Study On Several Epidemic Dynamic Models

In this paper, we consider three kinds of epidemic models:First,we consider an epidemic model with waning immunity which corresponds to transitions from the SIS to the SIR frameworks.For the case with the population size dependent contact rate,we show the difference of the dynamic behavior between the existence and the absence of imput of infectious individuals,and the sufficient conditions of global asymptotical stability of the endemic equilibriums are obtained,respectively.Futher,for the models with the bilinear and standard incidence rate,the sufficient and necessary conditions of global stability of the endemic equilibriums are proved by means of constructing a suitable Liapunov function andapplying Bendixson-Dulac theorem,respectively.Secondly,we propose an eco-epidemiological system,namely,prey with general density-dependence,sound predator and infected predator. Both simple mass action and the standard incidence are analyzed in detail. Two threshold parameters are found which control the development of the disease and the growth of the predator population.By combining with the LaSalle invariant set theorem,the comparison theorem of differential equation and the theory of limit systems,the global stable results of the predator-extinct equilibrium and disease-free equilibriums are obtained. By applying uniform persistence of infinite-dimensional dynamical system and Hopf Bifurcation theorem,we get the permanence of system and that a small amplitute periodic solution appears for the case of the standard incidence.Finally,we first formulate and study the global dynamics of the avian flu model with quarantine-adjusted incidence that has infectious force in latent period and infected period.Secondly, an epidemic model of two un-interaction species which describes the human avian flu is proposed and investigated. The results not only embody the importance of strengthen quarantine work to the recruitment and treatment for infections, but also show that the hunting and isolation are help to the epidemic control.