Behavior of a Discrete SIR Epidemic Model

I. Introduction Infections and infectious diseases are a great burden on many societies, including the countries. An epidemic may be described as a sudden outbreak of a disease that infects a substantial portion of the population in a region before it disappears. In the nineteenth century, recurrent waves of cholera killed millions in India. The influenza epidemic of 1918-1919 killed at least 20 million people overall, more than half a million in the United States. Epidemics of infectious diseases have been documented throughout history. Mathematical models provide an explicit framework within which to develop and communicate an understanding of infectious disease transmission dynamics. The earliest account of mathematical modeling of spread of disease was carried out in 1766 by Daniel Bernoulli. The SIR model is a simple model, due to Kermack and McKendrick, of an epidemic of an infectious disease in a population, [3]. In the theoretical studies of epidemic dynamical models, there are two kinds of mathematical models: the continuous-time models described by differential equations, and the discrete-time models described by difference equations. Analysis of steady states of the model and the stability for the epidemic model is of a great importance as it can help our society and direct us to determine and forecast the development trend of infection. The solution results can be used to describe the spread characteristics of infectious diseases, predict the status of the infection and evaluate the efficiency of the control strategies.