Backward Bifurcation and Associated Phenomena in Epidemic Models

The equilibrium phenomenon of backward bifurcation has in recent years been shown to occur in several deterministic differential equation models for the spread of infectious diseases. For simple epidemic models the disease is usually able to persist in the population when the basic reproduction ratio is greater than one, but dies out otherwise. However, for more complicated models in which backward bifurcation is exhibited the disease can, for certain parameter values, persist even when the basic reproduction ratio is less than one.Extended versions of deterministic models in which backward bifurcation is already known to be present are studied. By performing detailed equilibrium analyses we consider the possibility for the existence of multiple subcritical and supercritical endemic equilibria in these models, and the implications that this might have with regard to the dynamics of the system. In addition to extending the number of classes of the models, we examine the situation for which one or more of the parameters are functions of the population size. Corresponding stochastic models are also developed, and we look at the potential influence that the presence of backward bifurcation might exert on the probability of extinction for the disease. Our investigations are carried out using analytical and numerical methods, and also via computer simulations.We have been able to demonstrate the existence of, and then classify, a variety of differently shaped bifurcation diagrams for our extended models which, in conjunction with the results of our investigations into the stability of the endemic equilibria, would suggest the potential for extremely complex dynamic behaviour when the basic reproduction ratio is close to one. Some of the features of epidemic models in which backward bifurcation is known to be present have been identified. We have also found interesting links between the presence of backward bifurcation in the deterministic models and the probability of extinction in the stochastic versions.