Dynamical Model Of Infectious Diseases With The Incubation Period

This thesis studies several classes of epidemic dynamics models with incubation period and mainly focuses on the qualitative analysis for the SEIR infectious model.The dynamic behavior, such as the existence of the equilibrium point, the stability of the equilibrium point and system’s continued survivability, of infectious model is obtained by using the differential equation stability theory, functional differential equation theory and bifurcation theory. Particularly, Matlab software is applied to simulate the dynamic behavior of the models. We describe then in details as follows.The SEIR epidemic model with saturated incidence rate and with saturated recovery rate is considered. We obtain the reproduction number that determines disease destroy and disease survive continuously and the existing threshold conditions of all kinds of the equilibrium point; locally asymptotic stability of the endemic equilibrium is proved by using the general center manifold theory; the sufficient conditions of global asymptotic stability of the endemic equilibrium is also obtained by using the second additive compound matrix. Finally, we carry on the numerical simulation for the stability of the system behavior. The results obtained improve the main results of reference [1].The SEIR epidemic model with treatment and with incubation period is considered. The reproduction number that determines disease destroy and disease survive continuously is given, the existing threshold conditions of all kinds of the equilibrium point are obtained; then we get the conditions for the stability of equilibrium points. Finally, we carry on the numerical simulation. The results of this part generalize the main results of [2].The SEIRS epidemic model with continuous vaccination, which is more general, is studied. We get the reproduction number that determines disease destroy and disease survive continuously, the disease-free equilibrium and endemic equilibrium of disease. The stability of the disease-free equilibrium and endemic equilibrium are determined by using Routh-Hurwitz matrix discriminant conditions and Lasalle invariant principle. Finally, uniform continuous survivability of model is proved and its numerical simulation is also carried out.