Stability And HOPF Bifurcation Analysis For A SIR Model

The epidemical model was first proposed by Kermack-McKendrick, it is to an-alysis the dynamic characteristic of model to make pepole know the spread ofinfection mechanism and spread rule. Moreover, it predictes the developmenttendency of diseases and searches optimum strategy for disease controllingprimarily according to characteristics and all kinds of influence factors of epidemic.Thus, it has profound influence on epidemical research and is attracting more andmore researchers’ attention.In this paper, we study SIR epidemical model based on epidemical model thatproposed by Huang and Takeuchi, and consider susceptible population grows withthe Logistic growth. Here, what’s different with the past is that the liver sizedepends on not merely the susceptible population but also infected person, hence,compared with previous epidemical models, research for it has more practicalsignificance, but increases difficulties in analysis. We mainly analyze the stabilityand Hopf bifurcation for this kind SIR epidemical model in the paper.Firstly, we study the positivity and ultimate boundedness of the solution forepidemical model. On the basis, when the basic reproductive number satisfiescertain conditions, we prove the global attractivity of the disease-free equilibriumby comparison principle and Lyapunov-LaSalle’s invariance principle. Meanwhile,we also demonstrate the permanence of the model by means of the uniformpersistence theory for infinite-dimensional. Secondly, we analyze the distribution ofroot relating to the second degree transcendental Characteristic equation of themodel, and obtain the unstability of the trivial equilibrium. Particularly, we give theexistence condition at an interior equilibrium of a Hopf bifurcation by approach ofBeretta and Kuang. Moreover, we deduce several calculation formulas to determinethe properties of Hopf bifurcation in the help of Hassard’s center manifold theoremand normal form theory. Finally, we choose proper parameters to verify theaccuracy of theoretical analysis by numerical simulations.