Stability Analysis Of A Dynamical Model For Transmission Of Tuberculosis

Tuberculosis is an infectious disease caused by Mycobacterium tuberculosis. Tuberculosis’ infectivity is very strong, millions of people died of the illness every year in the world. Once infected with tuberculosis, oneself will endure pain and suffering in a long time, even lose their lives. So it has very important practical meaning to study the pathogenesis and the spread of tuberculosis, and to control the spread of tuberculosis.The mathematical model provides an effective tool for the spread of tuberculosis, and for assessment of medical intervention results. In this paper, we deal with the stability analysis of a dynamical model for the spread of tuberculosis with a general transmission rate. We arrive the basic reproduction number.Through Lyapunov functions we prove that when the basic reproduction ratio is lesser than one, the disease-free equilibrium is globally asymptotically stable. When the basic reproduction ratio is greater than one, we can also easily prove that the unique endemic equilibrium is globally asymptotically stable.The organizational structure of this paper is as follows:the first chapter is introduction, we introduce the medical background, as well as the main research achievements of tuberculosis transmission model. The second chapter is mathematical tools, we survey the basic knowledge of differential equation that related to this paper. The third chapter, we build a typical tuberculosis transmission SEI model. Then we arrive the disease-free equilibrium point, the basic reproduction number of the system, and proved that the disease-free equilibrium is globally stable; the unique endemic equilibrium is globally asymptotically stable under certain conditions. The fourth chapter is the improvement of the above model, we build a new model and analysis this model by using the similar method. The fifth chapter is a numerical simulation. Finally, the sixth chapter is the summary of the article.