Alexei Pokrovskii

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.

A time-dependent Preisach model

We study a new type of hysteresis nonlinearity: that arising in time-dependent Preisach systems. The Preisach model is amended to allow for state or time dependent threshold values. The paper presents a simple time-dependent Preisach model. The basic properties of this model, including finite approximations are considered.

Differential Equations with Hysteresis via a Canonical Example

Analysis of closed-loop system is needed and these systems are described by differential equations with hysteresis, and hysteresis terms are to be taken into account in various areas of differential equations, thus leading to numerous distinct branches of study, depending on the subject area, type of hysteresis operators that are used, etc. Operators of hysteresis nonlinearities often admit a simple “picture definition,” however their properties are quite different from the properties of more classical operators. The investigation of differential equations with hysteresis nonlinearities requires new mathematical methods. In return, methods that have been originally suggested for the analysis of differential equations with hysteresis appear to be useful in the classical theory of differential-operator equations. This chapter demonstrates the theory of differential equations with hysteresis via a simple canonical example. Essentially, the semi-linear Duffing oscillator is considered with the Preisach non-linearity. The chapter presents various results on existence and uniqueness, on properties of periodic motions, on the convergence of numerical solutions, etc. Moreover, it shows how these fuse with, and complement each other. Apart from results in traditional areas, the chapter also presents a version of the shadowing lemma specifically designed for the analysis of systems with hysteresis.

Memory of recessions

This paper reviews the evidence on the effects of recessions on potential output. In contrast to the assumption in mainstream macroeconomic models that economic fluctuations do not change potential output paths, the evidence is that they do in the case of recessions. A model is proposed to explain this phenomenon based on an analogy with water flows in porous media. Because of the discrete adjustments made by heterogeneous economic agents in such a world, potential output displays hysteresis with regard to aggregate demand shocks and thus retains a memory of the shocks associated with recessions.

Topological degree in analysis of canard-type trajectories in 3-D systems

Piecewise linear systems become increasingly important across a wide range of engineering applications spurring an interest in developing new mathematical models and methods of their analysis, or adapting methods of the theory of smooth dynamical systems. One such areas is the design of controllers which support the regimes of operation described by canard trajectories of the model, including applications to engineering chemical processes, flight control, electrical circuits design, and neural networks. In this article, we present a scenario which ensures the existence of a topologically stable periodic (cyclic) canard trajectory in slow-fast systems with a piecewise linear fast component. In order to reveal the geometrical structure responsible for the existence of the canard trajectory, we focus on a simple prototype piecewise linear nonlinearity. The analysis is based on application of the topological degree.

STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings fl(x)=1−|1−2x|l,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.

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.

Mixed moments of random mappings and chaotic dynamical systems

Some statistical characteristics of completely random mappings and of random mappings with an absorbing or an attracting centre are calculated. Results are applied to validation of some phenomenological models of computer simulations of dynamical systems.

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.

Qualitative and numerical investigations of the impact of a novel pathogen on a seabird colony

Understanding the dynamics of novel pathogens in dense populations is crucial to public and veterinary health as well as wildlife ecology. Seabirds live in crowded colonies numbering several thousands of individuals. The long-term dynamics of avian influenza H5N1 virus in a seabird colony with no existing herd immunity are investigated using sophisticated mathematical techniques. The key characteristics of seabird population biology and the H5N1 virus are incorporated into a Susceptible-Exposed-Infected-Recovered (SEIR) model. Using the theory of integral manifolds, the SEIR model is reduced to a simpler system of two differential equations depending on the infected and recovered populations only, termed the IR model. The results of numerical experiments indicate that the IR model and the SEIR model are in close agreement. Using Lyapunovs direct method, the equilibria of the SEIR and the IR models are proven to be globally asymptotically stable in the positive quadrant.