Optimal Antiviral Treatment Strategies of HBV Infection Model with Logistic Hepatocyte Growth

This study considers an optimal therapy strategy for HBV infection by incorporating two controls laws into a previous hepatitis B viral infection model with logistic hepatocyte growth. Our goal is to maximize the number of healthy cells and to minimize the cost of the therapy. In this context, the existence of an optimal control is proved. The optimal control is obtained by solving the optimality system which was composed of three nonlinear ODEs with initial conditions and three nonlinear adjoint ODEs with transversality conditions. The results were analysed and interpreted numerically using MATLAB. 1. Introduction Hepatitis B virus (HBV) infection is a major global public-health problem. Of the approximately 2 billion people who have been infected worldwide, more than 350 million are chronic carriers of HBV. Approximately 15–40% of infected patients will develop cirrhosis, liver failure, or hepatocellular carcinoma (HCC) which is the fifth most frequent cancer, killing 300？000–500？000 people each year. The treatment of chronic infection may be necessary to reduce the risk of cirrhosis and liver cancer. Chronically, infected individuals with persistently elevated serum alanine aminotransferase, a marker of liver damage, and HBV DNA levels are candidates for therapy. Although none of the available drugs can clear the infection, they can stop the virus from replicating and prevent liver damage such as cirrhosis and liver cancer. Treatments include antiviral drugs such as lamivudine, adefovir, tenofovir, telbivudine, and entecavir and immune system modulators such as interferon alpha. However, some individuals are much more likely to respond than others; this might be because of the genotype of the infecting virus or the patient’s heredity. The treatment works by reducing the viral load (the amount of virus particles as measured in the blood), which in turn reduces viral replication in the liver. Mathematical modelling and analysis of hepatitis B infections have been explored extensively over the last decade. One of the earliest models was proposed by Nowak et al. [1] who modeled the infection of healthy hepatocytes by free virions as a mass action process. This makes the viral basic reproduction number dependent on the homeostatic liver size. Gourley et al. [2], Min et al. [3], Eikenberry et al. [4], Ciupe et al. [5, 6], and Hews et al. [7] all replace this mass-action process with a standard incidence function to increase the richness of the dynamics. Optimal control theory has found wide-ranging applications in biological and ecological problems [8]. In