Dynamic Analysis Of The Discrete SIS Epidemic Model

With the rampant invasion of epidemic,more and more epidemic models were investigated.Epidemic models were divided into discrete models and continuous models.The fact that the epidemiological data was usually collected in discrete time units,made the discrete models a natural choice to describe a discrete transmission.To solve the discrete epidemic model was of uppermost priority in the process of the research of epidemic models.The research of discrete epidemic models was mainly focused on the equilibrium,persistence,and bifurcation theory.Based on a large number of epidemic model research,the SIS epidemic model with exponential incidence were structured.The discrete SIS epidemic model with exponential incidence was based on warehouse theory.The total population were divided into susceptible and infectious compartments.The effective control measures was implemented to reduce the transmission of the disease,and the incidence of disease was regulated as exponential incidence.Because the susceptible compartments were infected by the previous infection,there was a close relationship between the incidence of the disease and the previous infected.Following the main research contents and research directions of the discrete SIS epidemic model,the main work and contents of this paper were :First of all,based on accurate analysis and reasonable assumptions,a discrete SIS epidemic model with exponential incidence was proposed.the stability of the equilibrium point and the persistence of the model were discussed.It was proved by Stability theory that the disease-free equilibrium point was globally asymptotically stable and that the positive equilibrium point was locally asymptotically stable,and numerical simulation were conducted to demonstrate the theoretical results and show the complexity of the model dynamics.Secondly,on the basis of the constructed model,a new discrete model with time delay was proposed.the existence and stability of the equilibrium points ofthe new model were analyzed,the results showed that the flip bifurcation appeared in the new discrete model under the condition of unstable positive equilibrium point,it was proved by the central manifold theory that the perodic-2solution flip bifurcation was stable,finally,the results were verified by numerical simulation.Thirdly,a discrete SIS epidemic model with saturated cure rate and exponential incidence was discussed.the existence of the disease-free equilibrium point and two positive equilibrium points were proved,and the disease-free equilibrium point of the model was globally asymptotically stable,because of the complexity of the calculations,the stability of the positive equilibrium point was proved by the numerical simulation method.The results showed that the backward bifurcation appeared when the positive equilibrium point of the model was unstable.Finally,the discrete SIS epidemic model with exponential incidence was improved.the existence and stability of the disease-free equilibrium point and positive equilibrium point of the new model were discussed.The results showed that the neimark-sacker bifurcation was generated when the positive equilibrium point of the new model was unstable,the results were verified by numerical simulation method at last.