{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T07:20:22Z","timestamp":1763018422208,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T00:00:00Z","timestamp":1636329600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002766","name":"Beijing University of Posts and Telecommunications","doi-asserted-by":"publisher","award":["71871233","9182015","2020XD-A01-1"],"award-info":[{"award-number":["71871233","9182015","2020XD-A01-1"]}],"id":[{"id":"10.13039\/501100002766","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008778","name":"University of Science and Technology Beijing","doi-asserted-by":"publisher","award":["JDZX2015299"],"award-info":[{"award-number":["JDZX2015299"]}],"id":[{"id":"10.13039\/501100008778","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, a Caputo fractional-order HCV Periodic immune response model with saturation incidence, cell-to-cell and drug control was proposed. We derive two different basic reproductive numbers and their relation with infection-free equilibrium and the immune-exhausted equilibrium. Moreover, there exists some symmetry in the relationship between the two equilibria and the basic reproduction numbers. We obtain the global stability of the infection-free equilibrium by using Lyapunov function and the local stability of the immune-exhausted equilibrium. The optimal control problem is also considered and two control strategies are given; one is for ITX5061 monotherapy, the other is for ITX5061 and DAAs combination therapy. Matlab numerical simulation shows that combination therapy has lower objective function value; therefore, it is worth trying to use combination therapy to treat HCV infection.<\/jats:p>","DOI":"10.3390\/sym13112121","type":"journal-article","created":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T22:08:41Z","timestamp":1636409321000},"page":"2121","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Optimal Control of a Cell-to-Cell Fractional-Order Model with Periodic Immune Response for HCV"],"prefix":"10.3390","volume":"13","author":[{"given":"Xue","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4488-5990","authenticated-orcid":false,"given":"Yongmei","family":"Su","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1000-717X","authenticated-orcid":false,"given":"Huijia","family":"Li","sequence":"additional","affiliation":[{"name":"School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100700, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinjian","family":"Zhuo","sequence":"additional","affiliation":[{"name":"School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100700, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2429","DOI":"10.1056\/NEJMcp1006613","article-title":"Clinical practice. Chronic hepatitis C infection","volume":"364","author":"Rosen","year":"2011","journal-title":"New Engl. J. Med."},{"key":"ref_2","unstructured":"(2021, September 12). World Health Organization WHO. Available online: https:\/\/www.who.int\/health-topics\/hepatitis#tab=tab_1."},{"key":"ref_3","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-\u03b1 Therapy","volume":"282","author":"Neumann","year":"1998","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"922","DOI":"10.1038\/nature03153","article-title":"Modelling how ribavirin improves interferon response rates in hepatitis C virus infection","volume":"432","author":"Dixit","year":"2004","journal-title":"Nature"},{"key":"ref_5","first-page":"4752","article-title":"Memory effects and macroscopic manifestation of randomness","volume":"61","author":"Stanislavsky","year":"2000","journal-title":"Phys. Review. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1051\/mmnp\/201712503","article-title":"Dynamics of Hepatitis C Virus Infection: Mathematical Modeling and Parameter Estimation","volume":"12","author":"Rihan","year":"2017","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4761","DOI":"10.1016\/j.aej.2021.03.057","article-title":"Fractional-order delay differential equations for the dynamics of hepatitis C virus infection with IFN-\u03b1 treatment","volume":"60","author":"Rihan","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_8","first-page":"1","article-title":"Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate","volume":"2017","author":"Boukhouima","year":"2017","journal-title":"Int. J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"113064","DOI":"10.1016\/j.cam.2020.113064","article-title":"Dynamics of fractional order HIV infection model with antibody and cytotoxic T-lymphocyte immune responses","volume":"382","author":"Tamilalagan","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"109826","DOI":"10.1016\/j.chaos.2020.109826","article-title":"Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control","volume":"138","author":"Naik","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"111333","DOI":"10.1016\/j.chaos.2021.111333","article-title":"Numerical technique based on the interpolation with Lagrange polynomials to analyze the fractional variable-order mathematical model of the hepatitis C with different types of virus genome","volume":"152","author":"Khader","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"673240","DOI":"10.1155\/2013\/673240","article-title":"Hepatitis C Virus Adaptation to T-Cell Immune Pressure","volume":"2013","author":"Plauzolles","year":"2013","journal-title":"Sci. World J."},{"key":"ref_13","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_14","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":"Wodarz","year":"2003","journal-title":"J. Gen. Virol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.chaos.2005.05.003","article-title":"Viral infection model with periodic lytic immune response","volume":"28","author":"Wang","year":"2006","journal-title":"Chaos, Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"78","DOI":"10.5539\/jmr.v8n5p78","article-title":"Dynamics of a Viral Infection Logistic Model with Delayed Nonlinear CTL Response and Periodic Immune Response","volume":"8","year":"2016","journal-title":"J. Math. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1016\/j.matcom.2010.04.029","article-title":"A viral infection model with periodic immune response and nonlinear CTL response","volume":"80","author":"Ji","year":"2010","journal-title":"Math. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1002\/hep.21959","article-title":"Hepatitis C virus cell-cell transmission in hepatoma cells in the presence of neutralizing antibodies","volume":"47","author":"Timpe","year":"2008","journal-title":"Hepatology"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Xiao, F., Fofana, I., Heydmann, L., Barth, H., Soulier, E., Habersetzer, F., Doff\u00f6el, M., Bukh, J., Patel, A.H., and Zeisel, M. (2014). Hepatitis C Virus Cell-Cell Transmission and Resistance to Direct-Acting Antiviral Agents. PLoS Pathog., 10.","DOI":"10.1371\/journal.ppat.1004128"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1016\/j.jmaa.2014.10.086","article-title":"Modeling cell-to-cell spread of HIV-1 with logistic target cell growth","volume":"426","author":"Lai","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1650038","DOI":"10.1142\/S1793524516500388","article-title":"Dynamical analysis of a class of hepatitis C virus infection models with application of optimal control","volume":"09","author":"Mojaver","year":"2016","journal-title":"Int. J. Biomath."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.cnsns.2016.02.007","article-title":"Modeling the intracellular pathogen-immune interaction with cure rate","volume":"38","author":"Dubey","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.cnsns.2018.02.010","article-title":"Threshold dynamics of HCV model with cell-to-cell transmission and a non-cytolytic cure in the presence of humoral immunity","volume":"61","author":"Pan","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1007\/s12190-020-01420-w","article-title":"Modeling the cell-to-cell transmission dynamics of viral infection under the exposure of non-cytolytic cure","volume":"65","author":"Dhar","year":"2020","journal-title":"J. Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1007\/s13226-020-0489-2","article-title":"Hopf Bifurcation and Stability Switches Induced by Humoral Immune Delay in Hepatitis C","volume":"51","author":"Pan","year":"2020","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"103826","DOI":"10.1016\/j.rinp.2021.103826","article-title":"Stochastic modeling of within host dynamics of HCV model under therapy","volume":"22","author":"Badshah","year":"2021","journal-title":"Results Phys."},{"key":"ref_27","unstructured":"Elaiw, A.M. (2019). Global stability of virus dynamics model with immune response, cellular infection and holling type-ii. J. Korean Soc. Ind. Appl. Math., 39\u201363."},{"key":"ref_28","first-page":"1","article-title":"Analysis and Simulation of a Fractional Order Optimal Control Model for HBV","volume":"2020","author":"Yang","year":"2020","journal-title":"J. Funct. Spaces Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1002\/lt.24349","article-title":"Effect of scavenger receptor class B type I antagonist ITX5061 in patients with hepatitis C virus infection undergoing liver transplantation","volume":"22","author":"Rowe","year":"2016","journal-title":"Liver Transplant."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/S1473-3099(15)70050-2","article-title":"Ledipasvir-sofosbuvir with or without ribavirin to treat patients with HCV genotype 1 infection and cirrhosis non-responsive to previous protease-inhibitor therapy: A randomised, double-blind, phase 2 trial (SIRIUS)","volume":"15","author":"Bronowicki","year":"2015","journal-title":"Lancet. Infect. Dis."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Chang, K.C., Tung, S.Y., Wei, K.L., Shen, C.H., Hsieh, Y.Y., Chen, W.M., Chen, Y.H., Chen, C.H., Yen, C.W., and Xu, H.W. (2021). Real-world efficacy and safety of pangenotypic direct-acting antivirals against hepatitis C virus infection in Taiwan. Sci. Rep., 11.","DOI":"10.1038\/s41598-021-93095-x"},{"key":"ref_32","unstructured":"Kilbas, A., Srivastava, H., and Trujillo, J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science & Technology."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.jmaa.2006.10.040","article-title":"Global existence theory and chaos control of fractional differential equations","volume":"332","author":"Lin","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1515\/fca-2017-0082","article-title":"Erratum: The mean value theorems and a Nagumo-type uniqueness theorem for Caputo\u2019s fractional calculus","volume":"20","author":"Diethelm","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_35","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":"Watmough","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_36","first-page":"7","article-title":"Stability, Boundedness, and Lagrange Stability of Fractional Differential Equations with Initial Time Difference","volume":"2014","author":"Yakar","year":"2014","journal-title":"Sci. World J."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.physa.2007.01.010","article-title":"On fractional order differential equations model for nonlocal epidemics","volume":"379","author":"Ahmed","year":"2007","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1109\/TCST.2011.2153203","article-title":"Optimal Control of a Fractional-Order HIV-Immune System With Memory","volume":"20","author":"Ding","year":"2012","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.cam.2004.08.016","article-title":"Mathematical models of immune effector responses to viral infections: Virus control versus the development of pathology","volume":"184","author":"Wodarz","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"109592","DOI":"10.1016\/j.chaos.2019.109592","article-title":"Dynamics of fractional-order delay differential model for tumor-immune system","volume":"132","author":"Rihan","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/j.jtbi.2007.03.006","article-title":"Modeling hepatitis C virus dynamics: Liver regeneration and critical drug efficacy","volume":"247 2","author":"Dahari","year":"2007","journal-title":"J. Theor. Biol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2121\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:27:33Z","timestamp":1760167653000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2121"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,8]]},"references-count":41,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112121"],"URL":"https:\/\/doi.org\/10.3390\/sym13112121","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,8]]}}}