Global Stability For Two Infectious Diseases Models

Although, the prevalence of many infectious diseases is prevented or controlledby the advancement of science and technology and the improvement of medicallevel, e?ectively. The infectious diseases are still harmful to the people's health,seriously. Every year millions of people die of infectious diseases. Recently, moreand more scholars pay attention to the study of di?erential equation for infectiousdiseases, and the study of stability has been one of eternal subjects in di?erentialequation. In this thesis, we firstly study stability of a waterborne pathogen modelwith latent period. Then, we consider the stability and uniformly persistent of amodel for competition between antibiotic-sensitive and antibiotic-resistant bacteriawith nutrient recycling and antibiotic treatment in the chemostat.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, we consider the stability of a waterborne pathogen model withlatent period. We compute the basic reproductive number (R0), then we indentifythe global stability of the disease-free equilibrium and the endemic equilibrium byconstructing Liapunov functions. Then, we consider the final outbreak size. Finally,some numerical simulations are also given to illustrate our analytic results.In Chapter 3, a model for competition between antibiotic-sensitive and antibiotic-resistant bacteria with nutrient recycling and antibiotic treatment in the chemostatis considered. The su?cient and necessary condition for boundedness of solutionand existence of non-negative equilibria are derived. The extinction and uniformpersistence of antibiotic-sensitive and antibiotic-resistant bacteria are also carriedout. Finally, some numerical simulations are also given to illustrate our analyticresults.