Modeling And Dynamics Of Disease Spreading On Complex Networks

Although classic disease spread dynamic models make some modest success in predicting some particular disease behaviors, they are usually overly simplistic, and neglect certain im-portant aspects, which include, for example, multiple stages/groups, the number of contacts and other disease states. In this paper, we investigate some virus and epidemic spread models under complex network framework. We establish the global stability for multi-stage/group epi-demic models and the global dynamics for several spread models on complex networks. We also consider modeling problems for SIR disease spread on degree correlated networks. The paper is divided into five main chapters. The second chapter deals with multi-stage/group epidemic models on coupled networks. The third chapter contains several spread models on complex networks. The fourth chapter is concerned with edge-based network SIR disease modeling.In Chapter 2, first, we consider the existence and uniqueness and global stability of equilibria for a multi-stage waterborne disease model. Furthermore, we propose a general class of multi-stage cholera spread model. Under biologically motivated assumptions, the basic reproduction number is derived, and the global stability of equilibria is obtained by using global Lyapunov functions, Kirchhoff’s matrix tree theorem and LaSalle Invariance Principle. Next, we study the global dynamics of multi-group SEI animal disease models with indirect transmission. Under biologically motivated assumptions, the basic reproduction number is derived, and the global stability of the disease-free equilibrium is proved. On the other hand, since the weight matrix for weighted digraphs may be reducible, a new combinatorial identity, combined with global Lyapunov functions and Kirchhoff’s matrix tree theorem, is employed to show the global stability of the endemic equilibrium.In Chapter 3, by using the comparison principle for Ordinary Differential Equations and Kirchhoff’s matrix tree theorem in Digraph, we investigate the global dynamics of several spread models on complex networks. First, we obtain the global dynamics of a network water-borne diseases spread model with birth and death, and discuss the effect of various immuniza-tion strategies on disease spread. Second, we establish the global stability of a heterogeneous network disease model that incorporates balanced birth and death event and has infectious force both in the infected state and carrier state. When individual birth and death is ignored, the final epidemic size formula is obtained. The effect of various immunization strategies is also investigated and compared by numerical simulations. In the end, by utilizing Lyapunov functions and Kirchhoff’s matrix tree theorem, the global stability of the virose equilibrium for a network-based computer viruses spread model is discussed. Applying comparison principle, a new brief proof for the global stability of the virus-free equilibrium is provided.In Chapter 4, first, an edge-based SIR disease model on configuration networks and two node-based SIR disease models on degree correlated networks are revisited, and then two growing network models that will lead to the appearance of degree correlations are introduced. By employing continuous-time stochastic simulation algorithms, the predicted behavior of an edge-based and two node-based SIR models on correlated networks is compared with the ensemble average of 100 runs of stochastic simulations. The simulation results show that, on degree correlated networks, the edge-based SIR model that only utilizes the degree distribution agrees much better than the two node-based SIR models that also utilizes the degree correlation information with stochastic SIR simulations in many respects, including the initial exponential growth rate, the epidemic peak and the arrival time of the peak, which suggests that the Miller model developed for configuration model or uncorrelated networks may have wider applicability than it should have been.