A. A. Momoh, M. O. Ibrahim, A. Tahir / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-august 2012, pp.2248-2250 2248 | P a g e Modeling the Effects of Detection and Treatment of Latent Hepatitis B Infection on Transmission Dynamics of Hepatitis B Disease * A. A. Momoh, ** M. O. Ibrahim and *A. Tahir * Department of Mathematics, Modibbo Adama University of Technology, Yola. Adamawa State. Nigeria. ** Department of Mathematics, Usmanu Danfodiyo University, Sokoto. Sokoto State. Nigeria Abstract In this paper, we proposed an SVEIR model to understand the effect of detection and treatment of Hepatitis B at latent stage on the transmission dynamics of hepatitis B disease. Mathematical analysis was carried out that completely determines the global dynamics of the model. The impact of detection and treatment of Hepatitis B at latent stage on the transmission dynamics are discussed through the stability analysis of the disease free equilibrium. Keywords: Hepatitis B Virus (HBV), Mathematical Model, Latent Hepatitis, Stability analysis 1. Introduction Hepatitis B virus is the most common cause of cirrhosis and hepatocellular carcinoma in the world today. Of approximately 2 billion people who have been infected with HBV worldwide, more than 350 million, or about 5% of the world’s population are chronic carriers, and with an annual incidence of more than 50 million (Khan et al 2007 & Patel et al 2004 ). Hepatitis B virus accounts for 500,000 to 1.2 million death per year. Compartmental mathematical models have been widely used to gain insight into the spread and control of emerging and re-emerging human disease dating back to the pioneering work of Bernoulli in 1760 and likes of Ross, Kermack and McKendrick and others (Anderson et al 1982, 1991). The study of infectious diseases has been transformed by the use of mathematical models to gain insight into the dynamics of epidemics, to identify potential public health interventions, and to assess their impact (McCluskey et al 2003). Mathematical models were useful in informing policy during the foot and mouth disease outbreak in the United Kingdom in 2001, during the Severe Acute Respiratory Syndrome (SARS) outbreak in 2003 and in recent planning of responses to potential smallpox or pandemic influenza outbreak. In this paper, we propose an SVEIR model to understand the effect of detection and treatment of hepatitis B at latent stage on the transmission dynamics of the disease. We solved the system of ordinary differential equation and obtain our disease free equilibrium. Stability analysis was carried out on the disease free equilibrium using Routh-Hurtwitz Theorem. The model derivation is given the next sections. 2. Model Formulation As with any modeling endeavor, various assumptions about the underlying biology must be made. At this stage, we wish to clearly state some assumptions.  We assume that throughout the duration of the vaccines efficacy, latent Hepatitis B is completely undetectable.  We also assume that the birth and mortality rate are equal.  We assume that the efficacy of the vaccines wanes out at the rate .  Individuals enters the population with a recruitment rate of  . The natural mortality rate is denoted by  . A proportion, , c of individuals entering the population are vaccinated as infants and hence enter the vaccinated class, V. the remaining proportion, 1 c  , enter the susceptible class, S. the efficacy of the vaccines used in the vaccines used in the vaccination wanes over time at the rate of  and result in the movement of individuals from the vaccinated class to the susceptive class. We use the frequency-dependent description of disease transmission, with transmission coefficient .  We assume that assume that, upon infection with hepatitis B, an individual can either progress quickly to active hepatitis B or develop a latent infection and progress slowly to active hepatitis B. we denote the classes of latently infected and infected individuals by E and I respectively. The probability of a random individual exhibiting fast progression to active hepatitis B is denoted by .  latent infections progress slowly to active disease at the rate .  We assume that individuals in the latent hepatitis class moves to removed class at the rate  when treated. We also assume that active hepatitis B disease clears at the rate .  Model Diagram