Problems involving the capture of a moving entity by a trap occur in a variety of physical situations, the moving entity being an electron, an excitation, an atom, a molecule, a biological object such as a receptor cluster, a cell, or even an animal such as a mouse carrying an epidemic. Theoretical considerations have almost always assumed that the particle motion is translationally invariant. We study here the case when that assumption is relaxed, in that the particle is additionally subjected to a harmonic potential. This tethering to a center modifies the reaction-diffusion phenomenon. Using a Smoluchowski equation to describe the system, we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions. Interesting features emerge depending on the relative location of the trap, the attractive center, and the initial placement of the diffusing particle.

A theory of the spread of epidemics is formulated on the basis of pairwise interactions in a dilute system of random walkers (infected and susceptible animals) moving in n dimensions. The motion of an animal pair is taken to obey a Smoluchowski equation in 2n-dimensional space that combines diffusion with confinement of each animal to its particular home range. An additional (reaction) term that comes into play when the animals are in close proximity describes the process of infection. Analytic solutions are obtained, confirmed by numerical procedures, and shown to predict a surprising effect of confinement. The effect is that infection spread has a non-monotonic dependence on the diffusion constant and/or the extent of the attachment of the animals to the home ranges. Optimum values of these parameters exist for any given distance between the attractive centers. Any change from those values, involving faster/slower diffusion or shallower/steeper confinement, hinders the transmission of infection. A physical explanation is provided by the theory. Reduction to the simpler case of no home ranges is demonstrated. Effective infection rates are calculated and it is shown how to use them in complex systems consisting of dense populations. The purpose of the following is to construct an analytic theory of the transmission of infection in epidemics spread on the basis of a simple but exactly soluble model of interacting random walkers representing animals moving about on the terrain and infecting one another on encounter. Seminal contributions by Anderson and May and others [1][2][3][4], involving concepts such as mass action, SIR, and the basic reproductive rate R 0 , launched this field of research which derives its importance from human relevance as well as intellectual challenge. Spatial considerations were introduced into the investigations independently by various authors [2,[5][6][7][8][9][10][11][12] giving the studies a kinetic equation flavor. Missing from some of these studies were confinement features that arise in animal motion from home ranges and yet are clear and compelling in the light of field observations [12][13][14]. These and other issues have made it essential to undertake a fundamental study of the transmission of infection in terms of interacting random walks specially under confinement.Model and Method of Analysis-Our model starts with just two animals, one initially infected and the other initially uninfected (susceptible), respectively denoted by 1 and 2, performing random walks around respective attractive centers at R 1 and R 2 , with a diffusion constant D, there being the possibility of the uninfected individual getting infected at a rate proportional to C when the two occupy the same position. The central quantity that serves as the focus of our calculation is the joint proba- * Electronic address: kenkre@unm.edu † Electronic address: satomi@unm.edu bility density P (r 1 , r 2 , t) that the infected animal is at r 1 and the susceptible animal is at r 2 . Given this definition, P (r 1 , r...

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