Infectious Disease Model Of Complex Space-time Dynamic Analysis

The epidemic exists extensively in modern life.Since1920s,people had tried to study therules of the spread of the epidemic diseases,which presented the theoretical proof for makingthe strategies of predicting and treating diseases. However,the ability which models predictand control disease depends greatly on the assumptions made in the modeling process.Inorder to gain deeper insights into the mechanism of disease transmission,much attentionhas been focused on the design and analyses of mathematical models.In modeling of com-municable diseases, the incidence rate is considered to play a key role.To the study of theepidemic models, people are interested in the issue that the parameters determine the diseaseto prevail or die out in a population.Many models represent the transmission dynamic ofdiferently infectious diseases by developing mathematical models using systems of ordinarydiferential equations,partial diferential equations and function diferential equations.In chapter2,we investigate a reaction difusion SIR model with time delay and obtainthe conditions for Hopf bifurcation and Turing bifurcation.Furthermore,we analyze the sta-bility of the equilibria,i.e.the conditions of the disease extinction or persistence. Throughtheoretical derivation,We conclude that when the model with time delay and difusion,therewill be periodic solutions of time delay and difusion.In chapter3,pattern formation of a spatial S-I epidemic model is investigated.Throughthe analysis,efects of various parameters on the stability of equilibrium model. Further-more,according to the dispersion relation formula,we discuss the changes of the wavelength,aswell as the conditions of the spatial pattern formation.Our obtained results may be helpful tounderstand the mechanism of the spatial-temporal epidemics and have potential applicationof control of epidemics.In chapter4,we present a bilinear incidence rate spatial epidemic model.From themathematical analysis and prove,we obtain the conditions for Turing bifurcation. Further-more,Turing instability and the pattern of conditions are discussed.And we analyse The equi-librium stability of the model space,which shows that it is useful to use reaction-difusion model to reveal the spatial dynamics.