kinetics of acute hepatitis B virus infection in humans. J. Exp. Med. 193, 847\u2013853 (2001)","journal-title":"J. Exp. Med."},{"key":"1226_CR55","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198504184.001.0001","volume-title":"Viral Dynamics","author":"MA Nowak","year":"2000","unstructured":"Nowak, M.A., May, R.M.: Viral Dynamics. Oxford University Press, Oxford (2000)"},{"key":"1226_CR56","doi-asserted-by":"publisher","first-page":"5460","DOI":"10.1126\/science.272.5270.1904","volume":"272","author":"R Ahmed","year":"1996","unstructured":"Ahmed, R., Gray, D.: Immunologycal memory and protective immunity. Understanding their relation. Science 272, 5460 (1996)","journal-title":"Science"},{"key":"1226_CR57","volume-title":"The Mathematical Theory of Optimal Processes","author":"LSV Pontryagin","year":"1986","unstructured":"Pontryagin, L.S.V., Boltyanskii, G.R., Gamkrelidze, V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Gordon and Breach Science, New York (1986)"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-018-01226-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s12190-018-01226-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-018-01226-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,12]],"date-time":"2024-07-12T20:31:47Z","timestamp":1720816307000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s12190-018-01226-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,28]]},"references-count":57,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2019,6]]}},"alternative-id":["1226"],"URL":"https:\/\/doi.org\/10.1007\/s12190-018-01226-x","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,11,28]]},"assertion":[{"value":"18 September 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 November 2018","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}