The Studies Of Travelling Wave Solutions In Epidemic Dynamics System With Diffusion

In this paper, we propose three types SI epidemic models with different capacity todiffuse in accordance with the symptoms of different type disease based on diseases spreadprocess that spread from pathogens to the surrounding environment at constant velocity.Then, we investigate the condition of the existence of travelling wave solutions of epidemicmodels, and forecast the minimum invasion speed of the infectious diseases. For a diseasesatisfying such epidemic models, as long as people take various methods to control theinvasion speed of infectious diseases, and made it smaller than the minimum invasion speedof the epidemic, the epidemic will die out; or if we make the diffusion speed of the susceptibleis greater than the minimum invasion speed of epidemic, we also can control the infectiousdiseases. It is sure that our research have a certain significance for the prevention and controlof infectious diseases.In chapter 2, we consider this situation that the susceptible and the infectious havedifferent birth rate, and the disease has a nonlinear incidence rate, and we also assumeonly the infectious have the capacity to diffluse. Then, we build a spatially extended SIepidemic model. By using the Shooting argument, construct W set, Lyapunov function andWazewski theorem, LaSalle's Invariance principle, we study the existence of a heteroclinicorbit connecting two equilibrium points in R~3 which corresponds to a travelling wave so-lution connecting the disease-free and endemic equilibria for the reaction-diffusion systemand obtain the expression of the minimum invasion speed of diseases cff. At the same time,numerical simulations are given to verify the theoretical results.In chapter 3, we study a type SI epidemic model with nonlinear incidence rate underthe condition that the infectious and the susceptible have the same capacity to diffuse. And analyze the condition that the existence of traveling wave solutions of this epidemic model,obtain the minimum invasion speed and the condition that the model exists the travellingwave solutions which are exponential damped oscillately or monotonously. We also validateour result through computer.In chapter 4, basing on the pathogenesis of rabies, we establish a SI epidemic modelof rabies with birth rate of Logistic type under the situation that only the infectious havethe capacity to diffuse. We analyze the existence of traveling wave solutions and obtain theminimum invasion speed of diseases and the condition that the model exists the travellingwave solutions which are exponential damped oscillately or monotonously. We also validateour results with simulation as we did in the previous section.