Research On SEIR Tuberculosis Epidemic Model With Time Delay

In this thesis, we assume that the disease tuberculosis has infective force in infectedperiod and latent period, and several SEIR tuberculosis delayed models with differentincidence rates are established.To have different incidence rates of TB model of the global asymptotical stability of thedisease-free equilibrium and the global asymptotical stability of the endemic equilibrium anddisease of persistent by comprehensive application differential equation theorem. The basicreproductive number which determines whether the disease extinct or not is gotten.Firstly, the SEIR tuberculosis model with bilinear incidence is established. By using thefixed point theory, differential equation stability theory and LaSalle invariant set theory, weproved that the existence and the stability of the equilibrium and the persistence of system.The disease-free equilibrium is globally stable if the basic reproduction number is no morethan the unity, the disease will die out, and unstable if the basic reproductive number is greatthan the unity. Specially, a unique endemic equilibrium is locally stable and the disease ispermanent if the basic reproductive number is great than the unity.Secondly, the tuberculosis model with nonlinear incidence is investigated.By linearizing method and Hurwitz criterion and LaSalle invariant set theorem, we proved theglobally asymptotical stability of the disease-free equilibrium and the local asymptoticalstability of the endemic equilibrium, and the basic reproductive number depends on timedelay is obtained. Moreover, we discussed the persistence of disease.Finally, the SEIR TB epidemic model with vertical transmission and vaccination is investigated. Then, the unique-existence and stability of both the disease-free equilibrium andendemic equilibrium are studied respectively. By analyzing the formulae of the basicreproductive number, the reasonability of the prevention and control measures are discussed.Numerical simulations are provided to illustrate the theoretical results. The study in this thesis provides theoretical and quantitative basis with the control and prevention of TB.