Two Kinds Of Infectious Disease Model With Double Time-delays Hopf Branch

In this paper, the stability of equilibrium of two classes of double delayed infectious disease models are discussed. Infectious disease research has an important role in the social health, but the work of considering the double delay is less for the infectious disease and the discussion on the Hopf bifurcation of double delayed infectious disease models is a blank. So some work has been done in this article.For a kind of double delays and strong incidence rate of SIS model suggested in this paper, using the method of characteristic root, the stability of the equilibrium of endemic disease is analyzed. By regarding the delays of decubation and incubation as the bifurcation parameters in turn, the stability of systematic zero solution is discussed and we obtain the sufficient conditions of system existing Hopf bifurcation. Method:Firstly, making the incubation delay τ1=0and regarding the decubation delay of diseases τ2as the bifurcation parameter, by analyzing the nature of the characteristic roots of corresponding characteristic equation, we get that the system occur Hopf bifurcation phenomenon near the equilibrium of diseases when the delay change; Secondly, fixed τ2in the stable interval, the equilibrium of diseases will be stable asymptotically and form the endemic from a practical significance. When regarding the incubation delay τ1as the bifurcation parameter is discussed again, it is found that the stability of the equilibrium of diseases will change at the first critical value of τ1. Meanwhile, the system has a small amplitude of periodic solution, and the diseases coexist in the form of periodic oscillations. It illustrates that the equilibrium of systematic endemic disease change from stable to periodic oscillations under certain conditions after adding incubation and the diseases will happen periodically, this will be in favor of researching the disease control strategies in the cycle and has practical significance. In addition, by using the center manifold theorem and the normal form method, the formula to determine the bifurcation direction, the stability of bifurcation periodic solutions, and period are obtained. The results obtained fill the blank of double delayed SIS infectious disease models in the research direction of Hopf bifurcation.In this artical, a class of double delays and strong phase structure of SIR model is presented. Because the impact of coefficients of ec2on characteristic equation, the methods of solving this kind of problem proposed by E.Beretta, Y.Kuang are used. This chapter adopt the method of regarding the delays of mature period and decubation as the bifurcation parameters in turn to discuss the infectious disease model which has the double delays and coefficients of ec2for the first time, and the sufficient conditions of periodic oscillations of diseases are given. The results established fill the blank of double delays and strong phase structure of SIR infectious disease models in the research direction of Hopf bifurcation.Finally, it is the conclusion and prospect of this article.