Study Of Epidemic Dynamics On Complex Networks

Infectious diseases are always a serious threat to human’s health and lives, soit is of great practical signifcance to study epidemic transmission and then to takeefective measures to prevent and control them. An important research method isepidemiology, with mathematical modeling as an analytical approach. The tradi-tional models, based on uniformly mixing populations, are unable to describe epi-demic propagation in large-scale social contact networks with distinct heterogeneity.Actually, the fact that most population-based epidemic spreads through physical in-teractions lays contact networks as practical grounds to model contagion dynamics.In the last decade, spurred by the availability of real data and the maturation ofnetwork theory, there has been a burst of research on network-based epidemic trans-mission. To better understand and control epidemics, in this doctoral thesis, basedon existing research works, we establish diferent epidemic models on complex net-works via mean-feld approximation, and do a deep and profound research on them,by means of epidemic dynamics, the theory of complex networks, and qualitativetheory and stability methods of ordinary diferential equations. The main researchresults are as follows:Firstly, we propose an epidemic SIS model with nonlinear infectivity on het-erogeneous networks, and analyze its dynamical behaviors. By constructing a seriesof comparison equations and iterative sequences, we study the global attractivity ofthe disease-free equilibrium and the endemic equilibrium. The method used in theproof is much more intuitional and concise than those used in previous work.Secondly, we generalize a theorem proposed by Lajmanovich and Yorke in1976, which can be used to prove the persistence of epidemic models with multi-compartments on heterogeneous networks. Then we establish an SIRS and an SIR(with birth and death of individuals) quantitative models on networks, and ana-lyze the epidemic threshold and the global behaviors. We fnd that the averageimmune period cannot infuence the threshold, but its increase will quickly decreasethe morbidity and the fnal infected density. Thirdly, we present a generalized epidemic system on heterogeneous networks,which includes many models as special cases, such as SIS, SIR, SEIS, SIRS andSEIRS models. We verify the global stability of the model in theoretical and nu-merical methods, and do sensitivity analysis in terms of model parameters. Weconclude that the basic reproduction number is in proportion to the heterogeneityof the network, the infected time have much more efects on the threshold than thelatent time, and the nodes with higher degree would be more likely to be infected.Fourthly, the network weights usually represent the intimacy between two con-nected nodes. As an epidemic propagates, people will take corresponding measuresto avoid being infected, so the weights will decrease as infected size increases, whichtrigger us to propose ‘adaptive weights’. We present an SIS model with a reason-able rule of birth and death, and analyze the efects of fxed and adaptive weightson epidemics. We see that the adaptivity of weights cannot change the threshold,but it will induce the disease to decay quickly and suppress the endemic to a lowlevel.Fifthly, we establish a new epidemic model on two separated contact networksof populations (human network and animal network) and cross-species (vectors),where vectors can interact with animals and humans respectively, thus transmitdiseases from one network to another. We compute the basic reproduction number,prove the global stability of the equilibria, and examine the infuences of variousinfection rates and contact patterns on the epidemic threshold and the fnal infectedsize, thus elicit an efective method to control diseases.Finally, we develop a framework for studying the epidemic dynamics on interde-pendent networks. We study its global dynamical behaviors and identify the efectsof contact patterns and infectivity on the epidemic threshold and the fnal epidemicsizes. We fnd that if one of the contact patterns is heterogeneous, the reproductionnumber keeps increasing as network size increases, and inner contacts usually playa more important role in epidemic spreading than cross contacts. In particular,we show that the interdependent nature of such a network strongly infuences thepropagation characteristics, which induces epidemics much easier to transmit on aninterdependent network than a single or a bipartite network.