  A pair of random walkers, the motion of each of which in two dimensions is
confined spatially by the action of a quadratic potential centered at different
locations for the two walks, are analyzed in the context of reaction-diffusion.
The application sought is to the process of transmission of infection in
epidemics. The walkers are animals such as rodents in considerations of the
Hantavirus epidemic, infected or susceptible, the reaction is the transmission
of infection, and the confining potential represents the tendency of the
animals to stay in the neighborhood of their home range centers. Calculations
are based on a recently developed formalism (Kenkre and Sugaya, Bull. Math.
Bio. 76, 3016 (2014)) structured around analytic solutions of a Smoluchowski
equation and one of its aims is the resolution of peculiar but well-known
problems of reaction-diffusion theory in 2-dimensions. In the present analysis,
reaction occurs not at points but in spatial regions of dimensions larger than
0. The analysis uncovers interesting nonintuitive phenomena one of which is
similar to that encountered in the 1-dimensional analysis given in the quoted
article, and another specific to the fact that the reaction region is spatially
extended. The analysis additionally provides a realistic description of
observations on animals transmitting infection while moving on what is
effectively a 2-dimensional landscape. Along with the general formalism and
explicit 1-dimensional analysis given in Bull. Math. Bio. 76, 3016 (2014), the
present work forms a model calculational tool for the analysis for the
transmission of infection in dilute systems.
