IOSR Journal of Mathematics (IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 2 (Sep-Oct. 2012), PP 25-31 www.iosrjournals.org www.iosrjournals.org 25 | Page Stability Analysis at DFE of an Epidemic Model in the Presence of a Preventive Vaccine Md. Saiful Islam 1 , Dr. Md. Asaduzzaman 2 , Dr. Md. Nazrul Islam Mondal 3 Abstract: Various kinds of deterministic models for the spread of infectious disease have been analyzed mathematically and applied to control the epidemic. A vaccine is a biological preparation that improves immunity to a particular disease. In this paper, a deterministic model for the dynamics of an infectious disease in the presence of a preventive vaccine and natural death rate is formulated. The model has various kinds of parameter. In this paper, we try to present a model for the transmission dynamics of an infectious disease In order to control the epidemic by changing the value of the parameters. AMS: 92C60, 92D30, 92B05, 92D25. Keywords: Basic reproduction number, diseases free equilibrium, Infectious diseases, Stability analysis. I. Introduction: The spread of communicable diseases is often described mathematically by compartmental models. In 1927, Kermack and McKendrick proposed, as a particular case of a more general model presented in their seminal work [1]. Many epidemiological models have a disease free equilibrium (DFE) at which the population remains in the absence of disease [2]. The classical SIR models are very important as conceptual models (similar to predator-prey and competing species models in ecology). The SIR epidemic modeling yields the useful concept of the threshold quantity which determines when an epidemic occurs and formulas for the peak infective fraction and the final susceptible fraction [3]. There are two major types of control strategies available to curtail the spread of infectious diseases: pharmaceutical interventions (drugs, vaccines) and non- pharmaceutical interventions (social distancing, quarantine). Vaccination, when it is available, is an effective preventive strategy. Arino et al. [4] introduced vaccination of susceptible individuals into an SIRS model and also considered vaccinating a fraction of newborns. Buonomo et al. [5] studied the traditional SIR model with 100% efficacious vaccine. Effective vaccines have been used successfully to control smallpox, polio and measles. In this paper, we try to present a model for the transmission dynamics of an infectious disease with a preventive vaccine. In order to control or eradicate the disease, we discus about various parameters used in this model except the natural death rater  (which is not controlled by human). II. Model formulation: In our model, we have divided the population into three compartments (susceptible, vaccinated susceptible and infectious) depending on the epidemiological status of individuals. We denote the population of those who are susceptible as S, who are vaccinated susceptible as V and those who subsequently infected as I. The model transfer diagram indicating the possible transitions between these compartments is shown in Fig 1. 1 Department of Computer Science and Engineering, Jatiya Kabi Kazi Nazrul Islam University, Trishal, Mymensingh-2220, Bangladesh. E-mail: saifulmath@yahoo.com. 2 Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh. E-mail: md_asaduzzaman@hotmail.com. 3 Department of Population Science and Human Resource Development, University of Rajshahi, Rajshahi-6205, Bangladesh, E-mail: nazrul_ru@yahoo.com. SI  S  S  VI c I  S V I V   Fig 1: Model Structure.