Tuberculosis Model With Koch Phenomenon

Tuberculosis(TB) is a chronic infectious disease, which is caused by tubercle bacillus. It is a serious threat to human’s life and health. The Koch phenomenon of tuberculosis transmission shows that the reactions of a body to the initial infection and reinfection with tubercle bacilli are different, that is, there are great differences in following aspects, such as the response way of immunity system, the distribution way of tubercle bacilli in host body, the possibility of processing active TB and the infectious ablility after infected. Koch phenomenon has a substantial effect on the regular pattern and mode of TB spread.In this thesis, tuberculosis dynamics models considering Koch phenomenon are formulated according to the spreading mechanism of tuberculosis. The effects of TB spread pattern have been studied with taking the different prevention and controlling measures. Firstly, the dynamical behaviors are analyzed in parameter spaces by using the theories and methods of the qualitative and stability of differential equation. The basic reproductive number expressions of established models are obtained. The sufficient and necessary conditions of the existence of the disease-free equilibrium and the endemic equilibria are found out. Furthermore, the bifurcation points of qualitative structure are exhibited. All the analytical results of models show that the Koch phenomenon plays an important role in the tuberculosis spread mode, that is, the disease may persist even if the basic reproductive number is less than unity and there exists three equilibria in the model if the basic reproductive number exceeds unity. The steady-state number of the infective cases depends on the initial size of infective number. Secondly, we give the examples for some parameter values. The results of numerical simulation not only test the analytical results and show the stability of each equilibrium, but also provide certain statistical basis in terms of the different prevention levals and controlling strategies against variable epidemics.