Stability Analysis And Optimal Control For A Epidemic Model

Epidemic dynamics is a kind of important theoretical quantitative research methods on epidemic. There have been a lot of research achievements on epidemic models. While most scholars assume that the population is changeless, and use differential system to describe the infectious disease model. However, in real life the population mobility is very big in a region. Birth rate, natural mortality and mortality due to illness, etc, have great influence on the total population. Meanwhile the spread and popular time of disease also will be affected deeply. Then assuming the population unchanged regardless of situation is not scientific and practical. In the early days of the outbreaks, without good treatment measures and vaccine for infectious disease, the best way to control infectious diseases is segregation. As a kind of effective method, isolation control has been widely used in infectious diseases. But theoretical research on this method is rare. Research results are obtained under controlling the disease toll without considering isolation control cost. Yet any control need to be paid, and sometimes we will control it excessively (For example, when no high intensity control disease can also quickly eliminate, we take high intensity control. Then this will cause waste of resources). Given these, this model considers the input and the output of population, birth rate, natural mortality and mortality due to illness, to improve SIR epidemic model, and establishes differential-algebraic system. This thesis analyzes the SIR epidemic model and uses the optimal control theory to give the optimal control strategy for controlling epidemic. The main contents are the following:1. Using the differential-algebraic equation this thesis establishes epidemic model with constant input, and analyzes the model. This thesis discusses the existence and stability of the system’s disease-free equilibrium and endemic equilibrium. We not only get a sufficient condition of the stability of the equilibrium point, but also find the basic reproductive number of our system.2. For the new outbreak infectious disease, usually due to the lack of effective vaccines and therapy, isolation is the most effective method to controlling epidemic. Based on SIR epidemic model, we isolate infective I, and Introduce the control variables of the isolation strategy. Then we discuss the application of isolation control in infectious disease and use extreme theory to give optimal control scheme. Moreover numerical simulation, we compare the optimal control scheme and constant control schemes. By this we verify the superiority of the optimal control. Also this thesis point out that in the initial days disease we should try to strengthen quarantine.