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Register a new userChaos in a spatial epidemic model
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We investigate an interacting particle system inspired by the gypsy moth, whose populations grow until they become sufficiently dense so that an epidemic reduces them to a low level. We consider this process on a random 3-regular graph and on the $d$-dimensional lattice and torus, with $dgeq2$. On the finite graphs with global dispersal or with a dispersal radius that grows with the number of sites, we prove convergence to a dynamical system that is chaotic for some parameter values. We conjecture that on the infinite lattice with a fixed finite dispersal distance, distant parts of the lattice oscillate out of phase so there is a unique nontrivial stationary distribution.
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We introduce the effect of site contamination in a model for spatial epidemic spread and show that the presence of site contamination may have a strict effect on the model in the sense that it can make an otherwise subcritical process supercritical. Each site on $mathbb{Z}^d$ is independently assigned a random number of particles and these then perform random walks restricted to bounded regions around their home locations. At time 0, the origin is infected along with all its particles. The infection then spread in that an infected particle that jumps to a new site causes the site along with all particles located there to be infected. Also, a healthy particle that jumps to a site where infection is presents, either in that the site is infected or in the presence of infected particles, becomes infected. Particles and sites recover at rate $lambda$ and $gamma$, respectively, and then become susceptible to the infection again. We show that, for each given value of $lambda$, there is a positive probability that the infection survives indefinitely if $gamma$ is sufficiently small, and that, for each given value of $gamma$, the infection dies out almost surely if $lambda$ is large enough. Several open problems and modifications of the model are discussed, and some natural conjectures are supported by simulations.
We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${mathbb Z}^d,dge 3$, when there is no extinction of the infection. For this, we derive percolation estimates (using dynamic renormalization techniques) for a locally dependent random graph in correspondence with the epidemic model.
We investigate the behavior in $N$ of the $N$--particle entropy functional for Kacs stochastic model of Boltzmann dynamics, and its relation to the entropy function for solutions of Kacs one dimensional nonlinear model Boltzmann equation. We prove a number of results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, and obtain a bound showing that the entropic rate of convergence can be arbitrarily slow. Results proved here show that one can in fact use entropy production bounds in Kacs stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open, and the upper bounds obtained here show that this problem is somewhat subtle.
Motivated by models of cancer formation in which cells need to acquire $k$ mutations to become cancerous, we consider a spatial population model in which the population is represented by the $d$-dimensional torus of side length $L$. Initially, no sites have mutations, but sites with $i-1$ mutations acquire an $i$th mutation at rate $mu_i$ per unit area. Mutations spread to neighboring sites at rate $alpha$, so that $t$ time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius $alpha t$. We calculate, for some ranges of the parameter values, the asymptotic distribution of the time required for some individual to acquire $k$ mutations. Our results, which build on previous work of Durrett, Foo, and Leder, are essentially complete when $k = 2$ and when $mu_i = mu$ for all $i$.
We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this `cycling chaos manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. This behaviour can be robust (i.e., structurally stable) for systems with symmetries and provides robust examples of non-ergodic attractors in such systems; we examine bifurcations of this state. We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This is a novel scenario for the blowout bifurcation and causes us to introduce the idea of set supercriticality to recognise such bifurcations. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors. The model we consider is a 9th order truncated ODE model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map.
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