EQUILIBRIUMS AND STABILITY OF AN SVIR EPIDEMIC MODEL MD. SAIFUL ISLAM Department of Computer Science and Engineering, Jatiya Kabi Kazi Nazrul Islam University, Trishal, Mymensingh, Bangladesh ABSTRACT An epidemic model is a simplified means of describing the transmission of communicable disease through individuals. Compartmental model is one of the easiest way to analyzed communicable diseases. In this paper a nonlinear mathematical deterministic compartmental SVIR model for the dynamics of infectious disease including the role of a preventive vaccine is proposed and analyzed. The model has various kinds of parameter such as natural birth rate, natural death rate and dieses related death rate. Also incoming immigrants are considered in this model. A model for the transmission dynamics of an infectious disease has been presented and analyzed the stability of equilibrium points of this model. KEYWORDS: Endemic Equilibrium Basic Reproduction Number, Diseases Free Equilibrium, Infectious Diseases, Stability INTRODUCTION Mathematical modeling is one of the most important materials to analyze the characteristic of an infectious disease. One of the early triumphs of mathematical epidemiology was the formulation of a simple model by Kermack and McKendrick in 1927 [1]. The Kermack-McKendrick model is a compartmental model based on relatively simple assumptions on the rates of flow between different classes of members of the population [2]. Various kinds of deterministic models for the spread of infectious disease have been analyzed by mathematical modeling to control the epidemic. Epidemiological models have two kinds of equilibrium points. One of them is disease free equilibrium (DFE) at which the population remains in the absence of disease and other is endemic equilibrium [3]. There are two major types of control strategies available to curtail the spread of infectious diseases: pharmaceutical interventions (drugs, vaccines etc) and non-pharmaceutical interventions (social distancing, quarantine). Vaccination is important for the elimination of infectious disease in pharmaceutical interventions. Arino et al introduced vaccination of susceptible individuals into an SIRS model and also considered vaccinating a fraction of newborns [4]. Buonomo et al studied the traditional SIR model with 100% efficacious vaccine [5]. The epidemic models with vaccination have been investigated recently by some authors [6-12]. Effective vaccines have been used successfully to control smallpox, polio and measles. In this paper an SIR type disease has been considered when a vaccination program is in effect. MODEL FORMULATION Let us now consider an SIR type disease when a vaccination program is in effect and there is a constant flow of incoming immigrants. We define ) (t S , ) (t V , ) (t I , ) (t R and ) (t N be the number of susceptible, vaccinated, infective, recovered and total population respectively at time t . We model new infections using the simple mass-action law, so that in general there are SI α new infections in unit time when α is the rate of contact that is sufficient to transmit the disease. We also assume a constant recovery rate γ . The vaccine has the effect of reducing the susceptibility to infection by a BEST: International Journal of Humanities, Arts, Medicine and Sciences (BEST: IJHAMS) ISSN 2348-0521 Vol. 3, Issue 1, Jan 2015, 1-10 © BEST Journals