Stability And Hopf Bifurcation Of SIS Model With Ecological And Epidemic Sense With Double Time Delay

In this paper, it is studied that the stability of ecological epidemic SIS model with double delay and the properties of Hopf bifurcation.First of all, considering that predator-prey system has certain growth period and the recovered predators are still susceptible to some disease,so the recovery and the growth of a certain delay predator-prey system is added based on the ecological research of manifold disease SI model,which Jia - Fangzhang made in 2008, so as to set up ecological manifold disease SIS model with double delay; In addition,the equilibrium existence condition of the model is discussed, and character-states are analyzed by applying the characteristic root method and Hurwitz criterion.Second, delay of the system is discussed in six different conditions. First, when both of the two time-delay are zero,the positive equilibrium point of the system is stable; Second and third, the two time-delay can be transformed into a single time-delay system after setting one of the delay as zero; The fourth kind of circumstance is to make two time-delay equal, so that double time-delay can be turned into single time-delay; Based on the second and third kind of circumstances, the fifth and sixth situation is that a time-delay is taken as branch parameters by fixing another time-delay into stable area. Through analysis and research, we got the the sufficient condition for the existence of the Hopf bifurcation under the second, third, fourth, fifth, and sixth cases. This part is a further research of Jia-Fangzhang etc, and the results have generalized their conclusions. Its biological significance lies in that susceptible predators, infected predator predator-prey system will coexist in the form of periodic oscillation if the time-delay is controlled within the scope of certain. What is more, with the help of the center manifold theorem and specification and type theory, Hopf bifurcation direction, stability and changes the size of the cycle are discussed when the predator-prey system growth period is shorter than the predator pregnancy.Finally, MATLAB software is used to simulate the above conditions, so as to verify the validity of the main conclusion.