Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids
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Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids Kalyan Manna1 · Siddhartha P. Chakrabarty1
Received: 23 November 2014 / Revised: 20 February 2015 / Accepted: 10 May 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract This paper discusses the global dynamics of a model for chronic hepatitis B virus infection by incorporating delay(s) in the model. Two models, one with a one delay and the other with two delays, are analyzed for global properties in terms of the basic reproduction number. By constructing the appropriate Lyapunov functions, it is shown that the uninfected (infected) steady state is globally asymptotically stable when the basic reproduction number is less than or equal to (more than) one for both the one-delay and two-delay models. Finally, illustrative numerical simulations are presented. Keywords Hepatitis B · Delay model · Global stability · Lyapunov functions · Numerical simulation Mathematics Subject Classification
92B05
1 Introduction Hepatitis B virus (HBV) infection is a widely prevalent hepatic condition resulting from infection of the liver cells (Ciupe et al. 2007; Lewin et al. 2002). HBV is typically classified as acute or chronic, but potentially could lead to more serious long-term complications such as liver cirrhosis and hepatocellular carcinoma (HCC) (Ribeiro et al. 2002). The process of replication of HBV is fairly complex and has been elucidated in detail by Lewin et al. (2002) and Rebeiro et al. (2002). A prototype of the hepadnaviridae family of viruses, the HBV, is an enveloped virus which contains a 3.2 kilobase (kb) partially double-stranded (ds) open
Communicated by Eduardo Souza de Cursi.
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Siddhartha P. Chakrabarty [email protected] Kalyan Manna [email protected]
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Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, Assam, India
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K. Manna, S. P. Chakrabarty
circular DNA genome, that is enclosed in a nucleocapsid. The process of infectivity of the hepatocytes results in the DNA genome being transformed to covalently closed circular DNA (cccDNA). The cccDNA, with potentially multiple copies, is located in the nucleus of the infected cells and acts as a template for the production of mRNA and the consequent HBV transcription. The pre-genomic mRNA then undergoes reverse transcription into the DNA by the virally encoded polymerase which leads to the formation of HBV using the reverse transcriptase function. The persistence of virus for chronic HBV patients is contingent on the availability of the cccDNA copies, which is a major cause of post-therapeutic relapse. Several therapeutic protocols have been approved for HBV treatment by taking the life cycle of the HBV infection into account, particularly nucleoside analogs (NAs) which inhibits the process of reverse transcription (Dahari et al. 2009; Murray et al. 2006). A combination therapy of lamivudine (LMV) with interferon (IFN) alpha-2b or pegylated interferon (PE
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