Stability Problem For Coupled Systems Of Differential Equations On Networks

In this paper,stability problem for coupled systems of differential equations on networks are investigated.Many phenomena in the real world,such as biological and artificial neural networks,coupled systems of nonlinear oscillators on lattices,complex ecosystems and spread of infectious diseases in heterogeneous population,can be modeled by coupled systems of nonlinear differential equations on networks.Many processes in nature have memory effects.It means that the future depends not only on the present but also on the past.Fractional differential equation theory is an important tool to describe and simulate these processes.Although the stability analysis for coupled systems of integral differential equations on networks has made lots of important achievements,but similar study for coupled systems of fractional differential equations on networks is quite lacking.One of the main works of this paper is to study the stability of coupled systems of fractional differential equations on networks.In this part,using results from graph theory,we develop a systematic approach that allows one to construct global Lyapunov functions for fractional coupled systems from building blocks of individual vertex systems.The approach is applied to the study on several fractional population models and fractional epidemic models,and which is shown to be useful.Infectious disease model has been a hot research topic.Especially due to the development of complex network theory,the epidemic model based on complex network has been paid more and more attention recently.In the other part of this paper,we study the epidemic spreading of an SIRS model with population effects on networks.We find that dynamics of the network-based SIRS model is completely determined by a threshold0 R.If 0R(27)1,then the disease-free equilibrium is globally asymptotically stable and the disease dies out.Otherwise,if 0R(29)1,the disease-free equilibrium becomes unstable and in the meantime there exists uniquely an endemic equilibrium which is globally asymptotically stable and means that the disease will outbreak.Numerical simulations are given to illustrate the main theoretical results.Our research uncovered the relationship between the threshold 0R and the transmission ability of epidemic disease and the topological structure of the network.