Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and without Confinement

TL;DR

This paper develops a mathematical theory for epidemic spread among animals modeled as interacting random walkers with confinement, revealing a non-monotonic relationship between infection transmission and animal mobility or confinement extent.

Contribution

It introduces an analytic framework combining diffusion, confinement, and infection reactions, predicting optimal conditions for infection spread and extending to dense populations.

Findings

Infection spread depends non-monotonically on diffusion and confinement parameters.

Optimal diffusion and confinement levels maximize infection transmission.

The theory can be applied to complex, dense population systems.

Abstract

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.