Spatio-temporal Complexity Of Epidemic Models

Epidemic dynamics is to study disease spread, predict the trends of the disease, find outthe key factors in the process of the spread, and prevent the disease by the best methods. Andspatial epidemiology is the study of spatial variation in disease risk or incidence, includingthe spatial patterns of the population. The spread of diseases in human populations canexhibit large scale patterns, underlining the need for spatially explicit approaches. In thispaper, we mainly investigate the Turing pattern, spiral wave structures, and noise effect ofspatial extended epidemic systems. In chapter 1, we give the significance of investigatingepidemic models, development of this filed and the main work in this thesis.In chapter 2, we investigate a spatial S-I model with non-linear incidence rates SpIq, andobtain the conditions for transcritical bifurcation, Hopf bifurcation and Turing bifurcation.In particular, the exact Turing domain is found in the parameter space. For parameters arein that domain, a series of numerical simulations reveal that the model has rich dynamics.We obtain not only a stripe-like pattern but also a spot pattern, or coexistence of the two.In chapter 3, the spatiotemporal complexity of a spatial epidemic model with nonlinearincidence rate, which includes the behavioral changes and crowding effect of the infectiveindividuals, is investigated. Based on both theoretical analysis and computer simulations,we find out when, under the parameters which can guarantee a stable limit cycle in thenon-spatial model, spiral and target waves can emerge. Moreover, two different kinds ofbreakup of waves are shown. Specifically, the breakup of spiral waves is from the core andthe breakup of target waves is from the far-field, and both kinds of waves become irregularpatterns at last. Our results reveal that the spatiotemporal chaos is induced by the breakupof waves. The results obtained confirm that diffusion can form spiral waves, target waves orspatial chaos of high population density.In chapter 4, we present novel numerical evidence of complicated phenomenon controlled by noise in a spatial epidemic model. The number of the spot is decreased as the noiseintensity being increased, which we show by performing a series of numerical simulations.Moreover, when the noise intensity and temporal correlation are both large enough, themodel dynamics exhibits a noise controlled transition from spotted pattern to stripe growth.In addition to that, we show in details the number of the spotted and stripe pattern, withthe identification of a wide range of noise intensity and temporal correlation.In chapter 5, pattern formation of a spatial model with cross diffusion of the susceptibleis investigated. We compute Hopf and Turning bifurcations for the model. In particular,the exact Turning domain is delineated in the parameter space. When the parameters arein that domain, a series of numerical simulations reveals that the typical dynamics of theinfecteds class typically involves the formation of isolated groups, i.e., striped, spotted orlabyrinthine patterns. Furthermore, spatial oscillatory and anti-phase dynamics of differentspatial points were also found. These results demonstrate that cross diffusion of susceptiblesmay have great inffuence on the spread of the epidemic.