Research Article Mathematical Modeling of Transmission Dynamics and Optimal Control of Vaccination and Treatment for Hepatitis B Virus Ali Vahidian Kamyad, 1 Reza Akbari, 2 Ali Akbar Heydari, 3 and Aghileh Heydari 4 1 Department of Mathematics Sciences, Ferdowsi University of Mashhad, Mashhad, Iran 2 Department of Mathematical Sciences, Payame Noor University, Iran 3 Research Center for Infection Control and Hand Hygiene, Mashhad University of Medical Sciences, Mashhad, Iran 4 Payame Noor University, Khorasan Razavi, Mashhad, Iran Correspondence should be addressed to Reza Akbari; r9reza@yahoo.com Received 30 December 2013; Accepted 26 February 2014; Published 9 April 2014 Academic Editor: Chris Bauch Copyright © 2014 Ali Vahidian Kamyad et al. is 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. Hepatitis B virus (HBV) infection is a worldwide public health problem. In this paper, we study the dynamics of hepatitis B virus (HBV) infection which can be controlled by vaccination as well as treatment. Initially we consider constant controls for both vaccination and treatment. In the constant controls case, by determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady-state solutions of the model. Next, we take the controls as time and formulate the appropriate optimal control problem and obtain the optimal control strategy to minimize both the number of infectious humans and the associated costs. Finally at the end numerical simulation results show that optimal combination of vaccination and treatment is the most effective way to control hepatitis B virus infection. 1. Introduction Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus. It is a major global health problem. It can cause chronic liver disease and chronic infection and puts people at high risk of death from cirrhosis of the liver and liver cancer [1]. Infections of hepatitis B occur only if the virus is able to enter the blood stream and reach the liver. Once in the liver, the virus reproduces and releases large numbers of new viruses into the blood stream [2]. is infection has two possible phases: (1) acute and (2) chronic. Acute hepatitis B infection lasts less than six months. If the disease is acute, your immune system is usually able to clear the virus from your body, and you should recover completely within a few months. Most people who acquire hepatitis B as adults have an acute infection. Chronic hepatitis B infection lasts six months or longer. Most infants infected with HBV at birth and many children infected between 1 and 6 years of age become chronically infected [1]. About two-thirds of people with chronic HBV infection are chronic carriers. ese people do not develop symptoms, even though they harbor the virus and can transmit it to other people. e remaining one-third develop active hepatitis, a disease of the liver that can be very serious. More than 240 million people have chronic liver infections. About 600 000 people die every year due to the acute or chronic consequences of hepatitis B [1]. Transmission of hepatitis B virus results from exposure to infectious blood or body fluids containing blood. Possible forms of transmission include sexual contact, blood trans- fusions and transfusion with other human blood products (horizontal transmission), and possibly from mother to child during childbirth (vertical transmission) [3]. e most important influence on the probability of developing carriage aſter infection is age [4]. Children less than 6 years of age who become infected with the hepatitis B virus are the most likely to develop chronic infections: 80–90% of infants infected during the first year of life develop chronic infections; 30–50% of children infected before the age of 6 years develop chronic infections. Hindawi Publishing Corporation Computational and Mathematical Methods in Medicine Volume 2014, Article ID 475451, 15 pages http://dx.doi.org/10.1155/2014/475451