Global Stability For Two Kinds Of Tuberculosis Models

Tuberculosis, an infectious disease caused by Mycobacterium tuberculosis, isone of the leading causes of infectious mortality in our country. Almost half ofChinese are mycobacterium tuberculosis carrier. Models are presented to analyzethe spread mechanism of tuberculosis, and then, detailed qualitative analysis aboutglobal stability of the equilibria of the models are carried out.In Chapter 1, we mainly introduce the background and development situation ofthe subject, and give some theoretical tools and preliminaries serving the discussionin the paper.In Chapter 2, a tuberculosis model with various latent periods is presentedand studied. The basic reproduction number R₀ is obtained by the next generationmatrix. There is a disease-free equilibrium and it is globally asymptotically stablewhen R₀≤1. Then, if R₀ > 1, there is only one endemic equilibrium, which is alsoglobally asymptotically stable.In Chapter 3, a two-strain tuberculosis model with general contact rate whichallows tuberculosis patients with the drug sensitive of strain to be treated is pre-sented. The model includes both drug-sensitive and drug-resistant strains. Globalstability of disease-free equilibrium, boundary equilibrium and endemic equilibriumis studied. Analytical results of the model show that the quantities R₁ and R₂, whichrepresent the basic reproduction numbers of the sensitive and resistant strains, re-spectively, provide the threshold conditions which determine the competitive out-comes of the two strains. Numerical simulations are also conducted to con?rm andextend the analytic results.