Stability And HOPF Bifurcation Analysis For An Infectious Disease Model With Stage-Structure

The infectious diseases are still existing in a large area of the world now, breaking out from time to time, and taking many negative impact to people's normal production and life. Therefore, the research of the development of the diseases has important practical significance for its prevention and control.In this paper, we will investigate a kind of phased infectious diseases model with time delay. Taking time delay ? as the parameter, we will analyze the stability and Hopf bifurcation of the model.First, discussing the existence of the equilibriums, we find out the three equilibriums of the model. And we will focus on the two non-zero equilibriums of the system: disease-free equilibrium and the positive equilibrium.In the following, the disease-free equilibrium is analyzed firstly. We can give a condition. When it doesn't hold, the disease will continue, that is, it can't be eradicated; otherwise, the disease may be eradicated. However, the stability is also related with the time delay ?. By discussing the characteristic equation of the system at the disease-free equilibrium, we can get a series of values of ?. Proving the transversal condition at these values, we will determine the stability of the equilibrium, from which, we can know whether the disease can be eliminated at last.Then, we will analyze the stability of the positive equilibrium using the similar method. Through the characteristic equation of the system at the positive equilibrium, we can get a range of critical values of ?. Discussing the trend of the eigenvalues, we will know the distribution of the all the eigenvalues to determine the stability of this equilibrium. Furthermore, we will find that the system produces Hopf bifurcation at the equilibrium when ? crosses the critical values. Next, using the normal form method and the center manifold theory, we can determine the direction of the Hopf bifurcation, the stability of the bifurcating periodic solution and the period size.Finally, some numerical examples about the positive equilibrium will be given to illuminate our theoretical analysis.