Abstracts
Stability and bifurcations in a model of antigenic variation in malaria
Konstantin Blyuss
University of Keele
Interaction of malaria variants within human host is investigated using a recently proposed model for cross-reactive immune response. This model is based on the framework of modelling multiple epitopes and hence is able to capture many important features of the immune response. In the case of vanishing decay rate of a long-lasting immune response, we show that the system exhibits the so-called bifurcations without parameters due to the existence of a hypersurface of equilibria in the phase space. This explains observed patterns of temporal dynamics of malaria variants. When the decay rate of the long-lasting immune response is different from zero, the hypersurface of equilibria degenerates, and a multitude of other steady states are born, many of which are related by a permutation symmetry of the system. Robustness of the fully symmetric state of the system is investigated by means of numerical computation of transverse Lyapunov exponents. These results indicate that for vanishing decay of specific immune response, the fully symmetric state is not robust in the substantial part of the parameter space, and instead all variants develop their own temporal dynamics contributing to the overall time evolution. At the same time, if the decay rate of the long-lasting immune response is increased, the fully symmetric state can become robust provided the growth rate of the long-lasting immune response is large enough.
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Dynamics of prion related diseases
Nikolai Brilliantov
University of Leicester
We study the dynamics of prion related diseases, as it follows from the nucleated polymerization model. Analyzing the elementary processes of prion fibril formation, we formulate a set of differential equations for the number of fibrils, their total mass, and the number of prion monomers. In difference to previous studies we analyze this set explicitly taking into account the time-dependence of the prion monomer concentration. We develop an analytical theory of the disease dynamics and perform numerical experiments. The theoretical results agree with experimental data, whereas the generally accepted hypothesis of constant monomer concentration leads to a fibril growth dynamics, which is not in agreement with observations. The obtained size distribution of the prion fibril aggregates is significantly shifted toward shorter lengths as compared to earlier studies, which leads to an enhanced infectivity of the prion material. Finally, we analyze the effect of filtering of the inoculated material on the incubation time and the disease dynamics.
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Models of adaptation and measurement of physiological fitness
Alexander Gorban
University of Leicester
Many different environmental factors affect living creatures. We can represent adaptation as distribution of a (hypothetical) internal adaptation resource for compensation of damaging factors or not favorable amount of some external resources. H. Selye called this hypothetical adaptation resource the ``adaptation energy" [1].
Since 1987 we developed the system of adaptation models based on this idea and applied them to various physiological problems [2]. The following effect was predicted and confirmed by thousands of experiments: the level of physiological fitness in a group (population) could be measured by the effective dimension of this group as a dataset in the space of those parameters. The values of physiological parameters are less sensitive. In other words, correlation are more sensitive than values of parameters. For high level of stress, the effective dimension usually decreases (and the scattering of data increases). This effect was studied theoretically and experimentally for groups of people, experimental populations of rats, and even for plants. Especially important are the relatively rare situations when this rule ``high stress - small dimension" is violated. This violation appears when the damaging factors demonstrate strong sinergetic effect, and significantly (superlinear) amplify each other.
In the talk we present the ``factors - resources" models and their application for various physiological and medical problems.
1. Selye H. Experimental evidence supporting the conception of ``adaptation energy", Am. J. Physiol. 123 (1938), 758-765.
2. Sedov K.R., Gorban A.N., Petushkova (Smirnova) E.V., Manchuk V.T., Shalamova E.N., Correlation adaptometry as a method for the population screening, Vestn Akad Med Nauk SSSR, 1988 (10), 69-75 (Proceedings of the USSR Academy of Medical Sciences).
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Stabilization in a predator-prey system posed on non coincident spatial domains
Michel Langlais
Universite Victor Segalen Bordeaux 2
A two-component phenomenological predator-prey system posed on non coincident domains is studied. Both discrete and continuous spatial structures are considered. The main departure from standard predator-prey systems arises in the parameterization of the numerical functional response to predation. It involves a continuous integral kernel in the case of a continuous spatial structure, and a discrete one for a discrete spatial structure. The stabilization to zero of the predator species via an internal control distributed on a small subdomain and acting either on predators or reducing the prey species below some sustainable threshold is analyzed.
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Competition of predation and infection in prey-predator systems with infected prey
Horst Malchow (a), Ivo Siekmann (a) & Ezio Venturino (b)
(a) University of Osnabrueck
(b) University of Torino
A class of prey-predator models with infected prey is investigated: Predation terms are either of Holling type II or III, infection is either modelled by mass action or frequency-dependent. It is shown that the key for understanding the model behaviour is the competition of predators versus infection. In the presented models the predator is not susceptible to the infection and the infection of the prey has no influence on the ability of the predator of catching the prey. However, the prey population can be seen as a resource which both the predators and the infection depend on. The competition for this resource is strong. The principle of competitive exclusion holds as long as there is no destabilisation by a Hopf bifurcation.
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Spatiotemporal patterns in reaction-diffusion models of biological control
Andrew Morozov & Sergei Petrovskii
University of Leicester
Biological control of harmful species has been a challenging problem for ecology during several decades. Mathematical modelling might explain possible causes of numerous failures of biological control observed in practice. Here we consider a conceptual model of control of a pest species (the target species) by its predator (the control agent) and study spatiotemporal patterns arising in the system. We apply the reaction-diffusion approach and study the processes in 1-D and 2-D environments. The local kinetics of species interactions is considered to be of ‘exitable’ type, i.e. a supercritical perturbation of locally-stable coexistence state results in a single population cycle of large amplitude. We studied the system both analytically and by means of extensive numerical simulations. We consider and classify the patterns of dynamics arising in the system when the homogeneous state with low pest density experiences a local heterogeneous perturbation (e.g. due to some environmental and/or demographic noise). The main question we were interested in was: whether or not the control agent could suppress the initial pest outbreak. We show that there is a critical difference between the properties of spatially homogeneous system and the corresponding system in space. Moreover, the use of different dimensionality of space in the model (1-D or 2-D environment) would predict different outcomes of biological control: an increase of dimensionality of impedes the biological control of pests. Also, we obtained that to provide the most effective control, the values of diffusivity of the control agent and that of the target species should be of the same order. This is contrary to what was believed earlier when fast spreading control agents were supposed to be best candidates to guarantee a successful biological control.
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Population dispersal: Fat tails revisited
Sergei Petrovskii
University of Leicester
Populations do not remain fixed in space. Their distribution changes continuously due to the impact of environmental factors, such as wind, and/or due to self-motion of individuals. A cornerstone for understanding mechanisms of dispersal is identification of factors affecting the dispersal curve, in particular, its rate of decay at large distance. The standard random walk approach resulting in a dispersal curve with a `thin' Gaussian tail was eventually opposed by the theory of Levy flights, which predicts a more realistic `fat' tail with a lower rate of decay. However, here we argue that the Gaussian large distance asymptotics is more an artefact of an oversimplified description of the dispersing population rather than an immanent property of the random walk diffusion. Specifically, we show that, when some inherent population structure is taken into account, diffusion results in a dispersal curve with either exponential or power law rate of decay. Our theoretical results appear to be in a very good agreement with some available data.
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Spatiotemporal Patterning in Cyclic Populations
Jonathan Sherratt
Heriot-Watt University and Maxwell Institute
Spatial data is becoming increasingly available for populations undergoing multi-year cycles.
In many cases, this indicates spatial patterning in abundance, as well as temporal oscillations.
Mathematical models play akey role in understanding when patterns of different types will occur.
My talk will address this form the particular case of cyclic predator-prey systems, and will be in three parts. I will begin by describing numerical simulations showing that appropriate conditions at the edge of a domain or at the edge of an obstacle can generate either regular patterns of periodic travelling wave form, or irregular patterns that appear to be chaotic. In the second part, I will show how to predict the transition between the two pattern types as parameters are varied, using new numerical methods for calculating the eigenvalue spectrum of periodic waves. Finally, I will describe how the method of reduction to normal form can be used to procide analytical predictions of the speed and wavelength of regular spatiotemporal patterns, for cases in which the population cycles are of low amplitude. Although I will show results for only predator-prey models, the methods that I will discuss represent a toolkit that can be applied to patterning in a wide range of other models for cyclic populations.
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An SIR contact infection spread: From network to a continuum description and back
Igor Sokolov
Humboldt University – Berlin
We analyze the epidemic spread via a contact infection process in an immobile population within the Susceptible-Infected-Removed (SIR) model on a network. Starting from regular lattices we present both the results of stochastic simulations assuming different numbers of individuals (degrees of freedom) per cell, as well as the solution of the corresponding deterministic equations. For the last ones we show that the appropriate system of nonlinear partial differential equations allows for a complete separation of variables and present the approximate analytical expressions for the infection wave in different ranges of parameters. Comparing these results with the direct Monte-Carlo simulations we discuss the domain of applicability of the PDE models and their restrictions. We moreover discuss the situation of networks more complex than simple lattices (small world and scale-free
ones) as well as the case of heterogeneous populations.
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