Stability And Delayed Feedback Stabilization Of The Infectious Disease Model In The Small-world Network

In recent years, networks became a subject of interest of scientists.Compared with the rules of network with random network, complex networks more closer to the real-world complex networks, especially the small-world networks and scale-free networks. Therefore, the model of infectious diseases basing on the complex network becomes more and more important. Compared with the rules of the network, the spread of the virus on complex network has a smaller threshold, the threshold on the scale-free network even disappeared,which reflected the complex network vulnerability. In this paper, we mainly study an improvement epidemic model on small-world networks, and discuss its dynamic control.Firstly we reviews the scientific theory of network development, network topology characterizing the basic parameters such as degree distribution, average path length and clustering coefficient, several important models such as regular network, random network,small world network and scale-free networks. Then we introduce the classical compartment model and Epidemic Model on small-world networks, and the corresponding results are briefly introduced.This paper mainly introduced an improved model of the spread of infectious diseases on the small-world networks, based on the original model adds removed crowd.To the low dimensional model we give those conclusions: The Threshold of whether the epidemic threshold, the stability of the disease-free equilibrium point and disease equilibrium point. Then we find that for low dimensional model, there is a threshold:If α=β/δ <1, the total amount of infection area is zero; If α=β/δ >1,the total amount of infection will regional infinity. However, the equilibrium point of the high-dimensional nonlinear model are no longer stable, so we consider to add delay and feedback mechanisms, and give the boundary conditions making the equilibrium point stability.Finally, we summarize the paper and present the further development direction in the future.