| Summary: | This paper is an investigation of the possible behaviors of epidemics when there is immigration with some infectives and when vaccination is in effect. Infective immigrants from outside the population may introduce communicable disease into a host population. To help controlling disease one might introduce mass vaccination. First the model of SVIS type disease is discussed to describe the behavior of an epidemic disease when spreading into a population with immigrants and when a vaccination policy is in effect. Even in this simple version of the model, backward bifurcation and multiple endemic steady states can be observed with some sets of parameter values; by mathematical analysis the condition for a stability change and for having a backward bifurcation is proved to be identical in this case. Also the sufficient condition for backward bifurcation is stated explicitly. In the case of forward bifurcation, there can be exchange of stability and Hopf bifurcation occurs in such cases - numerical examples are provided to help understanding. Next we assume a disease with no immunity gained by recovery from infection, so the model is of SVIR type. In the case of no disease fatalities several special cases are discussed; stability analysis of steady states in each case and threshold values determining existence of endemic equilibria are presented if there are any. The proof that backward bifurcation cannot occur with any set of nonnegative parameter values in SVIR model is presented. On the other hand some numerical examples of backward bifurcation with negative immigration rates, i.e. outgoing immigration from host population, are shown. When there is a backward bifurcation bringing the vaccination-reduced reproductive number down to one may not be sufficient to control the disease. Also with some parameter values stability changes can be observed even with forward bifurcation. We would be able to know the minimum portion of susceptible class to be vaccinated in order to have a control over a disease by knowing the bifurcation point of backward bifurcation. If so we can avoid a sudden jump in the number of infectives due to an unstable middle branch of bifurcation. === Science, Faculty of === Mathematics, Department of === Graduate
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