Dynamic Analysis Of Several Infectious Disease Models

Infectious diseases are caused by a variety of pathogens.With the further spread of pathogens in the population,infectious diseases would be widespread.As small as individuals,it is dangerous to the health,living conditions and even life safety of individuals;As large as the society,it would have a significant impact on the economic and political development of the whole society.Since the mathematical model of smallpox was put forward in the 1760 s,researchers have been trying to build mathematical models based on epidemiological characteristics,and to study the principal elements controlling the development of diseases like infection and recovery rate,and make further judgments on the subsequent spread of the disease.So far,more and more mathematical models have been used to test the problems in infectious disease control,such as predicting the impact of vaccination strategy on common infections and determining the best control strategy for pandemic influenza.This paper is divided into three parts:Firstly,taking the classical infectious disease model as the basic framework,the dynamic behavior of a class of SIRS infectious disease model with nonlinear incidence and piecewise therapeutic function is constructed,which takes the limitation of medical resources as the main research object.Under certain conditions,it is found that the basic reproductive number 0 less than 1 is no longer a sufficient condition for the disease to die out due to the emergence of backward bifurcation of the system.The existence and stability of equilibria,the existence of forward and backward bifurcation,Hopf bifurcation and Bogdanov-Taken bifurcation are obtained,and then some numerical simulations are presented to illustrate the theoretical results.Secondly,based on the research work and results in the first part,the nonlinear recovery function is further introduced to focus on the role of basic medical resources(number of beds)in the dynamics of infectious diseases.The mathematical analysis results show that the parameters (9(The ratio of beds to population)in nonlinear recovery function can be used as a key parameter,which determines the dynamic behavior of the mathematical model.By analyzing the stability of the model,it is proved that in certain specific conditions,limit cycles,saddle-node bifurcation,pitchfork bifurcation and Bogdanov-Taken bifurcation could be found in this system.Thirdly,an SI-SEIR type avian influenza epidemic model with psychological effect,nonlinear recovery rate and saturation inhibition effect is formulated to study the transmission and control of avian influenza virus.On the premise of setting the basic reproductive number as the threshold,the existence and local stability of the equilibrium are obtained and the global stability of the equilibrium are further proved by means of geometric method and the construction of Lyapunov function and Dulac function.Theoretical analysis are carried out to show the role of the saturation inhibition effect,psychological effect and effective medical resources in this model,and numerical simulations are also given to verify the results.