Michael J.A. O’Callaghan

Lyapunov functions for SIR and SIRS epidemic models

In 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.

Memory Effects in Population Dynamics : Spread of Infectious Disease as a Case Study

Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to depend on the system state, and thus it disregards the history and, therefore, memory. Memory can be introduced into a mathematical model using a phenomenon known as hysteresis. We illustrate this idea using a simple SIR compartmental model that is applicable in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. For the purposes of illustrating these ideas in an application to biology, we consider a rather simple case study and develop a model from first principles. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects.

Uptake and partitioning of zinc in Lemnaceae

Macrophytes provide food and shelter for aquatic invertebrates and fish, while also acting as reservoirs for nutrients and trace elements. Zinc accumulation has been reported for various Lemnaceae species. However, comparative accumulation across species and the link between zinc accumulation and toxicity are poorly understood. Morphological distribution and cellular storage, in either bound or soluble form, are important for zinc tolerance. This study shows differences in the uptake and accumulation of zinc by three duckweed species. Landoltia punctata and Lemna minor generally accumulated more zinc than Lemna gibba. L. minor, but not L. gibba or L. punctata, accumulated greater concentrations of zinc in roots compared to fronds when exposed to high levels of zinc. The proportion of zinc stored in the bound form relative to the soluble-form was higher in L. minor. L. punctata accumulated greater concentrations of zinc in fronds compared to roots and increased the proportion of zinc it stored in the soluble form, when exposed to high zinc levels. L. gibba is the only species that significantly accumulated zinc at low concentrations, and was zinc-sensitive. Overall, internal zinc concentrations showed no consistent correlation with toxic effect. We conclude that relationships between zinc toxicity and uptake and accumulation are species specific reflecting, among others, zinc distribution and storage. Differences in zinc distribution and storage are also likely to have implications for zinc bioavailability and trophic mobility.

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm.

Seasonality is a complex force in nature that affects multiple processes in wild animal populations. In particular, seasonal variations in demographic processes may considerably affect the persistence of a pathogen in these populations. Furthermore, it has been long observed in computer simulations that under seasonal perturbations, a host–pathogen system can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal perturbation increases. In this paper, we develop a seasonally perturbed Susceptible-Infected-Recovered model of avian influenza in a seabird colony. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host– pathogen system. We give a computer-assisted exposition of the existence of chaos in the model using methods that are based on the concept of topological hyperbolicity. Our approach elucidates the geometry of the chaos in the phase space of the model, thereby offering a mechanism for the persistence of the infection. Finally, the methods described in this paper may be immediately extended to other infections and hosts, including humans.

Bat collisions with civil aircraft in the Republic of Ireland over a decade suggest negligible impact on aviation safety

Globally, collisions between wildlife and aircraft are a serious threat to aviation safety. While reported collisions have increased in recent years, the impact of these collisions on air safety is rarely quantified. Here, we report all bat collisions (bat strikes) with civil aircraft known to have occurred in the Republic of Ireland over the 10-year interval, 2006–2015. Morphological and/or DNA identification techniques were used to identify chiropteran specimens to species level. In total, carcasses or remains from five bat strikes—three Leisler’s Nyctalus leisleri, one Soprano Pipistrelle Pipistrellus pygmaeus, and one Natterer’s Myotis nattereri—were recovered. The collisions are discussed in the context of (a) the rate of chiropteran strikes in relation to the number of civil aircraft movements and (b) possible damage caused to aircraft. Overall, however, bat strikes with aircraft appear to have a negligible effect on civil aircraft safety.

Memory and adaptive behavior in population dynamics: anti-predator behavior as a case study

Memory allows organisms to forecast the future on the basis of experience, and thus, in some form, is important for the development of flexible adaptive behavior by animal communities. To model memory, we use the concept of hysteresis, which mathematically is described by the Preisach operator. As a case study, we consider anti-predator adaptation in the classic Lotka-Volterra predator-prey model. Despite its simplicity, the model allows us to naturally incorporate essential features of an adaptive system and memory. Our analysis and simulations show that a system with memory can have a continuum of equilibrium states with non-trivial stability properties. The main factor that determines the actual equilibrium state to which a trajectory converges is the maximal number achieved by the population of predator along this trajectory.