Infectious Disease Dynamics Model Of Behavior Analysis

In the paper, the dynamical behaviors of several epidemic models for host-parasite are investigated. In the second part of this paper, we consider the global dynamics of a microparasite model with more general incidences. The incidence characterize continuous transitions from the bilinear incidence to the standard incidence. Firstly, the extinction of both host population and parasite population occurs only under standard incidence by the methods of differential qualitative theory. Furthermore, we obtain the existence conditions of the endemic equilibrium and prove it is global whenever it exists.In the third part of this paper, we consider an epidemic model with the nonlinear incidence of a sigmoidal function. Firstly, a threshold condition of the endemic equilibria is obtained. The dynamical analysis of the model shows that the non-linearity of the incidence rate may lead to Allee effect. Secondly, it is shown that the epidemic model undergoes a supercritical Hopf bifurcation a subcritical Hopf bifurcation under the some conditions. Furthermore, it is shown that the model undergoes a Bogdanov-Takens bifurcation which means that it exhibits a saddle-node bifurcation, a Hopf bifurcation and a homoclinic bifurcation. Thirdly, by numerical simulations, We find two approaches that the model admits two limit cycles. First, a limit cycle emerges from a Hopf bifurcation, while another limit cycle occurs from a homoclinic bifurcation. Secondly, a branch of limit cycles is born from a Hopf bifurcation at first. Then there exists a second bifurcation from a degenerate limit cycle such that two limit cycles emerge. We find also three patterns of limit cycle bifurcations from a degenerate limit cycle. Our analysis shows that a little change of parameter shape of the incidence could lead to quite different bifurcation structures.In the fourth part of this paper, pattern dynamical behaviors of the model with space effect in the second part are analyzed. Applying the standard linear stability analysis, we" obtain the given steady state is stable in the reaction-diffusion system with bilinear incidence. However, the stability of the given steady state with standard incidence is obtained under the some conditions and Turing instability is given under the other conditions in the reaction-diffusion system. Furthermore, analyzing the behaviors of the amplitude equations of the model, we get the conditions of Turing pattern occurrence in the fixed parameter values.In the firth part of this paper, we analyze the spatiotemporal patterns in an epidemic model with the Allee effect. Using the linear stability theory, we obtain the conditions of existence for Turing instability and Hopf bifurcation. By means of analytic analysis and numerical simulations, we show that the model exhibits Turing peaks, Turing holes, labyrinths, spiral waves, standing waves and chaotic pattern. There are two cases if we increase the infectious dose. One case is that Turing peaks, holes and labyrinths occur if diffusion rate is greater than a certain value, and the patterns undergo transitions from spots, stripes to holes and labyrinths, which indicates that the spatial distribution of infected individuals changes from sparse to dense, then to sparse. The other case is that spiral waves, standing waves and labyrinths happen if diffusion rate is less than the value, and the patterns range from traveling wave to static wave and labyrinths. If the diffusion rate of susceptible individuals is taken as bifurcation parameter, we find that stable spiral waves occur whenever the diffusion rate is more than or less than the diffusion rate of infectious individuals and the increase of number of defects for spiral waves leads to chaotic pattern.