The Study For Several Epidemic Models With A Varying Total Population Size

According to the report of the World Health Organization, infective diseases are still the first killer for the human. The human are confronted with the menace of infective diseases for long-term. For the experimental methods are not permitted on studying infective diseases, the theoretic analysis and simulation technology are required for mechanism of epidemic, law of spread and tendency of the epidemic diseases. The dynamic models have been taken an important role in studying infective disease.In this paper, we construct four kinds of mathematical models in epidemiology and to analyze the asymptotic behavior of these models:Firstly, we study the global dynamics of an SEIR epidemic model with general contact rate that incorporates constant recruitment and have infectious force in the latent, infected and recovered period. By means of Liapunov function and LaSalle's invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. The local asymptotical stable results of the epidemic equilibrium is proved by Hurwitz criterion. Furthermore, sufficient conditions for the global asymptotical stable results are obtained by means of the compound matrix theory.Secondly, we consider an SEIR epidemic model with the general contact rate that susceptible and exposed individuals have constant recruitment. In the model, the latent, infected and recovered period are also infective. The local asymptotical stable results of the epidemic equilibrium is proved by Hurwitz criterion. Furthermore, using the compound matrix theory we obtain sufficient conditions for the global asymptotical stable of the unique endemic equilibrium .Finally, the combination of SIR and SIS model, SEIR and SEIS epidemic model are studied. The global asymptotical stable results of the disease-free equilibrium is discussed by means of Liapunov function and LaSalle's invariant set theorem. Moreover, using Dulac function and the compound matrix theory we obtained the global stability of the unique endemic equilibrium of the SIR and SIS model, SEIR and SEIS model, respectively.