Hopf Bifurcation Analysis In A Delayed Epidemic Model

Traditionally, the evolution theory of wildlife infections mainly focuses on thestudy of interactions between subjects and their parasites. However, lots of studieshave shown that, a high level of predation will also cause severe mortality rate ofamong the predators. Such mortality should be virulence-dependent because thepopulation size of predators depends on the number of available prey. It has beenshown that the evolutionary branching mediated by predator can occur within a widerange of subjects’ lifetime, and there are various types of virulence transmissionsexchanges. The importance of predation in explaining the existing polymorphismand biodiversity of pathogen strains in wildlife could not be neglected.In this paper, we discuss an ecological biological model with delay effect. Bycombining ecology and epidemiology, this model can explore the laws of diseasedistribution from both ecology change and environment change. According to thelaws of disease distribution, we predict how the disease happens and changes. Andthen, we can find out the corresponding countermeasures.Main contents of this paper are as follows:Firstly, we introduce the background of the topic and the source of this model.We will summarize current study of this field and the main goals of this paper.Some theoretical knowledge will also be explained.Secondly, we give the characteristic equation and explore the stability of thepositive equilibrium and the existence of a local Hopf bifurcation. We candetermine the nature of the Hopf bifurcation. By the theory of normal form andcenter manifold, on explicit formula for determining the Stability and direction ofperiodic solution bifurcation from Hopf bifurcation is decided.Finally, we perform numerical simulation. An example is given in the simulinksimulation to verify the theoretical results.