Space And The Randomness Of The Power System Of The Host - Pathogen

Recently, spatial theory has been widely applied to the epidemiology, and there have been a substantial increase of spatial multi-strain approaches to study disease spreading. In spatial context, strain mutation has been regarded as a key factor for most infectious disease, such as human influenza A, avian flu virus, Dengue virus and HIV virus. Hence, the study on the spatial dynamics of multi-strain systems plays a central role on practicability in disease. Traditionally, the research is mainly based on the spatial lattice model, but which lacks of the mathematical tractability. So using recently developed spatial moment equations—pair approximation method is necessary, which can give some analytical description for the lattice model.In chapter 2, we present a generalized baseline SI model for one host and two infec-tious strains, and then compare the dynamics both the spatial and non-spatial models. Our results show that mean-field model, pair approximation model, Gillespie algorithm-based simulations, and spatially explicit models give similar qualitative results. In particular, the temporal evolution of the spatial model can be successfully approximated by pair approx-imation. In addition, simulation results obtained from the spatially explicit model show that mutation plays a major role in multi-strain coexistence. In chapter 3, We consider a Susceptible-Infective-Recovered-Susceptible (SIRS) model with strain mutation and cross-immunity both in a non-spatial model and a lattice-structured setting. A family of correlated equations of both pair approximation and mean-field method are presented. By parameter-izing the basic reproductive numbers of the strains, we find that although the qualitative results of the pair approximation model are similar to those of the corresponding non-spatial model, the spatial model predicts coexistence over a wider range parameters than the non-spatial mode, particularly when the strain evolution tends to a larger basic reproduction number. Our results also show that the cross immunity is more sensitive on the spatial structure model, and a strong cross-immunity causes a pronounced effect on resident strain and mutated strain.In general, the methodology of epidemics is only based on deterministic model or stochastic model. Both of them have promoted the development of the epidemiology, but traditional single theory cannot give an exact explanation for the disease outbreak. In view of the recent research, our chapter 4 gives a framework for the integration between the deterministic model and the stochastic model. We mainly study an avian influenza epi-demics with external environmental transmission. Furthermore, a technique is provided by formulating the models as master equations and using van Kampen system-size expansion to provide analytical expressions for spectrum of stochastic fluctuations, which shows how the amplification of noise varies with the environmental transmission. Our results also show that the distribution of the resonant stochastic fluctuations can be captured by the changing skewness, which can indicate the impending prevalence of epidemics.