The Study Of Epidemiological Models With Nonlinear Incidence Rates

In this paper,we study four kinds of epidemic models with nonlinear incidence rates.First, we consider an SEIR epidemic model with nonlinear incidence rate and constant immigration, which includes susceptibles,exposeds and infectives. It has been shown that this model has only unique endemic equilibrium. The local asymptotical stable results of the epidemic equilibrium was proved by using the Hurwitz criterion and the global asymptotical stable results of its by means of the geometric approach developed by Li and Muldowney .Secondly, we study two kinds of SIQR epidemic models. The first one is the SIQR model with general form nonlinear saturated infectivity and constant inflows, which includes susceptibles,infectives and recovered, and the second one is the SIQR model with nonlinear incidence ratesβI~pS~q. For the first model, we reduce the four-dimensional model to a two-dimensional asymptotical autonomous system by means of a transformation of variables. Furthermore, we prove the global asymptotical stability of the epidemic equilibrium by means of Dulac's function and the theory of limit systems. For the second model, we analyse the stability of the equilibria by using the Hurwitz criterion, and obtained the existence of periodic solutions by Hopf bifurcation for some parameter values. Furthermore,using Dulac's function and the theory of limit systems,we prove the global asymptotical stability of the epidemic equilibrium when 0 < p≤1.Finally, we formulate a kind of SIRI model with the vertical transmission,general form nonlinear saturated infectivity and constant immigration which includes new susceptibles and infectives. It is also found that the system exists only one equilibrium. Using the Routh-Hurwitz criterion, we prove the local asymptotical stability of the epidemic equilibrium. For the important cases of mass action incidence and standard incidence,applying Bendixson-Dulac theorem, the existence of the periodic solutions of the three-dimensional system is excluded, thereby the global stability of the endemic equilibrium is proved provided the endemic equilibrium exists.