Study Of Epidemic Dynamical Models With Incubation Period On Complex Networks

Diseases have been a part of human life since the appearance of human society.They evolve along with the life of human beings and threaten them. In particular, infectious diseases bring human not only physical pain but also the loss of property and serious damage to the environment. It is necessary to study epidemic transmission by building mathematical models and then to make reasonable policy to prevent and control them. An individual in a society is always different from another in many ways, so it is more realistic significance to analyze epidemic spreading by using mathematical models on complex networks. Moreover, these methods can be generalized to analyze spreading behaviour of computer viruses and rumours. Since most infectious diseases have incubation period, a delay is introduced to the models in the thesis to denote the incubation period of disease, then the dynamical behaviour of infectious diseases on complex networks, the influence of network topologies on epidemic spreading and stability of equilibria are studied. The thesis is divided into the following three parts:1. Some epidemic diseases are transmitted not only by infective individuals, but also by infective vectors(e.g., mosquitoes, fleas). To study this spreading phenomena,a model on complex networks is presented and a delay is introduced to denote average incubation period of disease in a vector. The model in this thesis is closer to reality than the model without delay. The threshold of the model is obtained by using the existence of positive equilibrium. Epidemic threshold bears no relation to the delay, but the simulation shows the delay will effect the epidemic spreading. Finally, two major immunization strategies, i.e. uniform immunization and targeted immunization, are investigated and compared.2. An SIS model with vector and delay on heterogeneous networks is proposed in the thesis and the contacts between individuals and the vector are assumed to be heterogeneous. The epidemic threshold is calculated by limit system theory and some analytical methods, and it is find that the epidemic threshold will increase if the delay become bigger. Numerical simulations show that the epidemic threshold may change along with other factors, such as the infectivity function, the heterogeneity of networks, and the degrees of nodes.3. An SEIR and an SEIS models with birth and death on heterogeneous networks are built and a delay which denote average incubation period of disease in an individual is added. Theoretical computation shows that the incorporation of delay will change the value of the epidemic threshold, and numerical simulations state that the delay affects the spreading speed of the disease and the change of average infectious force. the stability of equilibria of SEIR model and the stability of disease-free equilibrium of SEIS model are proved by constructing reasonable Lyapunov functions and applying relevant knowledge of graph theory.