{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,24]],"date-time":"2026-10-24T22:10:25Z","timestamp":1792879825737,"version":"4.5.0"},"reference-count":46,"publisher":"Public Library of Science (PLoS)","issue":"10","license":[{"start":{"date-parts":[[2021,10,12]],"date-time":"2021-10-12T00:00:00Z","timestamp":1633996800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS ONE"],"abstract":"<jats:p>In this paper, a new mathematical model is formulated that describes the interaction between uninfected cells, infected cells, viruses, intracellular viral RNA, Cytotoxic T-lymphocytes (CTLs), and antibodies. Hence, the model contains certain biological relations that are thought to be key factors driving this interaction which allow us to obtain precise logical conclusions. Therefore, it improves our perception, that would otherwise not be possible, to comprehend the pathogenesis, to interpret clinical data, to control treatment, and to suggest new relations. This model can be used to study viral dynamics in patients for a wide range of infectious diseases like HIV, HPV, HBV, HCV, and Covid-19. Though, analysis of a new multiscale HCV model incorporating the immune system response is considered in detail, the analysis and results can be applied for all other viruses. The model utilizes a transformed multiscale model in the form of ordinary differential equations (ODE) and incorporates into it the interaction of the immune system. The role of CTLs and the role of antibody responses are investigated. The positivity of the solutions is proven, the basic reproduction number is obtained, and the equilibrium points are specified. The stability at the equilibrium points is analyzed based on the Lyapunov invariance principle. By using appropriate Lyapunov functions, the uninfected equilibrium point is proven to be globally asymptotically stable when the reproduction number is less than one and unstable otherwise. Global stability of the infected equilibrium points is considered, and it has been found that each equilibrium point has a specific domain of stability. Stability regions could be overlapped and a bistable equilibria could be found, which means the coexistence of two stable equilibrium points. Hence, the solution converges to one of them depending on the initial conditions.<\/jats:p>","DOI":"10.1371\/journal.pone.0257975","type":"journal-article","created":{"date-parts":[[2021,10,13]],"date-time":"2021-10-13T01:48:31Z","timestamp":1634089711000},"page":"e0257975","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":16,"title":["Lyapunov function and global asymptotic stability for a new multiscale viral dynamics model incorporating the immune system response: Implemented upon HCV"],"prefix":"10.1371","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0616-1732","authenticated-orcid":true,"given":"Hesham A.","family":"Elkaranshawy","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5013-3157","authenticated-orcid":true,"given":"Hossam M.","family":"Ezzat","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nermeen N.","family":"Ibrahim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2021,10,12]]},"reference":[{"issue":"13","key":"pone.0257975.ref001","doi-asserted-by":"crossref","first-page":"589","DOI":"10.12998\/wjcc.v6.i13.589","article-title":"Update on global epidemiology of viral hepatitis and preventive strategies","volume":"6","author":"M Jefferies","year":"2018","journal-title":"World Journal of Clinical Cases"},{"key":"pone.0257975.ref002","volume-title":"Hepatitis: Fact sheets","year":"2019"},{"issue":"S3","key":"pone.0257975.ref003","doi-asserted-by":"crossref","DOI":"10.1002\/hep.510260715","article-title":"Therapy of hepatitis C: Meta-analysis of interferon alfa-2b trials","volume":"26","author":"RL Carithers","year":"1997","journal-title":"Hepatology"},{"key":"pone.0257975.ref004","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1056\/NEJM199811193392101","article-title":"Interferon alfa-2b alone or in combination with RIBAVIRIN as initial treatment for chronic Hepatitis C","volume":"339","author":"JG McHutchison","year":"1998","journal-title":"New England Journal of Medicine"},{"key":"pone.0257975.ref005","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1053\/gast.2002.33023","article-title":"Impact of pegylated interferon alfa-2b and ribavirin on liver fibrosis in patients with chronic hepatitis C","volume":"122","author":"T Poynard","year":"2002","journal-title":"Gastroenterology"},{"key":"pone.0257975.ref006","first-page":"125","article-title":"Potential treatment options and future research to increase hepatitis C virus treatment response rate. Hepatic Medicine","volume":"2","author":"S Uprichard","year":"2010","journal-title":"Evidence and Research"},{"key":"pone.0257975.ref007","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1111\/liv.14011","article-title":"Hepatitis C: Is eradication possible?","volume":"39","author":"A Lombardi","year":"2019","journal-title":"Liver International"},{"key":"pone.0257975.ref008","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1111\/liv.12063","article-title":"Best strategies for global HCV eradication","volume":"33","author":"LM Hagan","year":"2013","journal-title":"Liver International"},{"key":"pone.0257975.ref009","doi-asserted-by":"crossref","DOI":"10.1201\/9781420083583","volume-title":"Differential equations and mathematical biology","author":"DS Jones","year":"2009"},{"key":"pone.0257975.ref010","first-page":"7187602","article-title":"Mathematical modelling for the role of cd4+t cells in tumor-immune interactions","author":"AM Makhlouf","year":"2020","journal-title":"Computational and Mathematical Methods in Medicine"},{"issue":"9","key":"pone.0257975.ref011","doi-asserted-by":"crossref","first-page":"239","DOI":"10.3390\/v9090239","article-title":"Fighting cancer with mathematics and viruses","volume":"9","author":"D Santiago","year":"2017","journal-title":"Viruses"},{"issue":"13","key":"pone.0257975.ref012","doi-asserted-by":"crossref","first-page":"6542","DOI":"10.1073\/pnas.93.13.6542","article-title":"The dynamics of hepatitis b virus infection","volume":"93","author":"RJ Payne","year":"1996","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"5255","key":"pone.0257975.ref013","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":"AS Perelson","year":"1996","journal-title":"Science"},{"issue":"3","key":"pone.0257975.ref014","doi-asserted-by":"crossref","first-page":"2708","DOI":"10.3934\/mbe.2020148","article-title":"A mathematical model for the novel Coronavirus epidemic IN WUHAN, CHINA","volume":"17","author":"C Yang","year":"2020","journal-title":"Mathematical Biosciences and Engineering"},{"issue":"6227","key":"pone.0257975.ref015","doi-asserted-by":"crossref","DOI":"10.1126\/science.aaa4339","article-title":"Modeling infectious disease dynamics in the complex landscape of global health","volume":"347","author":"H Heesterbeek","year":"2015","journal-title":"Science"},{"issue":"5258","key":"pone.0257975.ref016","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":"MA Nowak","year":"1996","journal-title":"Science"},{"issue":"5386","key":"pone.0257975.ref017","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1126\/science.282.5386.103","article-title":"Hepatitis c viral dynamics in vivo and the antiviral efficacy of interferon- therapy","volume":"282","author":"AU Neumann","year":"1998","journal-title":"Science"},{"issue":"3","key":"pone.0257975.ref018","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1007\/s11901-011-0101-7","article-title":"Hepatitis C VIRAL Kinetics in the era of direct Acting Antiviral agents AND INTERLEUKIN-28B","volume":"10","author":"H Dahari","year":"2011","journal-title":"Current Hepatitis Reports"},{"issue":"6 Pt B","key":"pone.0257975.ref019","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.3851\/IMP2428","article-title":"Mathematical modelling of HCV infection: What can it teach us in the era of direct-acting antiviral agents?","volume":"17","author":"A Chatterjee","year":"2012","journal-title":"Antiviral Therapy"},{"issue":"1","key":"pone.0257975.ref020","doi-asserted-by":"crossref","first-page":"28","DOI":"10.3390\/v10010028","article-title":"Quantitative analysis of Hepatitis c NS5A viral protein dynamics on the Er surface","volume":"10","author":"M Knodel","year":"2018","journal-title":"Viruses"},{"issue":"4","key":"pone.0257975.ref021","doi-asserted-by":"crossref","first-page":"195","DOI":"10.3390\/v10040195","article-title":"A new model for the dynamics of hepatitis C infection: Derivation, analysis and implications","volume":"10","author":"P Aston","year":"2018","journal-title":"Viruses"},{"issue":"10","key":"pone.0257975.ref022","doi-asserted-by":"crossref","first-page":"282","DOI":"10.3390\/v9100282","article-title":"3D spatially resolved models of the intracellular dynamics of the hepatitis C genome replication cycle","volume":"9","author":"M Knodel","year":"2017","journal-title":"Viruses"},{"key":"pone.0257975.ref023","first-page":"14547393","article-title":"Innovative approximate analytical solution for standard model of VIRAL Dynamics: Hepatitis C With direct-acting agents as an implemented case","author":"HA Elkaranshawy","year":"2019","journal-title":"Mathematical Problems in Engineering"},{"issue":"3","key":"pone.0257975.ref024","doi-asserted-by":"crossref","DOI":"10.5539\/mas.v9n3p250","article-title":"The stability analyses of the mathematical models of hepatitis c virus infection","volume":"9","author":"MS Chong","year":"2015","journal-title":"Modern Applied Science"},{"issue":"6","key":"pone.0257975.ref025","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1007\/s00285-004-0291-5","article-title":"On the stability of a model of testosterone dynamics","volume":"49","author":"G Enciso","year":"2004","journal-title":"Journal of Mathematical Biology"},{"issue":"4","key":"pone.0257975.ref026","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1016\/j.bulm.2004.02.001","article-title":"Global properties of basic virus dynamics models","volume":"66","author":"A Korobeinikov","year":"2004","journal-title":"Bulletin of Mathematical Biology"},{"issue":"7","key":"pone.0257975.ref027","doi-asserted-by":"crossref","first-page":"1743","DOI":"10.1099\/vir.0.19118-0","article-title":"Hepatitis C VIRUS dynamics and pathology: The role of CTL and antibody responses","volume":"84","author":"D Wodarz","year":"2003","journal-title":"Journal of General Virology"},{"key":"pone.0257975.ref028","doi-asserted-by":"crossref","first-page":"3997","DOI":"10.12988\/ams.2015.54334","article-title":"Global analysis of a HCV model with CTL, antibody responses and therapy","volume":"9","author":"A Meskaf","year":"2015","journal-title":"Applied Mathematical Sciences"},{"issue":"3","key":"pone.0257975.ref029","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1002959","article-title":"Analysis of hepatitis C virus decline during treatment with the protease inhibitor danoprevir using a multiscale model","volume":"9","author":"L Rong","year":"2013","journal-title":"PLOS Computational Biology"},{"issue":"10","key":"pone.0257975.ref030","doi-asserted-by":"crossref","first-page":"3991","DOI":"10.1073\/pnas.1203110110","article-title":"Modeling shows that THE NS5A inhibitor daclatasvir has two modes of action and yields a shorter estimate of the hepatitis C virus half-life","volume":"110","author":"J Guedj","year":"2013","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"1","key":"pone.0257975.ref031","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.mbs.2013.04.012","article-title":"Mathematical analysis of multiscale models for hepatitis c virus dynamics under therapy with direct-acting antiviral agents","volume":"245","author":"L Rong","year":"2013","journal-title":"Mathematical Biosciences"},{"issue":"1","key":"pone.0257975.ref032","doi-asserted-by":"crossref","DOI":"10.1038\/s41598-017-09776-z","article-title":"The paradox of highly effective sofosbuvir-based combination therapy despite slow viral decline: Can we still rely on viral kinetics?","volume":"7","author":"TH Nguyen","year":"2017","journal-title":"Scientific Reports"},{"key":"pone.0257975.ref033","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.jtbi.2018.04.006","article-title":"A PDE multiscale model of hepatitis c virus infection can be transformed to a system of ODEs","volume":"448","author":"K Kitagawa","year":"2018","journal-title":"Journal of Theoretical Biology"},{"issue":"5","key":"pone.0257975.ref034","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1007\/s11538-018-00564-y","article-title":"Mathematical analysis of a transformed ode from a PDE multiscale model of hepatitis C virus infection","volume":"81","author":"K Kitagawa","year":"2019","journal-title":"Bulletin of Mathematical Biology"},{"key":"pone.0257975.ref035","doi-asserted-by":"crossref","unstructured":"Elkaranshawy HA, Ezzat HM, Ibrahim NN. Dynamical analysis of a multiscale model of hepatitis c virus infection using a transformed odes model. 42nd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC). 2020; 2451\u20132454.","DOI":"10.1109\/EMBC44109.2020.9176525"},{"key":"pone.0257975.ref036","volume-title":"Texts in Applied Mathematics","author":"M Martcheva","year":"2015"},{"issue":"4","key":"pone.0257975.ref037","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/BF00178324","article-title":"On the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations","volume":"28","author":"O Diekmann","year":"1990","journal-title":"Journal of Mathematical Biology"},{"key":"pone.0257975.ref038","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"P van den Driessche","year":"2002","journal-title":"Mathematical Biosciences"},{"key":"pone.0257975.ref039","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/978-1-4757-3667-0_13","article-title":"On the computation of R0 and its role on global stability","author":"C Castillo-Chavez","year":"2002","journal-title":"Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction"},{"issue":"1","key":"pone.0257975.ref040","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1137\/110826588","article-title":"Lyapunov functions and global stability For Age-structured HIV infection model","volume":"72","author":"G Huang","year":"2012","journal-title":"SIAM Journal on Applied Mathematics"},{"key":"pone.0257975.ref041","unstructured":"Hadi H. A mathematical model of hepatitis C virus infection incorporating immune response and cell proliferation. Master thesis, University of Texas at Arlington 2017."},{"key":"pone.0257975.ref042","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-68733-9","volume-title":"Killer cell dynamics","author":"D Wodarz","year":"2007"},{"key":"pone.0257975.ref043","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/j.amc.2014.05.015","article-title":"Global threshold dynamics in a five-dimensional virus model with cell-mediated, humoral immune responses and distributed delays","volume":"241","author":"J Wang","year":"2014","journal-title":"Applied Mathematics and Computation"},{"key":"pone.0257975.ref044","doi-asserted-by":"crossref","first-page":"647853","DOI":"10.1155\/2012\/647853","article-title":"Backward bifurcation of an epidemic model with infectious force in infected and immune period and treatment","author":"Y Xue","year":"2012","journal-title":"Abstract and Applied Analysis"},{"issue":"1","key":"pone.0257975.ref045","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1016\/j.jmaa.2015.09.048","article-title":"Bifurcations of a mathematical model for HIV dynamics","volume":"434","author":"J Luo","year":"2016","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"pone.0257975.ref046","unstructured":"Ramirez I. Mathematical Modeling of Immune Responses to Hepatitis C Virus Infection. Master thesis, East Tennessee State University 2014."}],"container-title":["PLOS ONE"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0257975","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T21:35:10Z","timestamp":1725917710000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0257975"}},"subtitle":[],"editor":[{"given":"Hussein Hassan","family":"Aly","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2021,10,12]]},"references-count":46,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10,12]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0257975","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,12]]}}}