Global Analysis for a Delay-Distributed SIR Epidemic Model

Abstract

One of most tools for describing incubation period in epidemiological dynamic models is distributed delays. In this paper, we propose a delay-distributed SIR epidemic model with a nonlinear incidence rate. Using Lyapunov theory and LaSalle's principle, we show global asymptotic stability of disease-free and endemic equilibria. This analysis extends and performs existing results. Setting problem The analysis of mathematical models describing spread and control of infectious disease is one of major area biology. Transmission of a disease is a dynamical process driven by interaction between susceptible and infective. The SIR infections disease models have been studied by many authors. The behavior of SIR models are greatly aected by way in which transmission between infected and susceptible individuals. It divided population being studied into three class labeled S, I and R, where S denotes number of individuals who are susceptible to disease, I denotes number of infectious individuals and R denotes number of individuals who had been infected and were removed from possibility of being infected again or spreading infection. One fundamental parameter governs spread of diseases, and is also related to long term behaviors and level of vaccination necessary for eradication. This parameter is called reproduction number, R0: R0 is dened by epidemiologists as the average number of secondary cases caused by an infectious individual in a totally susceptible population. When R0 > 1; disease can enter a totally susceptible population and number of cases will increase, whereas when R0 < 1; disease will always fail to spread. Therefore, in its simplest form R0 tells us whether a population is at risk from a given disease. Nowadays, results of many epidemiological research are presented in terms of basic reproduction number. The epidemicity of disease is closely related to stability of solutions of mathematical models. Given global asymptotic stability conditions of those equilibria is of utmost biological relevance. Much attention has been paid to analysis of stability of disease free equilibrium and endemic equilibrium of epidemic models. The fraction of papers that obtain global stability of these models is relatively few, especially models with time delays (2, 12, 15, 19).