Epidemiological models with nonlinear incidence rates lambda IpSq show a much wider range of dynamical behaviors than do those with bilinear incidence rates lambda IS. These behaviors are determined mainly by p and lambda, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.