Dynamic Analysis Of Two Classes Of Epidemic Models

Differential equation model has important application value in various fields,and has been widely used in the study of predation relationship and dynamics of network infectious diseases.In the natural environment,the number of population is not only related to the current state,but also to the previous state.And considering the in:fluence of the past on the present state,the delay differential equations play an important role in many fields.In this paper,by Lyapunov functional method,combining the comparative principle and LaSalle invariant principle,the dynamic behaviors of two classes of epidemic models that are predator-prey model with time delay and SIRS network model with vector transmission is studied.Specifieally,those include the positivity,the existence and stability of equilibria.This thesis is divided into three chapters.In Chapter one,the background of two models and the current research progress axe briefly introduced,and the main contents and working arrangements of this paper are also introduced.In Chapter two,the existing literature has shown that disease introduction into the predator group can destabilize the established prey-predator communities.Dealing with the epidemiological prey-predator is very important for us to understand the dynamical charac-teristics of population models.In this paper,we establish a new delayed SIS epidemiological prey-predator model with the assumptions that the disease is transmitted among the preda-tor species only and the infected predator consumes the prey according to Holling type ?functional response.The positivity of the solutions of the model,the existence of various equilibrium points,the stability and bifurcation are investigated at length.Using the in-cubation period as bifurcation parameter,we get the direction and stability of the Hopf bifurcation.Simulations are arranged to show the correctness and effectiveness of these theoretical resultsIn Chapter three,the classical epidemic models are modeled under the assumption by the law of mass action,but it is impossible in real life.Because individuals have differ-ent communicative abilities and contact ranges,and there are obvious differences among individuals,it is more realistic to construct networks to describe the transmission behav-ior of infectious diseases.Based on heterogeneous networks and considering that infectious diseases can be transmitted through vector contacting,a class of SIRS models with media transmission is established in this paper.The basic reproduction number R0 of the model is obtained by using regeneration matrix method.In addition,the existence and global stabili-ty of disease-free equilibrium are discussed while R0<1.On the contrary,the existence and uniqueness of positive equilibrium and its global attractiveness are analyzed while R0>1.Finall,numerical results demonstrate the correctness of the theoretical analysis results.