Stability And Persistent Of Epidemic Models With Two Discrete Time Delays

Three types of epidemic models with two time delays were introduced in this article. They are SEIR epidemic model,SIRS epidemic model and SIRS epidemic model with pulse vaccination.The stability of system’s disease-free equilibrium and the endemic equilibrium were researched.And the permanence of disease was considered.Part one, an SEIR epidemic model with latent period and recovered period was researched.Frist of all,using the theory of the limit system,a four-dimensional system of model was reduced to a two-dimensional system.A threshold which determines the outcome of the disease and two equilibriums were found out.Secondly,according to the theory of function’s limit,the global stability of disease-free equilibrium was obtained.The local asymptotically stability of endemic equilibrium was also to be proved. Using the method of rebuttal evidence and the comparison principle of differential equations,the permanence of disease was analyzed.At last, according to the theory of the limit system,the stability of equilibrium which in original system was illustrated.Using the Matlab software, the numerical simulation was presented and the effect of contact rate was analyzed.Part two, an SIRS epidemic model with two time delays in healing period and immune expiration period was researched.The threshold conditions which determines the epidemic die out or not were found out,the free-equilibrium and endemic equilibrium were calculated.By constructing Liapunov function and using invariance principle of Lasalle, the global stability of free-equilibrium was proved.We obtain the local asymptotically stability of endemic equilibrium by calculating the eigenvalue of Jacobian matrix at the endemic equilibrium.Using the method of rebuttal evidence and the comparison principle of differential equations,the permanence of disease was proved. Immune expiration period and the recover period for the influence of epidemic model by numerical simulation were analyzed at last.Part three,an SIRS epidemic model with two time delays and impulsive vaccination was researched. First,the disease-free periodic solution was obtained by using stroboscopic map, and the global stability of disease-free periodic solution was proved by using impulsive differential inequality. Second, the permanence of disease was proved by using the comparison principle of pulse vaccination.At last, the effect of vaccination rates is analyzed by the numerical simulation.