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International Journal of Differential Equations
Volume 2018 (2018), Article ID 6710575, 13 pages
Research Article

Mathematical Modeling of the Adaptive Immune Responses in the Early Stage of the HBV Infection

1Department of Mathematics, Faculty of Sciences and Technologies, Hassan II Mohammedia University, Mohammedia, Morocco
2Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, UAE

Correspondence should be addressed to Abdessamad Tridane

Received 29 August 2017; Revised 18 November 2017; Accepted 18 December 2017; Published 1 February 2018

Academic Editor: Ahmed M. Elaiw

Copyright © 2018 Karam Allali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The aim of this paper is to study the early stage of HBV infection and impact delay in the infection process on the adaptive immune response, which includes cytotoxic T-lymphocytes and antibodies. In this stage, the growth of the healthy hepatocyte cells is logistic while the growth of the infected ones is linear. To investigate the role of the treatment at this stage, we also consider two types of treatment: interferon- (IFN) and nucleoside analogues (NAs). To find the best strategy to use this treatment, an optimal control approach is developed to find the possibility of having a functional cure to HBV.