The Global Qualitative Behavior Of Epidemic Models

In this paper, the effect of population dispersal and nonlinear incidence rate on the spread of a disease is investigated. In the first part of this paper the effect of population dispersal between two patches on the spread of an epidemic disease is investigated. We obtain the uniqueness of endemic steady state when RQ > 1 and dispersal rates of susceptible and infective individuals are the same in either patch, when population dispersal exists between two patches, and establish a condition for a disease to be extinct in two pathes (i.e., for the disease free steady state to be locally stable ), under the assumption of three kind of birth rates. Numerical calculations show that, when the basic reproduction number .Ro > 1, two or three endemic steady states may arise within the two patches if the ratio of the dispersal rate of infective individuals to the dispersal rate of susceptible ones is sufficiently small in one patch or in both patches. The system can uadergoe multi-stable steady states under some certain conditions and the actual state the system settles into depends on the initial conditions. We also find by numerical calculations that certain population dispersal rates can intensify a disease or help eradicate it, especially, a disease can be extinct within two patches where population dispersal exists even though it can be endemic in either patch when they are isolated, provided reproduction numbers are not very large in single patch models. If a disease outbreaks in two patches (or one of the two patches), only preventing the infective population from dispersing will not decrease the possibility of the disease to be endemic, however, it can reduce the total number of infective individuals in steady state, if the initial number of infective individuals is very small.In the second part of this paper, the global qualitative behavior of an SIRS epidemic model with nonlinear incidence rate is studied under the assumption that the total number of the population varies. A threshold for the disease to be epidemic or to be eradicatedis established. The existence and locally asymptotic stability of all equilibria (including disease free equilibrium and endemic equilibria) are studied completely for all parameters. Some sufficient conditions for the disease free equilibrium or endemic equilibrium to be globally asymptotic stable are also obtained. In some special conditions, the disease can be epidemic periodically. When the conditions for the disease free equilibrium and endemic equilibrium to be locally asymptotic stable are related together, it is found that sometimes the system undergoes multi-stable steady states and the persistence or extinction of the disease depends on the initial conditions. The existence of limit cycles is also studied in this paper. The existence of saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation is studied clearly, and high codimension bifurcation may emerge for some special parameters.