The Quantitative And Qualitative Analysis Of A Tuberculosis Model

This paper uses the method of dynamics to establish two kinds of tuberculosis models withvertical transmission, and it mainly researches the case of vertical transmission in the latent class.We analyze the model in two aspects: the total population is constant or variational.Firstly, we formulated a kind of tuberculosis model whose total population is constant. Byusing the Lyapunov stability theory and LaSalle’s invariant set theory, we discuss the stability ofthe disease free equilibrium and the endemic equilibrium, which determined by the thresholdconditionR0. When the basic reproduction ratio is less than the unity, the disease-freeequilibrium is local asymptotically stable, and when the basic reproduction ratio is great thanunity, a unique endemic equilibrium exists and is local asymptotically stable. When the basicreproduction ratio is equal or less than unity, the disease-free equilibrium is globalasymptotically stable, and when the basic reproduction ratio is great than unity, the uniqueendemic equilibrium is global asymptotically stable.Than, we formulate a class of tuberculosis model with variational population size. The basicreproduction ratio is obtained, which determines the extinction of the disease or not. The stabilityof the disease free equilibrium and endemic equilibrium is discussed. The results providetheoretical and quantitative basis for the control and prevention of the disease.