On Synchronization And Epidemic Dynamics In Complex Networks

Complex network science provides a model and research platform for dynamical evo-lution of complex systems. In this thesis, two types of important dynamical processes in complex networks, including synchronization and epidemic spreading, have been studied. The main contents of the thesis are organized as follows:Chapter 2 studies the synchronization in delayed networks with multiple connections. For the same connection pattern between two layers of networks, we present efficient criteria for synchronization; for the different case, we give the sufficient condition for synchronization with small delays. For the general case, simulations show that the delays in networks can result in alternative synchronization phenomenon.In Chapter 3, the feedback synchronization of two systems is studied. We investigate the relations among the adaptive synchronization, linear feedback synchronization and the leading Lyapunov exponent of a unified chaotic system. We find that the constant determined by the adaptive synchronization system is equal to the leading Lyapunov exponent, which can be helpful to rapidly compute the leading Lyapunov exponent.From Chapter 4, we study the epidemic spreading in networks. Chapter 4 deals with the mean behavior of heterogenous infection rates in scale-free networks. The pro-posed epidemic spreading efficiency extends the constant infection rate in classical models. Through numerical simulations, we find a new phenomenon, that is, its threshold is larger than the epidemic threshold for the same network size, which shows that heterogeneous infection rates can inhabit epidemic outbreak.Chapter 5 focuses on the epidemic threshold in scale-free networks with competitive environment. Using heterogenous mean-field theory, we establish two-strain competitive models. We present theoretical analysis to determine the conditions of epidemic spread-ing, which shows that there exist two different types of epidemic threshold, i.e., one is caused by the network, the other is caused by another strain. Furthermore, we build the corresponding stochastic model, and perform numerical simulations by Monte Carlo method to verify the theoretical results. By using the mean-field approach in heteroge-neous populations and the eigenvalue analysis, Chapter 6 investigates strain dynamical behaviors, including strain coexistence, strain replacement and strain extinction, and ob-tains the corresponding threshold conditions. Theoretical and simulation analysis shows that the impact of the mutation mechanism on epidemic dynamics is stronger than the super-infection mechanism to change the original competing state, which is relevant to application of the epidemic control strategy.Chapter 7 studies impacts of awareness on epidemics. We study three types of aware-ness, including individual awareness, local awareness and global awareness. We find dif-ferent effects of these awareness on epidemic dynamics for both continuous and discrete models.Chapter 8 discusses the global properties of a discrete SIS model in heterogenous networks, including the global stability of the disease-free equilibrium and the global at-tractiveness of the endemic state.Finally, in Chapter 9, we summarize our work and propose some problems for further study in the future.