Continuous and inverse shadowing

By the Shadowing Lemma we can shadow any sufficient accurate pseudo-trajectory of a hyperbolic system by a true trajectory of a hyperbolic system. If we are interested in finite trajectories, at least from one side, then a pseudo trajectory usually has many possible shadows. Here we show that we can choose a continuous single-valued selector from the corresponding multi-valued operator "pseudo-trajectory ↦ the totality of possible shadows". We do this in the context of Lipschitz mappings which are semi-hyperbolic on some compact subset, which need not be invariant. We also prove that semi-hyperbolicity implis inverse shadowing with respect to a very broad class of nonsmooth perturbations

Lyapunov functions for SIR and SIRS epidemic models

AbstractIn this paper, we construct a new Lyapunov function for a variety of SIR and SIRS models in epidemiology. Global stability of the endemic equilibrium states of these systems is established

The impact of an emerging avian influenza virus (H5N1) in a seabird colony

Emerging and re-emerging pathogens pose a significant threat to the health status f man, of domestic animals and of wildlife. Therefore, it is important to know how such pathogens spread in immunologically naive host populations. In this Chapter, we study mathematical models for the long-term dynamics of an emerging pathogen H5N1 in a mixed population of marine birds. Seabirds are highly colonial and form densely-packed colonies containing thousands of individuals. A SEIR (Susceptible-Exposed-Infected-Recovered) model is developed that incorporates the population biology of seabirds and the H5N1 virus. By employing the theory of integral manifolds, it was possible to reduce the SEIR model to a simpler system of two differential equations based exclusively on the infected and recovered populations; this is termed the IR model. The SEIR and IR models are shown to agree closely. Moreover, by employing Lyapunov's direct method, it has been proved that the equilibria of the SEIR and IR models are globally asymptotically stable in the positive quadrant. The effect of seasonality is also investigated by considering a seasonally perturbed variant of the IR model. This system exhibits complex dynamics as the amplitude of the seasonal perturbation term is increased. A rigorous proof of the existence of chaos in the seasonally perturbed IR system is established, using methods from topological degree theory and split hyperbolicity. To our knowledge, this is the first time that this method has been used to prove the existence of chaos in an epidemiological model