Stability And Bifurcation Analysis Of A Kind Of Bacterial Infection Model For Tissue Inflammation Dynamics

The bifurcation problem is a very significant problem in the dynamic system theory,while it also has an important position in math.Bifurcation theory study the system that has an unstable structure,and analyze whether the parameters changes can influence the structure(including the number and behavior)of solutions of the system.This paper concerns a kind of the bacterial infection model for tissue inflammation dynamics.The Hopf bifurcation of the model is the focus of attention.The main contents of the paper are as follows:Firstly,we introduce the research background,significance and status of the paper,then we summarize the theoretical knowledge that has close connection with our paper.They are the preliminary knowledge in order to study better.Secondly,we analyze a mathematical model for an inflammatory response to bacterial infection of a homogeneous tissues.A detailed analysis of the Lauffenburger-Kennedy bacterial infection model is provided.Considering positivity and boundedness of the solutions,we construct the feasible region where the model is biologically relevant,then establish the conditions for the existence of multiple equilibria(a bacteria-free and two endemic compromised steady states).Specifically,asymptotic results of the steady states along with the existences of Hopf bifurcations are shown under certain conditions of the parameters,and the stability of bifurcating periodic solutions is analyzed.Finally,numerical results are used to provide illustrative examples of the different dynamical patterns observed in the model,which provide a further research direction.