Mathematical models of rumour propagation have traditionally used a `rumour as epidemic' approach. Such models are loosely based on a reaction-diffusion system and grossly simplify the connections between the individual people in question. In most cases they are thought of as all being infinitely and continuous connected. Instead, we will consider a population connected together in a given network architecture. How does the architecture of the network itself affect propagation? The population will have two subgroups, each with a different set of values such as Liberals and Conservatives. How does the distribution of these subgroups on the network affect rumour propagation?

Many physiological systems involve rings of similar neurons. Such neural networks can be mathematically modelled as a system of delay differential equations, where the coupling between neurons is often modelled as simple step functions or nonlinear sigmoidal functions. Here, we investigate a three-dimensional bi-directional symmetric ring neural network with delayed coupling and self feedback. This model was numerically analyzed to identify and compute the different types of periodic solutions arising from equivariant Hopf bifurcations. This work was performed with the bifurcation package DDE-BIFTOOL which also allowed the computation of the stability of the various periodic solutions. Secondary bifurcations and multistability near codimension two bifurcation points are also investigated. These results are complemented by numerical simulations of the system using XPPAUT.

In this talk we make explicit the connection between projected dynamical systems on Hilbert spaces and evolutionary variational inequalities. We give a novel formulation that unifies the underlying constraint sets for such inequalities, which arise in time-dependent traffic network and spatial price equilibrium problems. We provide a traffic network numerical example in which we compute the curve of equilibria.

This is joint work with Patrizia Daniele, University of Catania, and Anna Nagurney, Isenberg School of Management, University of Massachusetts.

In 1998 Smale formulated 18 problems for the 21st century, the sixth of which was: prove that there are finitely many relative equilibria for positive masses in the planar n-body problem. In joint work with Richard Moeckel, we have proven this result for n = 4.

It is well known that the Lorenz system

Broad classes of inverse problems in differential and integral equations can be viewed as seeking to approximate a target x of a metric space X by fixed points of contraction maps on X. Each contraction map depends on the parameters of the underlying system-for example, chemical reaction rates, population interaction rates, Hooke's constants-and such problems frequently appear in the parameter estimation literature. The "collage method" attempts to solve such inverse problems by finding a map T_c that sends the target as close as possible to itself. In this talk, after briefly introducing the framework which surrounds the collage method, I will discuss some recent applications, as well as some technical issues.

A special numerical technique has been developed for identification of solitary wave solutions of Boussinesq and Korteweg-de Vries equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed problem is transferred into a problem of the unknown coefficient from over-posed boundary data in which the trivial solution is excluded. The Method of Variational Imbedding is used for solving the inverse problem. The generalized sixth order Boussinesq equation is considered for illustration.

We will investigate a particular rational difference equation:

A general process of the immune system consists of effector stage and memory stage. It is not fully understood how the memory stage forms and how the system switches from the effector stage to the memory stage. Existing mathematical models of the immune system can describe either the effector stage or the memory stage, but not both. We formulate a mathematical model based on a recently developed cellular automata model for influenza A virus, and show that these two stages can be smoothly realized in our model. The numerical simulations of our model agree with the clinical observation very well.