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.

Summary This paper deals with an optimal control problem of a time‐delayed differential equation model that describes the interactions between hepatitis B virus (HBV) with HBV DNA‐containing capsids, liver cells (hepatocytes), and cytotoxic T‐lymphocyte immune response. Both the treatment and the intracellular delay are incorporated into the model. Furthermore, the existence of the optimal control pair is studied, and Pontryagin's minimum principle is used to characterize these 2 optimal controls. The first of them represents the efficiency of drug treatment in preventing new infections, whereas the second stands for the efficiency of drug treatment in inhibiting viral production. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are established to show the role of optimal therapy in controlling viral replication.

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In this paper, the global analysis of a HCV model with CTLs, antibody responses and therapy is studied. We incorporate into our model two treatments; the aim of the first one is to reduce the infected cells, while the second is to block the virus. We prove that the solutions with positive initial conditions are all positive and bounded. Moreover, we establish by using some appropriate Lyapunov functions that with the therapy the model becomes more stable than the one without treatment.

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