A weak interaction epidemic among diffusing particles

Summary

Abstract. – A multi-type system of n particles performing spatial motions given by a diffusion process on R^d 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].