| This thesis aims to establish and analyze the epidemic models with heterogeneity from the viewpoint of the differences between population and individual. In terms of population heterogeneity, two kinds of models are established:a travel-related epidemic model with spatial heterogeneity and a multiple parasites model with virulent heterogeneity. In terms of individual age-related difference, two models are proposed: an SEIR epidemic model with latent age and relapse age and a heroin epidemic model with susceptible age and relapse age. Based on the theories and methods related to dynamical systems, stability theory and functional differential equations, systematic analysis on global dynamics for these epidemic models are done to obtain their global dynamics respectively. Those results can provide a theoretical basis and guidance for making related strategies on the control of infectious diseases and heroin. The main contents and results are as follows.In chapter 1, the research background and significance of several epidemic models to be established are introduced. Main research methods on epidemic models are provided. Meanwhile the major of this paper as well as some preliminaries are briefly stated.In chapter 2, a travel-related epidemic model with spatial heterogeneity and general incidence rate is established and studied. We obtain three main relevant results. Firstly, by applying the theory of limiting systems, we obtain the corresponding limiting system of original model. It indicates that the dynamical behavior of original model is asymptotically the same as that of the limiting system. Secondly, by using the next generation matrix method, we get the basic reproduction number of the limiting system and establish its global dynamics based on analysis. Lastly, by making numerical simulations via Matlab software, we present the effects of spatial heterogeneity, general incidence rate and travel rate on the epidemic dynamical behaviors, which demonstrate that spatial heterogeneity and general incidence rate affect the final states of the system and travel-related infection intensifies the spread of diseases. All these results suggest that avoiding crowd and travel can effectively control the spread of diseases.In chapter 3, a multiple parasites model with virulent heterogeneity is studied. We investigate the effects of superinfection of parasite strains on the dynamical behaviors and obtain two main relevant results. Firstly, by applying Hurwitz criterion and the theory of Lyapunov stability theory, we analyze the dynamical behaviors of the model with two parasite strains. Moreover, numerical simulations reveal that superinfection makes the important effects on the coexistence of two parasite strains and the coexistence region. Secondly, by using a’bottom-up’way and the theory of Lyapunov stability, we obtain the existence and stability of equilibria of multiple parasites model. Furthermore, we apply them into the model with three parasite strains, and make some comparisons with the previous results which show some distinct phenomena.In chapter 4, an SEIR epidemic model with latent age and relapse age is established and studied. We obtain two main relevant. Firstly, by using the method of integrating along the characteristic lines, we derive a corresponding Volterra-type system. And, we obtain the asymptotic smoothness and uniform persistence of the system according to the theoretical knowledge of functional differential functions. Secondly, by using Hurwitz criterion and constructing Lyapunov functional, we derive the global dynamics of the system which completely depends on the basic reproduction number. Results reveals that individual age-related difference have important effects on the threshold value of epidemic model, i.e., the basic reproduction number, thus affect the spread of disease. This can be applied to guide health workers to adopt more practical strategies to control the spread of diseases.In chapter 5, a heroin epidemic model with susceptible age and relapse age is established and studied. We obtain three main relevant results. Firstly, according to the Volterra-type formula, we get the corresponding Volterra-type epidemic model and obtain the basic reproduction number as well as two possible equilibria for the system. Secondly, by applying the method of semi-flow of functional differential equations, we obtain the existence of a global attractor and the uniform persistence for the system. Thirdly, by analyzing the characteristic equations and constructing Volterra-type Lyapunov functional, we get local and global dynamic properties of the system. Our results imply that susceptible age and relapse age are factors influencing the contact rate and relapse rate of heroin epidemic model respectively, and thus decreasing the contact rate between susceptible and heroin users and the relapse rate of heroin users in treatment can effectively control the spread of heroin. These can provide related departments with a theoretical basis for the control of drugs including heroin.At the end, the main contents of this thesis are briefly summarized and some worthwhile future works related to our results are put forward. |
