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. 2018 Nov 19;7(4):35.
doi: 10.3390/ht7040035.

Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids

Jaouad Danane  1 , Karam Allali  2
Affiliations

Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids

Jaouad Danane et al. High Throughput. .

Abstract

We model the transmission of the hepatitis B virus (HBV) by six differential equations that represent the reactions between HBV with DNA-containing capsids, the hepatocytes, the antibodies and the cytotoxic T-lymphocyte (CTL) cells. The intracellular delay and treatment are integrated into the model. The existence of the optimal control pair is supported and the characterization of this pair is given by the Pontryagin's minimum principle. Note that one of them describes the effectiveness of medical treatment in restraining viral production, while the second stands for the success of drug treatment in blocking new infections. Using the finite difference approximation, the optimality system is derived and solved numerically. Finally, the numerical simulations are illustrated in order to determine the role of optimal treatment in preventing viral replication.

Keywords: HBV virus; abaptive immune response; numerical simulation; treatment.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Figure 1
Figure 1
The evolution of the healthy cells (left) vs. time and a zoomed in region (right).
Figure 2
Figure 2
The evolution of the hepatitis B virus (HBV) infected cells vs. time.
Figure 3
Figure 3
The evolution of HBV capsids vs. time.
Figure 4
Figure 4
The free HBV virions as function of time.
Figure 5
Figure 5
The evolution of antibodies as function of time.
Figure 6
Figure 6
The behavior of the cytotoxic T-lymphocyte (CTL) immune response as function of time.
Figure 7
Figure 7
The optimal control η1 (left) and the optimal control η2 (right) versus time.
See this image and copyright information in PMC

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