On Dynamics For Several Epidemic Models

Infectious diseases threaten the human health and have an abominable influence on society.Preventing and restraining the spread of infectious diseases effectively is a big challenge in the field of infections.There is a long history in studying epidemic model with compartments.Take into account population reproduction,spread rule and social factors,the mathematical model is formulated.And the primary cause of diseases transmission is analyzed,which provides the reliable theoretical basis for control diseases.Therefore,modeling reasonably,understanding the spread of diseases and presenting controlling measures,are the currently main areas of infectious diseases dynamics.The purpose of this thesis is to establish the dynamic models of infectious diseases propagation and to analyze their dynamic behaviors.By considering the heterogeneity of the population with general age structure and relapse,we propose a multi-group SEIR model with distributed delays.Firstly,the non-negativity and uniform boundedness of the solution are demonstrated.We also obtain the basic reproduction number R₀.Secondly,the dynamic behaviors of the model are analyzed.By constructing the Lyapunov functional,we find that if the basic reproduction number R₀?1,the disease-free equilibrium is globally asymptotically stable;by using the graph theory of Lyapunov functional,we can find that when R₀>1,the disease equilibrium is globally asymptotically stable.Thirdly,we apply our main results to a single group SEIR epidemic model and investigate its dynamic behaviors.Finally,the numerical simulations are presented,which show that the heterogeneity and general incidence rate will not change the dynamic behaviors of the model.By considering the isolation and vaccination,a non-autonomous SEIQRS epidemic model with immunity and isolation is brought out.By comparison principle,we get the sufficient conditions of the extinction and strong persistence of the infectives I under the weak conditions.Furthermore,a relevant periodic model is simulated to verify our theories.We find that the isolation and vaccination have a good effect on the control diseases.By considering the latency and transient immunity,we set up a periodic SEIRS model with two delays in the periodic environment.Firstly,the general form of the basic reproduction ratio is identified by using the latest results of Zhao,which determines the local stability of the disease-free periodic solution.Secondly,by comparison principle,we show that if R₀<1,the disease-free periodic solution is globally attractive;by persistence theory,we prove that if R₀>1,the infectives are strong persistence and the disease periodic solution exists and is unique.