Book contents
- Frontmatter
- Contents
- Foreword
- List of participants
- Stochastic differential equations with boundary conditions and the change of measure method
- The Martin boundary of the Brownian sheet
- Neocompact sets and stochastic Navier-Stokes equations
- Numerical experiments with S(P)DE's
- Contour processes of random trees
- On a class of quasilinear stochastic differential equations of parabolic type: regular dependence of solutions on initial data
- Fluctuations of a two-level critical branching system
- Non-persistence of two-level branching particle systems in low dimensions
- The stochastic Wick-type Burgers equation
- A weak interaction epidemic among diffusing particles
- Noise and dynamic transitions
- Backward stochastic differential equations and quasilinear partial differential equations
- Path integrals and finite dimensional filters
- A skew-product representation for the generator of a two sex population model
- A nonlinear hyperbolic SPDE: approximations and support
- Statistical dynamics with thermal noise
- Stochastic Hamilton-Jacobi equations
- On backward filtering equations for SDE systems (direct approach)
- Ergodicity of Markov semigroups
A weak interaction epidemic among diffusing particles
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Foreword
- List of participants
- Stochastic differential equations with boundary conditions and the change of measure method
- The Martin boundary of the Brownian sheet
- Neocompact sets and stochastic Navier-Stokes equations
- Numerical experiments with S(P)DE's
- Contour processes of random trees
- On a class of quasilinear stochastic differential equations of parabolic type: regular dependence of solutions on initial data
- Fluctuations of a two-level critical branching system
- Non-persistence of two-level branching particle systems in low dimensions
- The stochastic Wick-type Burgers equation
- A weak interaction epidemic among diffusing particles
- Noise and dynamic transitions
- Backward stochastic differential equations and quasilinear partial differential equations
- Path integrals and finite dimensional filters
- A skew-product representation for the generator of a two sex population model
- A nonlinear hyperbolic SPDE: approximations and support
- Statistical dynamics with thermal noise
- Stochastic Hamilton-Jacobi equations
- On backward filtering equations for SDE systems (direct approach)
- Ergodicity of Markov semigroups
Summary
Abstract. – A multi-type system of n particles performing spatial motions given by a diffusion process on Rd and changing types according to a general jump process structure is considered. In terms of their empirical measure the particles are allowed to interact, both in the drift of the diffusions as well as in the jump intensity measure for the type motions. In the limit n → ∞ we derive a principle of large deviations from the McKean-Vlasov equation satisfied by the empirical process of the system. The resulting rate function is shown to admit convenient representations.
In particular, the set-up covers a measure-valued model for an epidemic of SIR-type among spatially diffusing individuals. The infection rate is then proportional to the number of infective individuals and their distances to the susceptible one.
INTRODUCTION
Purpose. The purpose of this report is to provide a multi-type extension, allowing weak interaction in both space and type, of the well-known results of Dawson and Gärtner (1987) [DG] regarding large deviations from the McKean-Vlasov limit for weakly interacting diffusions. This is achieved byintegrating more systematically the previous work Djehiche and Kaj (1994) [DK], in which a large deviation result is derived for a class of measure-valued jump processes, with the setting of the Dawson-Gärtner large deviation principle. Necessarilly, some aspects of such an extension will be mere notational rather than substantial. We will try to focus on those parts that are less evident and to point out some techniques from [DK] which can be used as an alternative to those of [DG].
- Type
- Chapter
- Information
- Stochastic Partial Differential Equations , pp. 162 - 180Publisher: Cambridge University PressPrint publication year: 1995